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Blog: https://johncarlosbaez.wordpress.com/

Last Notes

2026-06-16 15:54:54 UTC

##
Octonions and the Standard Model
Paul Schwahn and I have come out with a new paper about octonions and the Standard Model:
• [The Standard Model gauge group from the exceptional Jordan algebra](https://arxiv.org/abs/2606.15235 )
It builds on things I’ve discussed here, but it goes further. Let me explain a bit.
A bit is just a binary alternative: 1 or 0, true or false. That’s how it works in classical logic. We could also have a ‘trit’, meaning 3 alternatives.
In quantum physics we instead have qubits and qutrits.
Qubits and qutrits are usually described using complex numbers. The algebra of observables of a qubit is the [Jordan algebra](https://ncatlab.org/nlab/show/Jordan+algebra ) , consisting of self-adjoint complex matrices. Similarly, the algebra of observables of an qutrit is the Jordan algebra , consisting of self-adjoint complex matrices.
We can also study systems with more than 3 alternative ways to be. They work the same way, using the Jordan algebras with
But we can also do quantum mechanics using other number systems! The options have been mapped out, and the largest allowed number system for this purpose is the algebra of [octonions](https://math.ucr.edu/home/baez/octonions/ ).
A weird thing is that Jordan algebras built using octonions can describe qutrits, but not quantum systems with more than 3 alternative ways to be. The algebra of observables of an octonionic qutrit is the so-called ‘exceptional’ Jordan algebra , consisting of self-adjoint octonion matrices. What makes it exceptional is that is not a Jordan algebra when is bigger than 3.
So, there’s something special about octonionic qutrits—and it turns out that every symmetry in the gauge group of the Standard Model is a symmetry of an octonionic qutrit!
Not every symmetry of an octonionic qutrit is a symmetry of the Standard Model. But those that do have a simple description. They are those that restrict to give symmetries of an ordinary qutrit sitting inside the octonionic qutrit… and an ordinary qubit sitting inside that!
That sounds exciting, but also vague, so let me make it precise.
While lots of people say the gauge group of the Standard Model of particle physics is , in fact a certain subgroup of this acts trivially on all known particles. If we mod out by that, we’re left with a group called , which is
and this is the group I’m talking about.
We proved two theorems describing this group in terms of the symmetries of an octonionic qutrit. The group of automorphisms of the exceptional Jordan algebra is a 52-dimensional Lie group known affectionately as —so that’s what I mean by the symmetries of an octonionic qutrit.
Here’s our main result:
**Theorem 1.** *Suppose are Jordan subalgebras of such that*
*Then*
Here is the stabilizer of —that is, the subgroup of consisting of elements that map to itself—while is the identity component of the stabilizer of .
This ‘identity component’ business is rather sneaky, but it turns out that guys in are symmetries of an ordinary qutrit that can be described as *unitary* operators on , while also contains those symmetries that are described by *antiunitary* operators. The CPT symmetry of the Standard Model is antiunitary, for example.
Theorem 1 emerged from a related result, which grew out of the work of Todorov and Dubois-Violette:
**Theorem 2.** *Suppose are Jordan subalgebras of such that*
*Then*
Todorov and Dubois–Violette proved this for a certain standard choice of subalgebras and . Thus, the challenge in proving Theorem 2 was to show that every other choice can be mapped to this standard choice using the action of . This shows that the theorem is not an artifact of a specific choice, but rather a general fact.
How do we prove these results?
We start by constructing the octonion product from -invariant operations on and . We then use this description to reprove Todorov and Dubois–Violette’s special case of Theorem 2. Then we show that acts transitively on the set of subalgebras of that are isomorphic to . We also show every Jordan subalgebra of isomorphic to is contained in a unique Jordan subalgebra isomorphic to . This lets us prove that acts transitively on the set of pairs of Jordan subalgebra with , and . Theorem 2 then follows from Todorov and Dubois-Violette’s special case. We conclude by using these results to prove Theorem 1.
However, if you want to get into the details of the physics, the interesting part is how the strong force gauge group and the electroweak show up from the relation between octonionic qutrits, complex qutrits and complex qubits. You’ll see that in the proof of Lemma 4.
And if you want to get into the details of the math, the main interesting thing here is the use of Jordan algebra technology like ‘Peirce decompositions’ and ‘Jordan frames’ to figure out what it must be like when you have a Jordan algebra or sitting inside , where is some normed division algebra contained in a bigger normed division algebra .
What it all ‘really means’, if anything, is a question for later. It could be just a coincidence. Of course I hope not.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/06/todorov.png?w=450

2026-02-12 03:40:32 UTC

##
Stela C
One bad thing about archeologists is that some of the successful ones get a big head.
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/a-colossal-stone-head-from-the-olmec-civilization.jpg )
People used to think the Olmecs, who made these colossal stone heads, were contemporary with the Mayans. But in 1939, an archaeologist couple, [Marion](https://en.wikipedia.org/wiki/Marion_Stirling_Pugh ) and [Matthew Stirling](https://en.wikipedia.org/wiki/Matthew_Stirling ), found the bottom half of an Olmec stone that had part of a date carved on it!
It’s called [Stela C](https://en.wikipedia.org/wiki/Tres_Zapotes#Stela_C ):
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/estela_c_de_tres_zapotes.jpg )
The Stirlings guessed the date was 7.16.6.16.18. In the calendar used by the Olmecs and other Central American civilizations, this corresponds to September 3, 32 BC. That meant the Olmecs were extremely old—much older than the Mayans.
But the first digit was missing from the bottom half of the stone! All the Stirlings actually *saw* was 16.6.16.18. And the first digit was the most significant one! If it were 8 instead of 7, the date of the stone would be much later: roughly 362 AD, when the Mayans were in full swing.
The Stirlings guessed that the first digit must be 7 using a clever indirect argument. But perhaps because of the subtlety of this argument, and certainly because of the general skepticism among experts that the Olmecs were so old, few believed the Stirlings.
But then, 30 years later, in 1969, they were proven correct! A farmer found the other half of the stone and confirmed that yes, the missing digit was a 7. So the date on Stela C really is September 3, 32 BC.
That’s a wonderful story of delayed vindication. But it leaves two mysteries.
• First, how in the world could the Olmec calendar be so damn good that we can look at that date and know it meant September 3, 32 BC?
• Second, what clever argument did the Stirlings use to guess the missing digit?
You can only *fully* understand the answer to this if you know a bit about the Olmec way of counting time. Like the Mayans, they used the [Mesoamerican Long Count Calendar](https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar ). This identifies a day by counting how many days passed since the world was created. The count is more or less base 20, except that the second “digit” is in base 18, since they liked a year that was 18 × 20 = 360 years long. So,
7.16.6.16.18
means
7 × 144,000 + 16 × 7,200 + 6 × 360 + 16 × 20 + 18 = 1,125,698 days
after the world was created. Or, if you’re a Mayan, you’d say it’s
7 baktuns, 16 katuns, 6 tuns, 16 uinals and 18 kins
But then we have to ask: when did the Olmecs and Mayans think the world was created? [Experts believe they know](https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar#Correlations_between_Western_calendars_and_the_Long_Count ): September 6, 3114 BCE in the proleptic Julian calendar, where ‘proleptic’ means roughly that we’re extrapolating this calendar back to times long before anyone used this calendar.
But enough background. I asked my friend Gro-Tsen
>
how in the world could the Olmec calendar be so damn good that we can look at that date and know it meant September 3, 32 BC?
And while I’ve already given a kind of answer, I’ve skimmed over many subtleties. So, it’s worth reading [his answer](https://bsky.app/profile/gro-tsen.bsky.social/post/3meiqswj7b22a ):
>
I did the math. 🙋
> 👉 It’s Sept. 3, 32BCE (reminder: “32BCE” actually means “−31” 😒) in the proleptic Julian calendar = Sept. 1 prol. Gregorian.
> The Western equivalent of the Mesoamerican Long Count is the “Julian Date”. The Julian Date simply counts the number of days from an arbitrary remote reference point (Nov. 24, 4714BCE proleptic Gregorian). More practically, on 2000-01-01 it equaled 2 451 545 (at 12:00 UTC if we want to use fractional Julian dates).
> For example, today as I write is Julian Date 2 461 082 (well, 2 461 081.9 because it’s not yet noon UTC). And the date of Sept. 1, 32BCE [prol. Greg.] we’re talking about corresponds to Julian Date 1 709 981. More convenient than all this dealing with complicated calendar conventions.
> So to convert a Long Count date to the Western calendar, we first convert the Long Count to an integer (trivial: it’s already just an integer written in base 20-except-18-in-the-penultimate-digit), we add a constant (C) to get a Julian Date, and we convert to our messy calendars.
> BUT! What is this constant C? This is known as the “Mayan correlation”. For a long time in the 20th century there was a debate about its value: scholars could relate any two Mayan dates, but not situate them exactly w.r.t. our own calendar. Various values were proposed, … ranging from the (frankly rather ludicrous) 394 483 to 774 078, an interval of about 1000 years! (😅) The now accepted value for C is 584 283 (the “Goodman-Martínez-Thompson” or GMT correlation, not to be confused with Greenwich Mean Time or UTC 😁), first proposed in 1905.
> This C = 584 283 or “GMT” correlation value places the “Long Count epoch” 0.0.0.0.0 on August 11, 3114BCE in the proleptic Gregorian calendar (the day with Julian Date 584 283), although IIUC it’s not clear if this precise date held any particular importance to the Olmecs (or later Mayans).
> Maybe it was just arbitrary like the start of our own Julian Date (because, no, Julius Scalier didn’t think the world started on November 24, 4714BCE proleptic Gregorian).
> One Mayan inscription suggest that the Long Count was the truncation to the last 5 “digits” of an even longer count, and that a Long Count value such as 9.15.13.6.9 was in fact 13.13.13.13.13.13.13.13.9.15.13.6.9 in this Even Longer Count (why 13 everywhere? I don’t know!). But this may be one particular astronomer’s weird ideas, I guess we’ll never know.
> But back to the Mayan correlation constant C.
> Wikipedia suggests that this “GMT” value C = 584 283 for the Mayan correlation is now settled and firmly established. But between 1905 and now there was some going back and forth with various authors (including the three Goodman, Martínez and Thompson after which it is named) adding or removing a day or two (I think Goodman first proposed 584 283, then changed his mind to 584 280, but nobody really cared, Hernández resurrected the proposal in 1926 but altered it to 584 284, then Thompson to 584 285 in 1927, and then Thompson later said Goodman’s initial value of 584 283 had been right all long, and while this is now accepted, the confusion of ±3 days might still linger).
> The Emacs program’s calendar (M-x calendar) can give you the Long Count date (type ‘p m’ for “Print Mayan date”) and uses the GMT value C = 584 283. Today is 13.0.13.5.19. (You can also go to a particular Long Count date using ‘g m l’ but Emacs won’t let you go to 7.16.6.16.18 because its calendar starts on January 1, 1 prol. Gregorian = Julian Date 1 721 426 = Long Count 7.17.18.13.3. So close! This caused me some annoyance in checking the dates.)
> So anyway, 7.16.6.16.18 is
> (((7×20+16)×20+6)×18+16)×20+18 = 1 125 698 days
> after the Long Count epoch, so Julian Date 1 125 698 + 584 283 = 1 709 981 if we accept the GMT value of C = 584 283 for the Mayan correlation, and this is September 1, 32BCE in the proleptic Gregorian calendar, or September 3, 32BCE in the proleptic Julian calendar. (I write “proleptic” here, even though the Julian calendar did exist in 32BCE, because it was incorrectly applied between 45BCE and 9BCE, with the Pontiffs inserting a leap year every 3 years, not 4, and Augustus had this mess fixed.)
> Also, confusingly, if we use Thompson’s modified (and later disavowed) correlation of 584 285, then we get September 3, 32BCE in the proleptic Gregorian calendar, so maybe this could also be what was meant. Yeah, Julian Dates are a great way of avoiding this sort of confusion!
> PS: I wrote the pages
> [> http://www.madore.org/~david/misc/calendar.html](http://www.madore.org/~david/misc/calendar.html )
> (and also > [> http://www.madore.org/~david/misc/time.html](http://www.madore.org/~david/misc/time.html )> ) many years ago (starting on Long Cont 12.19.10.13.1), which I just used to refresh my memory on the subject.
All this is great. But it leaves us with the second puzzle: how in the world did the Stirlings guess the missing first digit of the date on the bottom half of Stela C?
Here’s the answer, as best as I can tell:
The Olmecs and Mayans used *two* calendars! In addition to the Mesoamerican Long Count, they also used one called the [Tzolkʼin](https://en.wikipedia.org/wiki/Tzolk%CA%BCin ). This uses a 260-day cycle, where each day gets its own number and name: there are 13 numbers and 20 names. And the bottom half of Stela C had inscribed not only the last four digits of the Mesoamerican Long Count digits, but also the Tzolkʼin day: 6 Etz’nab. This is what made the reconstruction possible!
Here’s why 7 was the only possible choice of the missing digit. Because the last four Long Count digits (16.6.16.18) are fixed, the total day count is B × 144,000 + 117,698, where B is the unknown baktun. The number of days in a baktun is
144,000 = 0 mod 20,
and there are 20 different Tzolkʼin day names, so changing the baktun never changes which Tzolkʼin day name you land. But there are 13 different Tzolkʼin day *numbers*, so a baktun contributes
144,000 ≡ –1 (mod 13) days to the Tzolkʼin day *number*.
This means that after the day
7.16.6.16.18 and 6 Etz’nab
the next day of the form
N.16.6.16.18 and 6 Etz’nab
happens when N = 7+13. But this is 13 × 144,000 days later: that is, roughly 5,094 years *after* 32 BC. Far in the future!
So, while 32 BC seemed awfully early for the Olmecs to carve this stone, there’s no way they could have done it later. (Or earlier, for that matter.)
Here is the Stirlings’ actual photo of Stela C:
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/stela_c.jpg )
This is from
• Matthew W. Stirling, *An Initial Series from Tres Zapotes*, Vera Cruz, Mexico. National Geographic Society Contributions, Technical Papers, Mexican Archaeological Series, Vol. 1, No. 1. Washington, 1940.
By the way, in this paper he doesn’t actually explain the argument I just gave. Apparently he assumes that expert Mayanists would understand this brief remark:
>
Assuming then that the number 6 adjacent to the terminal glyph represents the coefficient of the day sign, the complete reading of the date would be (7)-16-6-16-18, or 6 Eznab 1 Uo, since only by supplying a baktun reading of 7 can the requirements of the day sign 6 be satisfied.
I can’t help but wonder if this was much too terse! I haven’t found any place where he makes the argument in more detailed form.
**Puzzle 1.** What does “1 Uo” mean, and what bearing does this have on the dating of Stela C?
**Puzzle 2.** Why does the Tzolkʼin calendar use a 260-day cycle?
The second one is extremely hard: there are [several theories](https://en.wikipedia.org/wiki/Tzolk%CA%BCin#Origins ) but no consensus.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/stela_c.jpg?w=450
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/a-colossal-stone-head-from-the-olmec-civilization.jpg?w=450
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/02/estela_c_de_tres_zapotes.jpg?w=363

2026-01-28 02:29:08 UTC

##
Tiny Musical IntervalsMusic theorists have studied many fractions of the form
2i 3j 5k
that are close to 1. They’re called **[5-limit commas](https://en.wikipedia.org/wiki/Comma_(music) )**. Especially cherished are those that have fairly small exponents given how close they are to 1. I discussed a bunch here:
• [Well temperaments (part 2)](https://johncarlosbaez.wordpress.com/2024/01/18/well-temperaments-part-2/ ).
and I explained the tiniest named one, the utterly astounding ‘atom of Kirnberger’, here:
• [Well temperaments (part 3)](https://johncarlosbaez.wordpress.com/2024/01/18/well-temperaments-part-2/ ).
The **[atom of Kirnberger](https://golem.ph.utexas.edu/category/2024/02/the_atom_of_kirnberger.html )** equals
2161 · 3-84 · 5-12 ≈ 1.0000088728601397
Two pitches differing by this frequency sound the same to everyone except certain cleverly designed machines. But remarkably, the atom of Kirnberger shows up rather naturally in music—and it was discovered by a student of Bach! Read my article for details.
All this made me want to systematically explore such tiny intervals. Below is a table of them. Some have names but many do not—or at least I don’t know their names. I list these numbers in decimal form and also in **[cents](https://en.wikipedia.org/wiki/Cent_(music) )**, where we take the logarithm of the number in base 2 and multiply by 100. (I dislike this blend of base 2 and base 10, but it’s traditional in music theory.)
Most importantly, I list the **[monzo](https://en.xen.wiki/w/Monzo )**. This is the list of exponents: for example, the monzo of
2i 3j 5k
is
[i, j, k]
In case you’re wondering, this term was named after the music theorist [Joseph Monzo](https://en.xen.wiki/w/Joseph_Monzo ).
I also list the **[Tenney height](https://en.xen.wiki/w/Tenney_norm )**. This is a measure of complexity: the Tenney height of
2i 3j 5k
is
∣i∣ log2(2) + ∣j∣ log2(3) + ∣k∣ log2(5)
The table below purports to list only 5-limit commas that are close to 1 as possible for a given complexity. More precisely, it should list numbers of the form 2i 3j 5k that are > 1 and closer to 1 than any number with smaller Tenney height—except of course for 1 itself.<tbody><tr><th>Cents</th><th>Decimal</th><th>Name</th><th>Monzo</th><th>Tenney height</th></tr><tr><td>498.04</td><td>1.3333333333</td><td><a href="https://en.wikipedia.org/wiki/Perfect_fourth">just perfect fourth</a></td><td>[2, −1, 0]</td><td>3.6</td></tr><tr><td>386.31</td><td>1.2500000000</td><td><a href="https://en.wikipedia.org/wiki/Major_third">just major third</a></td><td>[−2, 0, 1]</td><td>4.3</td></tr><tr><td>315.64</td><td>1.2000000000</td><td><a href="https://en.wikipedia.org/wiki/Minor_third">just minor third</a></td><td>[1, 1, −1]</td><td>4.9</td></tr><tr><td>203.91</td><td>1.1250000000</td><td><a href="https://en.wikipedia.org/wiki/Major_second#Major_and_minor_tones">major tone</a></td><td>[−3, 2, 0]</td><td>6.2</td></tr><tr><td>182.40</td><td>1.1111111111</td><td><a href="https://en.wikipedia.org/wiki/Major_second#Major_and_minor_tones">minor tone</a></td><td>[1, −2, 1]</td><td>6.5</td></tr><tr><td>111.73</td><td>1.0666666667</td><td><a href="https://en.wikipedia.org/wiki/Semitone">diatonic semitone</a></td><td>[4, −1, −1]</td><td>7.9</td></tr><tr><td>70.67</td><td>1.0416666667</td><td><a href="https://en.wikipedia.org/wiki/Semitone">lesser chromatic semitone</a></td><td>[−3, −1, 2]</td><td>9.2</td></tr><tr><td>21.51</td><td>1.0125000000</td><td><a href="https://en.wikipedia.org/wiki/Syntonic_comma">syntonic comma</a></td><td>[−4, 4, −1]</td><td>12.7</td></tr><tr><td>19.55</td><td>1.0113580247</td><td><a href="https://en.wikipedia.org/wiki/Diaschisma">diaschisma</a></td><td>[11, −4, −2]</td><td>22.0</td></tr><tr><td>8.11</td><td>1.0046939300</td><td><a href="https://en.wikipedia.org/wiki/Kleisma">kleisma</a></td><td>[−6, −5, 6]</td><td>27.9</td></tr><tr><td>1.95</td><td>1.0011291504</td><td><a href="https://en.wikipedia.org/wiki/Schisma">schisma</a></td><td>[−15, 8, 1]</td><td>30.0</td></tr><tr><td>1.38</td><td>1.0007999172</td><td>unnamed?</td><td>[38, −2, −15]</td><td>76.0</td></tr><tr><td>0.86</td><td>1.0004979343</td><td>unnamed?</td><td>[1, −27, 18]</td><td>85.6</td></tr><tr><td>0.57</td><td>1.0003289700</td><td>unnamed?</td><td>[−53, 10, 16]</td><td>106.0</td></tr><tr><td>0.29</td><td>1.0001689086</td><td>unnamed?</td><td>[54, −37, 2]</td><td>117.3</td></tr><tr><td>0.23</td><td>1.0001329015</td><td>unnamed?</td><td>[−17, 62, −35]</td><td>196.5</td></tr><tr><td>0.047</td><td>1.0000271292</td><td>unnamed?</td><td>[−90, −15, 49]</td><td>227.5</td></tr><tr><td>0.0154</td><td>1.0000088729</td><td><a href="https://golem.ph.utexas.edu/category/2024/02/the_atom_of_kirnberger.html">atom of Kirnberger</a></td><td>[161, −84, −12]</td><td>322.0</td></tr><tr><td>0.0115</td><td>1.0000066317</td><td>unnamed?</td><td>[21, 290, −207]</td><td>961.3</td></tr><tr><td>0.00088</td><td>1.0000005104</td><td><a href="https://johncarlosbaez.wordpress.com/2026/01/28/tiny-musical-intervals/">quark of Baez</a></td><td>[−573, 237, 85]</td><td>1146.0</td></tr></tbody>
You’ll see there’s a big increase in Tenney height after the schisma. This is very interesting: it suggests that the schisma is the last ‘useful’ interval. It’s useful only in that it’s the ratio of two musically important commas, the syntonic comma and the Pythagorean comma, and life in music would be simpler if these were equal. All the intervals in this table up to the schisma were discovered by musicians a long time ago, and they all have standard names! But after the schisma, interest drops off dramatically.
The atom of Kirnberger has [such amazing properties](https://arxiv.org/abs/0907.5249 ) that it was worth naming. The rest, maybe not. But as you can see, I’ve taken the liberty of naming the smallest interval in the table the ‘quark of Baez’. This is much smaller than all that come before. It’s in bad taste to name things after oneself—indeed this is item 25 on the [crackpot index](https://math.ucr.edu/home/baez/crackpot.html )—but I hope it’s allowed as a joke.
Here is the [Python code](http://math.ucr.edu/home/baez/cultural/tuning/tiny_musical_intervals.py ) that should generate the above information. If you’re good at programming, please review it and check it! from math import log2log3 = log2(3) log5 = log2(5)commas = []max_exp_3 = 1200 max_exp_5 = 250for a3 in range(-max_exp_3, max_exp_3+1): for a5 in range(-max_exp_5, max_exp_5+1): if a3 == 0 and a5 == 0: continue### <code># Find a2 that minimizes |a2 + a3 * log2(3) + a5 * log2(5)|</code><code><code> target = -(a3 * log3 + a5 * log5) a2 = round(target) log2_ratio = a2 + a3 * log3 + a5 * log5 cents = abs(1200 * log2_ratio) if cents &gt; 0.00001: # non-trivial tenney = abs(a2) + abs(a3) * log3 + abs(a5) * log5 commas.append((tenney, cents, a2, a3, a5))</code></code>### <code>#Find Pareto frontier</code>commas.sort(key=lambda x: x[0]) # sort by Tenney heightfrontier = [] best_cents = float('inf') for c in commas: if c[1] < best_cents: best_cents = c[1] frontier.append(c)### <code># Print results</code>for tenney, cents, a2, a3, a5 in frontier: log2_ratio = a2 + a3 * log3 + a5 * log5 decimal = 2**log2_ratio if decimal < 1: decimal = 1/decimal a2, a3, a5 = -a2, -a3, -a5 print(f"{cents:.6f} cents | {decimal:.10f} | [{a2}, {a3}, {a5}] | Tenney: {tenney:.1f}") ###
Gene Ward Smith In studying this subject I discovered that tiny 5-limit intervals were studied by [Gene Ward Smith](https://thmuses.wordpress.com/2021/02/16/gene-w-smith-has-died-from-covid-19/ ), a mathematician I used to see around on sci.math and the like. I never knew he worked on microtonal music! I am sad to hear that he died from COVID-19 in January 2021.
I may just be redoing a tiny part of his work: if anyone can find details, please let me know. In his memory, I’ll conclude with this article from the [Xenharmonic Wiki](https://en.xen.wiki/w/Gene_Ward_Smith ):
> Gene Ward Smith (1947–2021) was an American mathematician, music theorist, and composer.
> In mathematics, he worked in the areas of Galois theory and Moonshine theory.
> In music theory, he introduced wedge products as a way of classifying regular temperaments. In this system, a temperament is specified by means of a wedgie, which may technically be identified as a point on a Grassmannian. He had long drawn attention to the relationship between equal divisions of the octave and the Riemann zeta function.[1][2][3] He early on identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the regular mapping paradigm.
> In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a crystal oscillator. The device in turn was controlled by HP 9800 series desktop computers, initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and pajara.
> Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note highschool scales, led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.
> Gene was among the first to consider extending the Tonnetz of Hugo Riemann beyond the 5-limit and hence into higher dimensional lattices. In three dimensions, the hexagonal lattice of 5-limit harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems. ####
> Historical interest> • > [> Usenet post from 1990 by Gene Smith on homomorphisms and kernels](https://en.xen.wiki/w/USENET_post_from_1990_by_Gene_Smith_on_homomorphisms_and_kernels )
> • > [> Usenet post from 1995 by Gene Smith on homomorphisms and kernels](https://en.xen.wiki/w/USENET_post_from_1995_by_Gene_Smith_on_homomorphisms_and_kernels )####
> See also > • > [> Microtonal music by Gene Ward Smith](https://en.xen.wiki/w/Microtonal_music_by_Gene_Ward_Smith )
> • > [> Hypergenesis58](https://en.xen.wiki/w/Hypergenesis58 )> (a scale described by Gene Ward Smith)####
> References > [1] Rusin, Dave. “> [> Why 12 tones per octave?](https://web.archive.org/web/20130731073247/http://www.math.niu.edu/~rusin/uses-math/music/12 )> ”
> [2] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: > [> OEIS: A117536](https://oeis.org/A117536 )> .
> [3] OEIS. Increasingly large integrals of the Z function between zeros: > [> OEIS: A117538](https://oeis.org/A117538 )> .
#Find
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/01/tonnetz_gro-tsen.jpg?w=450

2026-01-10 01:55:06 UTC

##
Sylvester and Clifford on Curved Space[](https://asd.gsfc.nasa.gov/blueshift/index.php/2015/11/25/100-years-of-general-relativity/ )
Einstein realized that gravity is due to the curvature of spacetime, but let’s go back earlier:
On the 18th of August 1869, the eminent mathematician [Sylvester](https://en.wikipedia.org/wiki/James_Joseph_Sylvester ) gave a speech arguing that geometry is not separate from physics. He later published this speech in the journal *Nature*, and added a footnote raising the possibility that space is curved:
> the laws of motion accepted as fact, suffice to prove in a general way that the space we live in is a flat or level space […], our existence therein being assimilable to the life of the bookworm in a flat page; but what if the page should be undergoing a process of gradual bending into a curved form?
Then, even more dramatically, he announced that the mathematician [Clifford](https://en.wikipedia.org/wiki/William_Kingdon_Clifford ) had been studying this!
> Mr. W. K. Clifford has indulged in more remarkable speculations as the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of three dimensions being in the act of undergoing in space of four dimensions (space as inconceivable to us as our space to the supposititious bookworm) a distortion analogous to the rumpling of the page.
This started a flame war in letters to *Nature* which the editor eventually shut off, saying “this correspondence must now cease”. Clifford later wrote about his theories in a famous short paper:
• William Clifford, [On the space-theory of matter](https://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter ), *Proceedings of the Cambridge Philosophical Society* **2** (1876), 157–158.
It’s so short I can show you it in its entirety:
> Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
> I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact
> (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
> (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
> (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
> (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.
> I am endeavouring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.
To my surprise, the following paper argues that Clifford did experiments to *test* his ideas by measuring the polarization of the skylight during a solar eclipse in Sicily on December 22, 1870:
• S. Galindo and Jorge L. Cervantes-Cota, [Clifford’s attempt to test his gravitation hypothesis](https://arxiv.org/abs/1807.09230 ).
Clifford did indeed go on such an expedition, and did indeed try to measure the polarization of skylight as the Moon passed the Sun. I don’t know of any record of him saying why he did it.
I’ll skip everything the above paper says about why the polarization of skylight was interesting and mysterious in the 1800s, and quote just a small bit:
> The English Eclipse Expedition set off earlier in December 1870, on the steamship H.M.S. Psyche scheduled for a stopover at Naples before continuing to Syracuse in Sicily. Unfortunately before arriving to her final call, the ship struck rocks and was wrecked off Catania. Fortunately all instruments and members of the party were saved without injury.
> Originally it was the intention of the expedition to establish in Syracuse their head-quarters, but in view of the wreckage the group set up their base camp at Catania. There the expedition split up into three groups. The group that included Clifford put up an observatory in Augusta near Catania. The leader of this group was William Grylls Adams, professor of Natural Philosophy at King’s College, London.
> In a report written by Prof. Adams, describing the expedition, we learn that the day of the eclipse, just before the time of totality, “… a dense cloud came over the Moon and shut out the whole, so that it was doubtful whether the Moon or the clouds first eclipsed the Sun […] Mr. Clifford observed light polarized on the cloud to the right and left and over the Moon, in a horizontal plane through the Moon’s centre [….] It will be seen from Mr. Clifford’s observations that the plane of polarization by the cloud…was nearly at right angles to the motion of the Sun”.
> As was to be expected, Clifford’s eclipse observations on polarization did not produce any result. His prime intention, of detecting angular changes of the polarization plane due to the curving of space by the Moon in its transit across the Sun´s disk, was not fulfilled. At most he confirmed the already known information, i.e. the skylight polarization plane moves at right angles to the Sun anti-Sun direction.
This is a remarkable prefiguring of [Eddington’s later voyage](https://en.wikipedia.org/wiki/Eddington_experiment ) to the West African island of Principe to measure the bending of starlight during an eclipse of the Sun in 1919. Just one of many stories in the amazing prehistory of general relativity!
https://johncarlosbaez.wordpress.com/wp-content/uploads/2026/01/spacetime_curvature.png?w=450

2025-12-29 15:33:01 UTC

##
Just Intonation (Part 6)In [this series](https://johncarlosbaez.wordpress.com/2023/11/17/just-intonation-part-5/ ) I’ve been explaining 12-tone scales in just intonation—or more precisely, ‘5-limit’ just intonation, where all the frequency ratios are integer powers of the primes 2, 3 and 5. There are various choices involved in building such a scale. A lot of famous mathematicians have tried their hand at it. Kepler, Descartes, Mersenne, Newton, Mercator, and Euler are among them. They didn’t agree on the best scale: they came up with different scales.
Given this, I can’t resist classifying all possible scales of this sort. Today we’ll see that by a certain precise definition, there are 174,240 such scales! It will take a bit of combinatorics to work this out. Among this large collection of scales we will also find smaller sets of scales with nice properties. But I still don’t know why those mathematicians chose the scales they did.
In studying this, and indeed in all my work on just intonation, I was greatly helped by this wonderful paper:
• Daniel Muzzulini, [Isaac Newton’s microtonal approach to just intonation](https://doi: 10.18061/emr.v15i3-4.7647 ), *Empirical Musicology Review* **15** (2021), 223–248.
It’s full of interesting diagrams:
[](http://math.ucr.edu/home/baez/cultural/tuning/newton_tuning_muzzulini.jpg )
Anyway, let’s get going!
In [Part 2](https://johncarlosbaez.wordpress.com/2023/11/06/just-intonation-part-2/ ) of this series, I examined the choices involved in building a just intonation scale. I described a general recipe for building such scales. These leads to 2 × 4 × 2 = 16 different scales, based on how you make the choices here:<tbody><tr><td>tonic</td><td>      </td><td>1</td></tr><tr><td>minor 2nd</td><td></td><td>16/15</td></tr><tr><td>major 2nd</td><td></td><td>10/9 or 9/8</td></tr><tr><td>minor 3rd</td><td></td><td>6/5</td></tr><tr><td>major 3rd</td><td></td><td>5/4</td></tr><tr><td>perfect 4th</td><td></td><td>4/3</td></tr><tr><td>tritone</td><td></td><td>25/18 or 45/32 or 64/45 or 36/25</td></tr><tr><td>perfect 5th</td><td></td><td>3/2</td></tr><tr><td>minor 6th</td><td></td><td>8/5</td></tr><tr><td>major 6th</td><td></td><td>5/3</td></tr><tr><td>minor 7th</td><td></td><td>16/9 or 9/5</td></tr><tr><td>major 7th</td><td></td><td>15/8</td></tr><tr><td>octave</td><td></td><td>2</td></tr></tbody>
One of these 16 scales is, I believe, the most popular just intonation scale of all:
[](https://math.ucr.edu/home/baez/cultural/just_intonation_1st_symmetric_scale_with_diminished_5th_removed_diagram.jpg )
The intervals between the notes come in 3 different sizes, which we will discuss soon. In [Part 4](https://johncarlosbaez.wordpress.com/2023/11/15/just-intonation-part-4/ ), I explained some reasons this scale is nice. For example, the intervals here are nearly palindromic! The first interval is the same as the last, and so on—except right at the middle of the scale, the ‘tritone’, where this symmetry is impossible because it would force to be a rational number.
In [Part 4](https://johncarlosbaez.wordpress.com/2023/11/15/just-intonation-part-4/ ), I also considered another less popular scale among the 16 generated by my recipe:
[](https://math.ucr.edu/home/baez/cultural/just_intonation_2nd_symmetric_scale_with_diminished_5th_removed_diagram.jpg )
In this one the intervals come in 4 different sizes! Let’s make up abbreviations for them. In order of increasing size, they are:
• c: the lesser chromatic semitone, with frequency ratio 25/24 = 1.041666…
• C: the greater chromatic semitone, with frequency ratio 135/128 = 1.0546875.
• d: the diatonic semitone, with a frequency ratio of 16/15 = 1.0666…
• D: the large diatonic semitone, with frequency ratio 27/25 = 1.08.
With this notation, the most popular just intonation scale is
d C d c d C d d c d C d
I’ll say this scale has **type** (2,3,7,0) since it has 2 c’s, 3 C’s, 7 d’s and 0 D’s. The less popular scale I mentioned is
d C d c d c D d c d C d
This scale has type (3,2,6,1). Arguably this scale is worse, because the large diatonic semitone is quite large compared to all the rest.
In 1665, when [Isaac Newton](https://en.wikipedia.org/wiki/Isaac_Newton ) was in college, he studied just intonation and created his own scale, which in my notation is
C d d c d C d d c d C d
In fact [Marin Mersenne](https://en.wikipedia.org/wiki/Marin_Mersenne ) had created the same scale in 1636, but Newton probably didn’t know this. This scale is very similar to what I’m calling the most popular one—recall that was
d C d c d C d d c d C d
The only difference is right at the start: instead of a d followed by a C, Newton put these intervals the other way around. So Newton’s scale has the same type, namely (2,3,7,0). But this change means his scale is not one of the 16 choices I described above, because its minor second is not 16/15: it’s a more complicated fraction, namely 135/128. Here is his scale:
[](https://math.ucr.edu/home/baez/cultural/just_intonation_newton_diagram.jpg )
Thinking about these things naturally led me to some questions, which I included as puzzles in [Part 4](https://johncarlosbaez.wordpress.com/2023/11/15/just-intonation-part-4/ ). Now let me finally answer them!
**Puzzle 1.** As we’ve seen, the most popular 12-tone just intonation scale is of type (2,3,7,0). That is, it has 2 lesser chromatic semitones, 3 greater chromatic semitones, 7 diatonic semitones, and no large diatonic semitones. By permuting these semitones we can get many other scales. How many different scales can we get this way?
**Answer.** We have a 12-element set and we’re asking: in how many ways can we partition it into a 2-element set, a 3-element set and a 7-element set? This is the kind of question that [multinomial coefficients](https://en.wikipedia.org/wiki/Multinomial_theorem#Ways_to_put_objects_into_bins ) were designed to answer. The answer is
█
**Puzzle 2.** Our second, less popular 12-tone just intonation scale is of type (3,2,6,1): it has 3 lesser chromatic semitones, 2 greater chromatic semitones, 6 diatonic semitones and 1 large diatonic semitone. How many other scales can we get by permuting these semitones?
**Answer.** By the same reasoning, we have
such scales. █
These puzzles were warmups for a bigger question:
**Puzzle 3.** How many 12-tone scales are there where the spacing between each pair of successive notes is either a lesser chromatic semitone, a greater chromatic semitone, a diatonic semitone, or a large diatonic semitone?
**Answer.** The only *types* of scales allowed are quadruples of nonnegative integers where
or equivalently,
The four numbers
span the 3-dimensional rational vector space with basis so they must obey one linear relation with integer coefficients (and others following from this one). This relation is
This says cD = Cd: the lesser chromatic semitone followed by the large diatonic semitone takes you up to a frequency 9/8 higher, just like the greater chromatic semitone followed by the diatonic semitone.
This means that if a type is allowed, so is if Furthermore, it means this move (and its inverse) can take you from any allowed type to *all* other allowed types.
So, let’s start with the type where the number of large diatonic semitones, is as small as possible. This is our friend
(2,3,7,0)
We can get all other allowed types by repeatedly adding 1 to the first and last component of this vector and subtracting 1 from the other components. Thus, these are all the allowed types:
(2,3,7,0)
(3,2,6,1)
(4,1,5,2)
(5,0,4,3)
We can now use the methods of Puzzles 1 and 2 to count the scales of each type. We get:
= 7,920 scales of type (2,3,7,0).
= 55,440 scales of type (3,2,6,1).
= 83,160 scales of type (4,1,5,2).
= 27,720 scales of type (5,0,4,3).
So, we get a total of
7,920 + 55,440 + 83,160 + 27,720 = 174,240 scales. █
This is a ridiculously large number of scales! But of course, not all are equally good. Let’s impose some extra constraints.
The whole point of just intonation was to make the third equal to 5/4, and we also want to keep the fourth at 4/3 and the fifth at 3/2, as we had in Pythagorean tuning. When it comes to the second, either 10/9 or 9/8 are considered acceptable in just intonation. I like 9/8 a bit better, so let’s do this:
**Puzzle 4.** How many 12-tone scales are there where the spacing between each pair of successive notes is either a lesser chromatic semitone, a greater chromatic semitone, a diatonic semitone, a large diatonic semitone, and:
• the second is 9/8
• the third is 5/4
• the fourth is 4/3
• the fifth is 3/2?
**Answer.** With these constraints there are 1,600 allowed scales. The idea is this:
• There are 4 ways to go from 1 up to 9/8 in two semitones, since only Cd, dC, cD and Dc multiply to 9/8.
• There are 2 ways to go from 9/8 up to 5/4 in two semitones, since only cd and dc multiply to 10/9.
• There is 1 way to go from 5/4 up to 4/3, since d is 16/15.
• There are 4 ways to go from 4/3 up to 3/2, since only Cd, dC, cD and Dc multiply to 9/8.
• There are 500 ways to go from 3/2 to 2 in five steps. Here we need to count ordered quintuples of c, C, d and D that multiply to 4/3. I did this with a computer.
So, we get 4 × 2 × 1 × 4 × 500 = 1,600 scales of this sort. █
All these scales have the second being the **[greater major second](https://en.wikipedia.org/wiki/List_of_intervals_in_5-limit_just_intonation )**, 9/8. But you might prefer the **[lesser major second](https://en.wikipedia.org/wiki/List_of_intervals_in_5-limit_just_intonation )**, 10/9. So let’s think about that:
**Puzzle 5.** What about the same question as before, but where we constrain the second to be 10/9 instead of 9/8?
**Answer.** Again there are 1600 scales. In Puzzle 4 our scales went up from 1 to 9/8 by choosing two semitones that multiply to 9/8, and then from 9/8 to 5/4 by choosing two that multiply to 10/9. Now the only difference is that we’re going things in the other order: we’re going up from 1 to 10/9 by choosing two semitones that multiply to 10/9, and then from 10/9 to 5/4 by choosing two that multiply to 9/8. So the overall count is the same as before. █
Since they differ only by switching some semitones, the 1,600 scales with a greater major second have the same distribution of types as the 1,600 with a lesser major second. Using a computer, I calculated that in each case there are
• 160 of type (2,3,7,0)
• 560 of type (3,2,6,1)
• 640 of type (4,1,5,2)
• 240 of type (5,0,4,3).
How can we pick out a smaller number of ‘better’ scales? We’ve imposed a lot of constraints on the tones from the first to the fifth, but none on the tones above that. To impose constraints on the higher tones, we can demand that our scale be palindromic, except that we can’t require that the interval from the fourth to the tritone equals the interval from the tritone to the fifth, because is irrational. So, I’ll call scales with the following properties **nearly palindromic**:
• the interval from 1 to ♭2 equals that from 7 to 8
• the interval from ♭2 to 2 equals that from ♭7 to 7
• the interval from 2 to ♭3 equals that from 6 to ♭7
• the interval from ♭3 to 3 equals that from ♭6 to 6
• the interval from 3 to 4 equals that from 5 to ♭6.
**Puzzle 6.** How many nearly palindromic 12-tone scales are there where the spacing between each pair of successive notes is either a lesser chromatic semitone, a greater chromatic semitone, a diatonic semitone, a large diatonic semitone, and:
• the second is 9/8
• the third is 5/4
• the fourth is 4/3?
**Answer.** There are 32 scales with these properties. First note that the above properties force other facts:
• the fifth is 3/4 × 2 = 3/2
• the minor sixth is 4/5 × 2 = 8/5
• the minor seventh is 8/9 × 2 = 16/9.
Thus, we have the following choices:
• There are 4 ways to go from 1 up to 9/8 in two semitones, since only Cd, dC, cD and Dc multiply to 9/8.
• There are 2 ways to go from 9/8 up to 5/4 in two semitones, since only cd and dc multiply to 10/9.
• There is 1 way to go from 5/4 up to 4/3, since d is 16/15.
• There are 4 ways to go from 4/3 up to 3/2, since only Cd, dC, cD and Dc multiply to 9/8.
and from then on, our choices are forced by the nearly palindromic nature of the scale.
There are thus a total of
4 × 2 × 1 × 4 = 32
choices. █
These 32 scales come in two kinds:
• the 16 scales discussed in [Part 4](https://johncarlosbaez.wordpress.com/2023/11/15/just-intonation-part-4/ ), where the minor second is the diatonic semitone, d = 16/15
• 16 others, where the minor second is the greater chromatic semitone, C = 135/128.
The most popular just intonation scale is of the first kind. Newton’s scale is of the second kind.
All 32 of these scales use the greater major second. A similar story holds with the lesser major second.
**Puzzle 7.** How many nearly palindromic 12-tone scales are there where the spacing between each pair of successive notes is either a lesser chromatic semitone, a greater chromatic semitone, a diatonic semitone, a large diatonic semitone, and:
• the second is 10/9
• the third is 5/4
• the fourth is 4/3?
**Answer.** By the symmetry we used to answer Puzzle 5, this question has the same answer as Puzzle 6: there are again 32 choices. █
These 32 scales again come in two kinds:
• 16 scales where the interval from the second to the minor third is the diatonic semitone, d = 16/15
• 16 others where the interval from the second to the minor third is the greater chromatic semitone, C = 135/128.
If you’ve made it this far, congratulations! I was lured in by how many famous mathematicians had studied this subject, and I wanted to lay out all the possibilities very explicitly so any future work would have a solid foundation. I’ll conclude by mentioning three more historically important just intonation scales. I’ll translate them from Muzzulini’s notation to mine, and hope I don’t make any mistakes.
Around 1660, [Nicolas Mercator](https://en.wikipedia.org/wiki/Nicholas_Mercator )—not the Mercator with the map, the one who discovered the power series for the logarithm—-created this scale:
c D d c d c D c d d C d
This is striking because it has two large diatonic semitones: it’s of type (4,1,5,2).
In 1694 the music theorist [William Holder](https://en.wikipedia.org/wiki/William_Holder ) came up with this scale:
c d D c d c D c d D c d
This has three diatonic semitones—the most possible! It’s of type (5,0,4,3).
[Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler ) came up with this scale in 1739:
c D c d d C d c d C d d
This has type (3,2,6,1).
It would be interesting to find out, if possible, why these authors chose the scales they did. They clearly didn’t share my fondness for nearly palindromic scales. Did they scan the universe of possibilities and try to pick a scale that was optimal in some way—or did they did they just make one up? Answering this would require some historical investigation.
For more on Pythagorean tuning, read this series:
• [Pythagorean tuning](https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ ).
For more on just intonation, read these:
• [Part 1](https://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1/ ): The history of just intonation. Just intonation versus Pythagorean tuning. The syntonic comma.
• [Part 2](https://johncarlosbaez.wordpress.com/2023/11/06/just-intonation-part-2/ ): Just intonation from the Tonnetz. The four possible tritones in just intonation. The small and large just whole tones. Ptolemy’s intense diatonic scale, and its major triads.
• [Part 3](https://johncarlosbaez.wordpress.com/2023/11/09/just-intonation-part-3/ ): Curling up a parallelogram in the Tonnetz to get just intonation. The frequency ratios of the four possible tritones: the syntonic comma, the lesser and greater diesis, and the diaschisma.
• [Part 4](https://johncarlosbaez.wordpress.com/2023/11/15/just-intonation-part-4/ ): Choices involved in just intonation. Two symmetrical 13-tone scales, and two 12-tone scales obtained from these by removing the diminished fifth. The four kinds of half-tone that appear in these scales: the diatonic, large diatonic, lesser chromatic and greater chromatic semitones.
• [Part 5](https://johncarlosbaez.wordpress.com/2023/11/17/just-intonation-part-5/ ): Frequency ratios between the four possible tritones in just intonation, and how they are related to frequency ratios between the four kinds of half-tone. The syntonic comma, lesser and greater diesis, diaschisma, and the relations they obey.
• [Part 6](https://johncarlosbaez.wordpress.com/2025/12/29/just-intonation-part-6/ ): Classifying all 174,240 12-tone scales where the intervals between successive notes are always diatonic, large diatonic, lesser chromatic and greater chromatic semitones. The scales of Isaac Newton, Nicolas Mercator, William Holder and Leonhard Euler.
For more on quarter-comma meantone tuning, read this series:
• [Quarter-comma meantone](https://johncarlosbaez.wordpress.com/2023/12/13/quarter-comma-meantone-part-1/ ).
For more on well-tempered scales, read this series:
• [Well temperaments](https://johncarlosbaez.wordpress.com/2024/01/11/well-temperaments-part-1/ ).
For more on equal temperament, read this series:
• [Equal temperament](https://johncarlosbaez.wordpress.com/2023/10/13/perfect-fifths-in-equal-tempered-scales/ ).
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/12/just_intonation_newton_diagram.jpg?w=450

2025-11-22 15:50:42 UTC

##
Beyond the Geometry of Music[](https://media.ed.ac.uk/media/Beyond%20the%20Geometry%20of%20Music/1_fxe5vt57 )
Yesterday I had a great conversation with Dmitri Tymoczko about groupoids in music theory. But at this Higgs Centre Colloquium, he preferred to downplay groupoids and talk in a way physicists would enjoy more. Click on the picture to watch his talk!
What’s great is that he’s not faking it: he’s really found deep ways in which symmetry shows up pervasively in music.
At first he tried to describe them geometrically using ‘[orbifolds](https://en.wikipedia.org/wiki/Orbifold )’, which are spaces in which some singular points have nontrivial symmetry groups, like the tip of a cone. But then he realized that the geometry was less important than the symmetry, which you can describe using ‘[groupoids](https://en.wikipedia.org/wiki/Groupoid )’: categories where every morphism is invertible. That’s why his talk is called “Beyond the geometry of music”.
I’m helping him with his work on groupoids, and I hope he explains his work to mathematicians someday without pulling his punches. I didn’t get to interview him yesterday, but I’ll try to do that soon.
For now you can read his books *[A Geometry of Music](https://dmitri.mycpanel.princeton.edu/geometry-of-music.html )* and *[Harmony: an Owner’s Manual](https://dmitri.mycpanel.princeton.edu/tonality-an-owners-manual.html )* along with [many papers](https://dmitri.mycpanel.princeton.edu/publications.html ). What I’ve read so far is really exciting.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/11/tymoczko_talk.jpg?w=450

2025-10-29 10:00:50 UTC

##
Applied Category Theory 2026[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/10/tallinn.jpg )
The next annual conference on applied category theory is in Estonia!
• [Applied Category Theory 2026](https://actconf2026.github.io/index.html ), Tallinn, Estonia, 6–10 July, 2026. Preceded by the Adjoint School Research Week, 29 June — 3 July.
The conference particularly encourages participation from underrepresented groups. The organizers are committed to non-discrimination, equity, and inclusion. The code of conduct for the conference is available [here](https://www.appliedcategorytheory.org/code-of-conduct/ ).
**Deadlines**
• Registration: TBA
• Abstracts Due: 23 March 2026
• Full Papers Due: 30 March 2026
• Author Notification: 11 May 2026
• Adjoint School: 29 June — 3 July 2026
• Conference: 6 — 10 July 2026
• Final versions of papers for proceedings due: TBA
**Submissions**
ACT2026 accepts submissions in English, in the following three tracks:<li>Research</li><li>Software demonstrations</li><li>Teaching and communication</li>
The detailed Call for Papers is available [here](https://actconf2026.github.io/cfp.html ).
Extended abstracts and conference papers should be prepared with LaTeX. For conference papers please use the EPTCS style files available [here](http://style.eptcs.org ). The submission link is [here](https://easychair.org/my/conference?conf=act2026 ).
Reviewing is single-blind, and we are not making public the reviews, reviewer names, the discussions nor the list of under-review submissions. This is the same as previous instances of ACT.
**Program Committee Chairs**
• Geoffrey Cruttwell, Mount Allison University, Sackville
• Priyaa Varshinee Srinivasan, Tallinn University of Technology, Estonia
**Program Committee**
• Alexis Toumi, Planting Space
• Bryce Clarke, Tallinn University of Technology
• Barbara König, University of Duisburg-Essen
• Bojana Femic, Serbian Academy of Sciences and Arts
• Chris Heunen, The University of Edinburgh
• Daniel Cicala, Southern Connecticut State University
• Dusko Pavlovic, University of Hawaii
• Evan Patterson, Topos Institute
• Fosco Loregian, Tallinn University of Technology
• Gabriele Lobbia, Università di Bologna
• Georgios Bakirtzis, Institut Polytechnique de Paris
• Jade Master, University of Strathclyde
• James Fairbanks, University of Florida
• Jonathan Gallagher, Hummingbird Biosciences
• Joe Moeller, Caltech
• Jules Hedges, University of Strathclyde
• Julie Bergner, University of Virginia
• Kohei Kishida, University of Illinois, Urbana-Champaign
• Maria Manuel Clementino, CMUC, Universidade de Coimbra
• Mario Román, University of Oxford
• Marti Karvonen, University College London
• Martina Rovelli, UMass Amherst
• Masahito Hasegawa, Kyoto University
• Matteo Capucci, University of Strathclyde
• Michael Shulman, University of San Diego
• Nick Gurski, Case Western Reserve University
• Niels Voorneveld, Cybernetica
• Paolo Perrone, University of Oxford
• Peter Selinger, Dalhousie University
• Paul Wilson, University of Southampton
• Robin Cockett, University of Calgary
• Robin Piedeleu, University College London
• Rory Lucyshyn-Wright, Brandon University
• Rose Kudzman-Blais, University of Ottawa
• Ryan Wisnesky, Conexus AI
• Sam Staton, University of Oxford
• Shin-Ya Katsumata, Kyoto Sangyo University
• Simon Willerton, University of Sheffield
• Spencer Breiner, National Institute of Standards and Technology
• Tai Danae Bradley, SandboxAQ
• Titouan Carette, École Polytechnique
• Tom Leinster, The University of Edinburgh
• Walter Tholen, York University
**Teaching & Communication**
• Selma Dündar-Coecke, University College London, Institute of Education
• Ted Theodosopoulos, Nueva School
**Organizing Committee**
• Pawel Sobocinski, Tallinn University of Technology
• Priyaa Varshinee Srinivasan, Tallinn University of Technology
• Sofiya Taskova, Tallinn University of Technology
• Kristi Ainen, Tallinn University of Technology
**Steering Committee**
• John Baez, University of California, Riverside
• Bob Coecke, University of Oxford
• Dorette Pronk, Dalhousie University
• David Spivak, Topos Institute
• Michael Johnson, Macquarie University
• Simona Paoli, University of Aberdeen
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/10/tallinn.jpg?w=450

2025-09-28 16:33:47 UTC

##
Negative MassI’ve been trying to lose weight, so I’ve been studying the physics of negative mass.
Basically it doesn’t exist. But physicists are have run into a serious problem. They think they can use astronomical measurements to put upper bounds on the total mass of all 3 kinds of neutrinos. But when they do this, the upper bound tends to come out *negative!*
This doesn’t really make sense. In fact, if you look at the details, you’ll see it’s the result of extrapolating an approximate formula beyond its realm of validity:
[](https://arxiv.org/abs/2407.10965 )
In more exact formula, equation (1), only *squares* of neutrino masses show up. Indeed in particle physics we never see masses, only squares of masses—so the sign of the mass never matters. But when the neutrino masses are positive and big enough, you can approximate the stuff in equation (1) by something involving the sum of the neutrino masses, getting equation (2).
So far so good. But then people consider what happens when the sum of the neutrino masses is *negative*. This is crazy, but the people doing it know this—or at least the smart ones do. If this theory fits the data best when the sum of neutrino masses is negative, it’s a sign that something has gone wrong somewhere.
The equations above are from here:
• Willem Elbers, Carlos S. Frenk, Adrian Jenkins, Baojiu Li, Silvia Pascoli [Negative neutrino masses as a mirage of dark energy](https://arxiv.org/abs/2407.10965 ), *Physical Review D* **111** (2025), 063534.
but also read this:
• Daniel Green and Joel Meyers, [The cosmological preference for negative neutrino mass](https://arxiv.org/abs/2407.07878 ).
This is a very interesting story, but let’s turn to something simpler: negative mass in Newtonian mechanics.
Suppose you have a positive mass star of mass m and a negative mass star of mass -m. Thanks to Newton’s laws of gravity they will each feel a force pointing outward. The positive mass star will accelerate away thanks to Newton’s other law, F = ma. But by the same rule, the negative mass star will accelerate *toward* the positive mass star, by the same amount!
So, the negative mass star will wind up chasing the positive mass star, faster and faster, never catching it.
That’s in Newtonian gravity… at least if we assume the equivalence principle, saying gravitational and inertial mass are equal.
**Puzzle.** What happens in Newtonian gravity if you have an ordinary positive mass star and a star with negative *gravitational* mass but positive *inertial* mass?
**Puzzle.** What happens in Newtonian gravity if you have an ordinary positive mass star and a star with negative *inertial* mass but positive *gravitational* mass?
What about general relativity? The famous physicist Bondi studied this question in 1957!
• Hermann Bondi, [Negative mass in general relativity](http://ayuba.fr/pdf/bondi1957.pdf ), *Reviews of Modern Physics* **29** (1957), 423–428.
He argued that essentially the same thing happens as in Newtonian mechanics: the two stars start accelerating, with the negative mass star chasing the positive mass star. One difference is that both stars approach but never exceed the speed of light.
But he seems to consider two infinitesimal masses, not thinking about the deeper question of how a finite-sized negative point mass would behave in general relativity.
What would a negative mass black hole be like? The only reasonable answer to this silly question is to take the Schwarzschild solution describing a black hole of mass m, and stick in a negative number for m.
You’ll get a solution of the equations of general relativity. I haven’t studied it carefully, but here’s what I know so far.
First, a negative mass black hole *not* a white hole.
A [white hole](https://en.wikipedia.org/wiki/White_hole ) is a time-reversed black hole. It still attracts you gravitationally! To see this, take a movie of someone throwing a rock up into the air. Then play this movie backwards. The rock still goes up and falls back down! The time reverse of gravitational attraction is not gravitational repulsion. It’s still attraction. In fact, if you’re near a white hole, it’s hard to tell it’s not a black hole…. until you see someone fall *out*.
A negative mass black hole is different. You’ll be pushed away from it, and this repulsive ‘force’ will become infinite as you approach the central singularity. But unlike an ordinary black hole, this is a naked singularity: it’s not hidden behind an event horizon, an imaginary surface such that if you cross it, you can never get out.
If you shine light at a negative mass black hole, it will be repelled and scatter away. Thus, calling it a ‘black hole’ is inaccurate. It would look roughly like some sort of strange mirrored sphere. Someone should simulate this thing and animate it.
II’ve had some good conversations about negative mass black holes [on Mastodon](https://mathstodon.xyz/@johncarlosbaez/115281018869867910 ), but I want to understand them better. So far I just know one paper on them, pointed out by Robert Low:
• Servando V. Serdio, Hernando Quevedo, [Singularity theorems in Schwarzschild spacetimes](https://arxiv.org/abs/2001.11376 ), *The European Physical Journal Plus* **135** (2020), 1–21.
This paper is very mathematical: for example, it points out that negative mass black hole spacetimes are not globally hyperbolic. But I would like to see a paper written by someone like John Wheeler or Kip Thorne.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/09/negative_neutrino_mass.jpg?w=450

2025-09-11 14:09:29 UTC

##
Categories for Public Health Modeling
How, exactly, can category theory help modeling in public health? I wrote a paper about this with two people who helped run Canada’s COVID modeling, together with a software engineer and a mathematician at the Topos Institute:
• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, [A categorical framework for modeling with stock and flow diagrams](https://arxiv.org/pdf/2211.01290 ), in *Mathematics of Public Health: Mathematical Modelling from the Next Generation*, eds. Jummy David and Jianhong Wu, Springer, 2023, pp. 175-207.
Anything you can with category theory, you can also do without it—just like you can cross the Alps without shoes. But categorical methods make public health modeling easier in a lot of ways.
The introduction lists a few of these ways. Then the paper goes on to provide details, including a long appendix showing actual Julia code for our software.
But here’s the basic idea:
Many people working on epidemiological modeling like diagrams because they provide easily understandable but informal steps towards a mathematically rigorous formulation of a model in terms of ordinary differential equations (ODEs). But ODEs are typically opaque to non-modelers—including the interdisciplinary members of the teams that typically are required for impactful models.
The tradition of modeling called [System Dynamics](https://en.wikipedia.org/wiki/System_dynamics ) places a premium on engagement with stakeholders, so it offers a modeling approach centered around *diagrams*. This approach commonly proceeds in a manner that depicts model structure using successively more detailed models.
The process starts with a [‘causal loop diagram’](https://en.wikipedia.org/wiki/Causal_loop_diagram ) illustrating causal connections and feedback loops:
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/09/causalloop_seir.jpg )
It then often proceeds to a ‘system structure diagram’, which distinguishes stocks from flows but still lacks quantitative information. The next step is to construct a [stock and flow diagram’](https://en.wikipedia.org/wiki/Causal_loop_diagram ):
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/09/seir_stockflow.jpg )
This diagram is visually identical to the system structure diagram, but it also includes formulae (at the bottom here).
The stock–flow diagram is treated as the durable end result of this modeling process, since it uniquely specifies a system of first-order ODEs. System Dynamics modeling typically then alternates between assessing scenario outcomes resulting from numerically integrating the ODEs, performing other analyses (e.g., identifying location or stability of equilibria), and elaborating the stock-flow diagram.
While each of the 3 types of diagrams in the System Dynamics tradition is recommended by visual accessibility, the traditional approach suffers from a number of practical shortcomings:
• **Monolithic models**: Models are traditionally built up in a monolithic fashion, leading ultimately to a single large piece of code. Drawn as a single diagram, a model can be extremely complex. For example, here is Canada’s main model of COVID during the pandemic, put together by Nathaniel Osgood and Xiaoyan Li, made using the commercially available software called AnyLogic:
[](https://math.ucr.edu/home/baez/epidemiology_categories/texfiles/Canadian_Covid_model.png )
Click to enlarge. If it looks like a huge mess, that’s part of the point.
Working with a single huge model like this inhibits independent simultaneous work by multiple modelers. Lack of model modularity further prevents effective reuse of particular model elements. If elements of other models are used, they are commonly copy-and-pasted into the developing model, with the source and destination then evolving independently. Such separation can lead to a proliferation of conceptually overlapping models in which a single conceptual change requires corresponding updates in several successive models.
• **The curse of dimensionality**: Modelers refine simple models by ‘stratifying’ them, subdividing stocks into smaller stocks. For example, the ‘infected’ stock might be stratified into ‘infected male’ and ‘infected female’. While stratification is a key tool for representing heterogeneity, stratification commonly requires modifications across the breadth of a model—stocks, flows, derived quantities, and many parameters. When stratification involves multiple dimensions of heterogeneity, it can lead to a proliferation of terms in the ODEs. For example, rendering a model characterizing both COVID-19 into a model also characterizes influenza would require that each COVID-19 state to be replicated for each stage in the natural history of influenza. Represented visually, this stratification leads to a multi-dimensional lattice, commonly with progression proceeding along several dimensions of the lattice. Because of the unwieldy character of the diagram, the structure of the model is obscured. Adding, removing, or otherwise changing dimensions of heterogeneity routinely leads to pervasive changes across the model.
• **Privileging ODE semantics**: The structure of causal loop diagrams, system structure diagrams and stock-flow diagrams characterizes general state and accumulations, transitions and posited causal relations—including induced feedbacks—amongst variables. Nothing about such a characterization restricts its meaning to ordinary differential equations; indeed, many other interpretations and uses of these diagrams are possible. However, existing software privileges an ODE interpretation for stock-flow diagrams, while sometimes allowing for secondary analyses in *ad hoc* way—for example, identifying causal loops associated with the model, or verifying dimensional homogeneity in dimensionally annotated models. Conducting other sorts of analyses—such as computation of eigenvalue elasticities or loop gains, analysis as a stochastic transition system, or other methods such as particle filtering, particle Markov chain Monte Carlo, or Kalman filtering—typically requires bespoke software for reading, representing and analyzing stock-flow models.
• **Divergence of model representations**: Although the evolution from causal loop diagrams to system structure diagrams to stock-flow models is one of successive elaboration and informational enrichment, existing representations treat these as entirely separate characterizations and fail to capture the logical relationships between them. Such fragmentation commonly induces inconsistent evolution. Indeed, in many projects, the evolution of stock-flow diagrams renders the earlier, more abstract formulations obsolete, and the focus henceforth rests on the stock-flow diagrams.
What is less widely appreciated is that beyond their visual transparency and capacity to be lent a clear ODE semantics, the 3 kinds of diagrams I mentioned each have a precise mathematical structure—a corresponding grammar, as it were. This algebraic structure, called the ‘syntax’ of tehse diagrams, can be characterized using category theory. Formalizing the syntax this way lends precise meaning to the process of ‘composing’ models (building them out of smaller parts), stratifying them, and other operations. Explicitly characterizing the syntax in software also allows for diagrams to be represented, manipulated, composed, transformed, and flexibly analyzed in software that implements the underlying mathematics.
Formalizing the mathematics of diagram-based models using category theory and capturing it in software offers many benefits. Our paper discusses and demonstrates just a few:
• **Separation of syntax and semantics.** Category theory gives tools to separate the formal structure, or ‘syntax’, of diagram-based models from the uses to which they are put, or ‘semantics’. The syntax lives in one category, which can then be mapped to various different semantic categories using various functors. This separation permits great flexibility in applying different semantics to the same model. With appropriate software design, this decoupling can allow the same software to support a diverse array of analyses, which can be supplemented over time.
• **Reuse of structure.** Category theory provides a structured way to build complex diagrams by composing small reusable pieces. Diagrams are morphisms in a monoidal category, and you build bigger diagrams by composing and tensoring these morphisms. With software support, modeling frameworks can allow for saving models and retrieving them for reuse as parts of many different models. For example, in public health a diagram representing contact tracing can be reused across diagrams addressing different pathogens.
• **Modular stratification.** A categorical foundation further supports a structured method to build stratified diagrams out of modular, reusable, largely orthogonal pieces. This method is called taking a ‘pullback’. In contrast to the global changes commonly required to a diagram and the curse of dimensionality that traditionally arises when stratifying a diagram, categorically-founded stratification methods allow for crisply characterizing a stratified diagram as built from simpler diagrams, one for each heterogeneity or progression dimension.
Our paper goes into detail about how all this works. Elsewhere we have longer lists of what’s bad about current modeling practice, and how we hope to improve it. But I hope this helps a bit.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/09/seir_stockflow.jpg?w=450
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/09/causalloop_seir.jpg?w=450

2025-06-20 12:14:18 UTC

##
Graphs With Polarities (Part 6)[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/06/homework_causal_loop_diagram.jpg )
I’ve been working with Adittya Chaudhuri on some ideas related to this series of blog articles, and now our paper is done!
• John Baez and Adittya Chaudhuri, [Graphs with polarities](http://math.ucr.edu/home/baez/polarities.pdf ).
**Abstract.** In fields ranging from business to systems biology, directed graphs with edges labeled by signs are used to model systems in a simple way: the nodes represent entities of some sort, and an edge indicates that one entity directly affects another either positively or negatively. Multiplying the signs along a directed path of edges lets us determine indirect positive or negative effects, and if the path is a loop we call this a positive or negative feedback loop. Here we generalize this to graphs with edges labeled by a monoid, whose elements represent ‘polarities’ possibly more general than simply ‘positive’ or ‘negative’. We study three notions of morphism between graphs with labeled edges, each with its own distinctive application: to refine a simple graph into a complicated one, to transform a complicated graph into a simple one, and to find recurring patterns called ‘motifs’. We construct three corresponding symmetric monoidal double categories of ‘open’ graphs. We study feedback loops using a generalization of the homology of a graph to homology with coefficients in a commutative monoid. In particular, we describe the emergence of new feedback loops when we compose open graphs using a variant of the Mayer–Vietoris exact sequence for homology with coefficients in a commutative monoid.
https://johncarlosbaez.wordpress.com/wp-content/uploads/2025/06/homework_causal_loop_diagram.jpg?w=450

2024-12-22 03:25:35 UTC

##
EpicyclesSome people think medieval astronomers kept adding ‘epicycles’ to the orbits of planets, culminating with the Alfonsine Tables created in 1252. The 1968 *Encyclopædia Britannica* says:
> By this time each planet had been provided with from 40 to 60 epicycles to represent after a fashion its complex movement among the stars.
But this is complete nonsense!
Medieval astronomers did *not* use so many epicycles. The [Alfonsine Tables](https://en.wikipedia.org/wiki/Alfonsine_tables ), which the *Brittanica* is mocking above, actually computed planetary orbits using the method in Ptolemy’s *Almagest*, developed way back in 150 AD. This method uses at most 31 circles and spheres—nothing like Britannica’s ridiculous claim of between 40 to 60 epicycles per planet.
The key idea in Ptolemy’s model was this:
[](http://math.ucr.edu/home/baez/astronomical/ptolemy_epicycles.gif )
The blue dot here is the Earth. The large black circle, offset from the Earth, is called a ‘deferent’. The smaller black circle is called an ‘epicycle’. The epicycle makes up for how in reality the Earth is not actually stationary, but moving around the Sun.
The center of the epicycle rotates at constant angular velocity around the purple dot, which is called the ‘equant’. The equant and the Earth are at equal distances from the center of the black circle. Meanwhile the planet, in red, moves around the center of the epicycle at constant angular velocity.
In the *Almagest*, Ptolemy used some additional cycles to account for how the latitudes of planets change over time. In reality this happens because the planets don’t all move in the same plane. Ptolemy also used additional ‘epicyclets’ to account for peculiarities in the orbits of Mercury and the Moon, and a mechanism to account for the precession of equinoxes—which really happens because the Earth’s axis is slowly precessing.
In a later work, the *Planetary Hypothesis*, Ptolemy eliminated some cycles by having the planets orbit in different planes (as they indeed do). On the other hand, he considered adding other cycles (or actually spheres) for physical purposes. Depending on how you define things, this setup either has more cycles than the *Almagest*, or a bit fewer.
Anyway, there was never a population explosion of epicycles to the wild degree that the 1968 *Brittanica* claimed. So, just because something is in an encyclopedia, or even an encyclopædia, doesn’t mean it’s true.
The Encyclopædia Britannica quote comes from their 1968 edition, volume 2, in the article on the Spanish king Alfonso X, which on page 645 discusses the Alfonsine Table commissioned by this king:
> By this time each planet had been provided with from 40 to 60 epicycles to represent after a fashion its complex movement among the stars. Amazed at the difficulty of the project, Alfonso is credited with the remark that had he been present at the Creation he might have given excellent advice.
In *The Book Nobody Read*, Owen Gingerich writes that he challenged Encyclopædia Britannica about the number of epicycles. Their response was that the original author of the entry had died and its source couldn’t be verified. Gingerich has also expressed doubts about the quotation attributed to King Alfonso X.
For the controversy over whether medieval astronomers used lots of epicycles, start here:
• Wikipedia, [Deferent and epicycle: history](https://en.wikipedia.org/wiki/Deferent_and_epicycle#History ).
and then go here:
• Wikipedia, [Deferent and epicycle: the number of epicycles](https://en.wikipedia.org/wiki/Deferent_and_epicycle#The_number_of_epicycles ).
Then dig into the sources! For example, Wikipedia says the claim that the Ptolemaic system uses about 80 circles seems to have appeared in 1898. It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro, who used either 77 or 79 orbs. So *some* theories used lots of epicycles—but not the most important theories, and nothing like the 240-360 claimed by the 1968 Brittanica.
Owen Gingerich wrote *The Book Nobody Read* about his quest to look at all 600 extant copies of Copernicus’ *De revolutionibus*. The following delightful passage was contributed by [pglpm](https://mathstodon.xyz/@npub138s…ydzl/113653412304655853 ) on Mastodon:
https://johncarlosbaez.wordpress.com/wp-content/uploads/2024/12/ptolemy_epicycles-1.gif?w=450

2024-11-28 02:08:08 UTC

##
Threats to US Climate AgenciesTrump’s cronies are already going after US government employees involved in the response to climate change. You can read about it here:
• Hadas Gold and Rene Marsh, [Elon Musk publicized the names of government employees he wants to cut. It’s terrifying federal workers](https://www.cnn.com/2024/11/27/business/elon-musk-government-employees-targets/index.html ), *CNN*, 27 November 2024.
> When President-elect Donald Trump said Elon Musk and Vivek Ramaswamy would recommend major cuts to the federal government in his administration, many public employees knew that their jobs could be on the line.
> Now they have a new fear: becoming the personal targets of the world’s richest man—and his legions of followers.
> Last week, in the midst of the flurry of his daily missives, Musk reposted two X posts that revealed the names and titles of people holding four relatively obscure climate-related government positions. Each post has been viewed tens of millions of times, and the individuals named have been subjected to a barrage of negative attention. At least one of the four women named has deleted her social media accounts.
> Although the information he posted on those government positions is available through public online databases, these posts target otherwise unknown government employees in roles that do not deal directly with the public.
> […]
> It appears [one] woman Musk targeted has since gone dark on social media, shutting down her accounts. The agency, the US International Development Finance Corporation, says it supports investment in climate mitigation, resilience and adaptation in low-income countries experiencing the most devastating effects of climate change. A DFC official said the agency does not comment on individual personnel positions or matters.
> Musk also called out the Department of Energy’s chief climate officer in its loan programs office. The office funds fledgling energy technologies in need of early investment and awarded $465 million to Tesla Motors in 2010, helping to position Musk’s electric vehicle company as an EV industry leader. The chief climate officer works across agencies to “reduce barriers and enable clean energy deployment” according to her online bio.
> Another woman, who serves as senior advisor on environmental justice and climate change at the Department of Health and Human Services, was another Musk target. HHS focuses on protecting the public health from pollution and other environmental hazards, especially in low-income communities and communities of color that are experiencing a higher share of exposures and impacts. The office first launched at Health and Human Services under the Biden administration in 2022.
> A senior adviser to climate at the Department of Housing and Urban Development was also singled out. The original X post said the woman “should not be paid $181,648.00 by the US taxpayer to be the ‘Climate advisor’ at HUD.” Musk reposted with the comment: “But maybe her advice is amazing.” Followed by two laughing emojis.
This revives fears that US climate change policies will be rolled back. Reporters are interviewing me again about the [Azimuth Climate Data Backup Project](https://math.ucr.edu/home/baez/azimuth_backup_project/ )—because we’re again facing the possibility that a Trump administration could get rid of the US government’s climate data.
From 2016 to 2018, our team backed up up 30 terabytes of US government databases on climate change and the environment, saving it from the threat of a government run by climate change deniers. 627 people contributed a total of $20,427 to our project on Kickstarter to pay for storage space and a server.
That project is done now, with the data stored in a secret permanent location. But that data is old, and there’s plenty more by now.
I don’t have the energy to repeat the process now. As before, I’m hoping that the people at NOAA, NASA, etc. have quietly taken their own precautions. They’re in a much better position to do it! But I don’t know what they’ve done.
First I got interviewed for this *New York Times* article about the current situation:
• Austyn Gaffney, [How Trump’s return could affect climate and weather data](http://math.ucr.edu/home/baez/azimuth_backup_project/article_NYT_20241114.pdf ), *New York Times*, 14 November 2024.
Then I got interviewed for a second article, which says a bit more about what the Azimuth Project actually did:
• Chelsea Harvey, [Scientists scramble to save climate data from Trump—again](http://math.ucr.edu/home/baez/azimuth_backup_project/article_SciAm_20241122.pdf ), *Scientific American*, 22 November 2024.
> Eight years ago, as the Trump administration was getting ready to take office for the first time, mathematician John Baez was making his own preparations.
> Together with a small group of friends and colleagues, he was arranging to download large quantities of public climate data from federal websites in order to safely store them away. Then-President-elect Donald Trump had repeatedly denied the basic science of climate change and had begun nominating climate skeptics for cabinet posts. Baez, a professor at the University of California, Riverside, was worried the information — everything from satellite data on global temperatures to ocean measurements of sea-level rise — might soon be destroyed.
> His effort, known as the Azimuth Climate Data Backup Project, archived at least 30 terabytes of federal climate data by the end of 2017.
> In the end, it was an overprecaution.
> The first Trump administration altered or deleted numerous federal web pages containing public-facing climate information, according to monitoring efforts by the nonprofit Environmental Data and Governance Initiative (EDGI), which tracks changes on federal websites. But federal databases, containing vast stores of globally valuable climate information, remained largely intact through the end of Trump’s first term.
> Yet as Trump prepares to take office again, scientists are growing more worried.
> Federal datasets may be in bigger trouble this time than they were under the first Trump administration, they say. And they’re preparing to begin their archiving efforts anew.
> “This time around we expect them to be much more strategic,” said Gretchen Gehrke, EDGI’s website monitoring program lead. “My guess is that they’ve learned their lessons.”
> [….]
> Much of the renewed concern about federal data stems from Project 2025, a 900-page conservative policy blueprint spearheaded by the Heritage Foundation that outlines recommendations for the next administration.
> Project 2025 calls for major overhauls of some federal science agencies. It suggests that Trump should dismantle NOAA and calls for the next administration to “reshape” the U.S. Global Change Research Program, which coordinates federal research on climate and the environment.
> The plan also suggests that the “Biden Administration’s climate fanaticism will need a whole-of-government unwinding.”
> A leaked video from the Project 2025 presidential transition project suggested that political appointees “will have to eradicate climate change references from absolutely everywhere.”
> Trump has previously distanced himself from Project 2025. In July, he wrote on the social media platform Truth Social that he knew “nothing about Project 2025,” did not know who was behind it and did not have anything to do with the plan.
> But since winning the 2024 presidential election, Trump has picked several nominees for his new administration that are credited by name in the conservative policy plan, reviving fears that Project 2025 could influence his priorities.
> Trump has also recently named Elon Musk and Vivek Ramaswamy to lead his new so-called Department of Government Efficiency, an external commission tasked with shrinking the federal government, restructuring federal agencies and cutting costs. The announcement has also ignited concerns about job security for federal scientists, including the researchers tasked with maintaining government datasets.
> “There are lots and lots of signs that the Trump team is attempting to decapitate the government in the sense of firing lots of people,” said Baez, who co-founded the Azimuth Climate Data Backup Project in 2016 and is currently a professor of the graduate division in the math department at University of California Riverside. “If they manage to do something like that, then these databases could be in more jeopardy.”
> Though federal datasets remained largely untouched under the first Trump administration, other climate-related information on federal websites did change or disappear, Gehrke pointed out. EDGI documented about a 40 percent decline in the use of the term “climate change” across 13 federal agencies it monitored during the first term.
> A better organized effort could result in more censoring under a second administration, she said.
> While groups like EDGI are gearing up for their next efforts, Baez says he has no immediate plans to revamp the Azimuth Climate Data Backup Project — although he hopes other groups will step up instead. One lesson he learned the first time is just how much data exists in the federal ecosystem and how much effort it takes to archive it, even with a dedicated group of volunteers.
> “We got sort of a little bit burnt out by that process,” Baez said. “I’m hoping some younger generation of people picks up where we left off.”
If you’re interested in doing this, and want to see what data we backed up, [you can see a list here](https://math.ucr.edu/home/baez/azimuth_backup_project/our_progress_20190412.pdf ).
[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2016/12/azimuth_logo.jpg )
https://johncarlosbaez.wordpress.com/wp-content/uploads/2016/12/azimuth_logo.jpg?w=450

2024-09-25 21:23:36 UTC

Here’s some basic information about the next big annual applied category theory conference—Applied Category Theory 2025—and the school that goes along with that: the Adjoint School.
[James Fairbanks](https://mae.ufl.edu/people/profiles/james-fairbanks/ ) will hold ACT2025 and the Adjoint School at the [University of Florida](https://www.ufl.edu/ ), in Gainesville, on these dates:
• Adjoint School: May 26–30, 2025.
• ACT 2025: June 2–6, 2025.
More information will eventually appear on a website somewhere, and I’ll try to remember to let you know about it!
https://johncarlosbaez.wordpress.com/2024/09/25/act-2025-and-the-adjoint-school/
https://johncarlosbaez.wordpress.com/wp-content/uploads/2024/09/university_of_florida.jpg?w=450

2023-12-13 13:10:35 UTC

##
Quarter-Comma Meantone (Part 1)I’ve spent the last few weeks drawing pictures of tuning systems, since I realized this is a good way to show off their mathematical beauty. Now I’ll start deploying them!
I’ve already written about the first two hugely important tuning systems in Western music:
• [Pythagorean tuning](https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ ).
• [Just intonation](https://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1/ ).
It’s time to introduce the third: ‘quarter-comma meantone’.
But first, remember the story so far!
[Pythagorean tuning](https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ ) may go back to Mesopotamia, but it was widely discussed by Greek mathematicians—perhaps including Pythagoras, whose life is mainly the stuff of legends written down centuries later, but more certainly Eratosthenes, and definitely Ptolemy. It was widely used in western Europe in the middle ages, especially before 1300.
The principle behind Pythagorean tuning is to start with some pitch and go up and down from there by ‘just fifths’—repeatedly multiplying and dividing the frequency by 3/2—until you get two pitches that are almost 7 octaves apart. Here I’ll do it starting with C:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_star_of_fifths.jpg )
But there are some problems. The highest tone is a bit *less* than 7 octaves above the lowest tone! Their frequency ratio is called the **[Pythagorean comma](https://en.wikipedia.org/wiki/Pythagorean_comma )**. And we get a total of 13 tones, not 12.
To deal with these problems, we can simply omit one of these two tones and use only the other in our scale. There are two ways to do this, which are mirror images of each other:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_star_of_fifths_without_diminished_fifth.jpg )
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_star_of_fifths_without_augmented_fourth.jpg )
Each breaks the symmetry of the scale. And each gives one fifth that’s noticeably smaller than the rest. It’s called a ‘wolf fifth’—because it’s so out of tune it howls like a wolf!
What can we do? One solution is simply to avoid playing this fifth. You’ve probably heard the old joke. A patient tells his doctor: “It hurts when I lift my arm like this.” The doctor replies: “So don’t lift your arm like that!”
This worked pretty well for medieval music, where the fifth and octaves were the dominant forms of harmony, and people didn’t change keys much, so they could avoid the wolf fifth. But in the late 1300s, major thirds became very important in English music, and soon they spread throughout Europe. A major third sounds perfectly in tune—or technically, ‘just’—when it has a frequency ratio of
But the major thirds in Pythagorean tuning are bigger than this!
Let’s see why. This will eventually lead us to the solution called ‘quarter-comma meantone’ tuning.
To go up a major third in Pythagorean tuning, we take any tone and go up 4 fifths, getting a tone whose frequency is
times as high. Then we go down 2 octaves to get a tone whose frequency is
times that of our original tone. This is called a **Pythagorean major third**. It’s close to the just major third, 5/4 = 1.25. But it’s a bit too high!
Let’s see what what these Pythagorean major thirds look like, and where they sit in the scale. To do this, let’s take our original ‘star of fifths’:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_star_of_fifths.jpg )
and reorder the notes so they form a ‘circle of fifths’:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_circle_of_fifths.jpg )
Here we see *two* wolf fifths, each containing one of the notes separated by a Pythagorean comma (namely G♭ and F♯). As we’ve seen, if we omit either one of these notes we’re left with a single wolf fifth. But this breaks the left-right symmetry of the above picture, so let’s leave them both in for now.
Now let’s draw all the Pythagorean thirds in blue:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_circle_of_fifths_with_pythagorean_thirds.jpg )
A pretty, symmetrical picture. But not every note has a blue arrow pointing out of it! The reason is that not every note has some other note in the scale a Pythagorean third higher than it. We could delve into this more….
But instead, let’s figure out what to do about these annoyingly large Pythagorean thirds!
Historically, the first really popular solution was to use ‘just intonation’, a system based on simple fractions built from the numbers 2, 3 (as in Pythagorean tuning) but also 5. It was discussed by Ptolemy as far back as 150 AD. But it became widely used from roughly 1300 to at least 1550—starting in England, and then spreading throughout Europe, along with the use of major thirds.
Just intonation makes a few important thirds in the scale be just, but not as many as possible. Around 1523 another solution was invented, with more just thirds: ‘quarter-comma meantone’. It became popular around 1550, and it dominated Europe until about 1690. Let’s see what this system is, and why it didn’t catch on sooner.
The idea is to tweak Pythagorean tuning so that all the Pythagorean thirds I just showed you become *just* thirds! To do this, we’ll simply take the Pythagorean system:
[](http://math.ucr.edu/home/baez/cultural/tuning/pythagorean_circle_of_fifths_with_pythagorean_thirds.jpg )
and shrink all the blue arrows so they have a frequency ratio of 5/4.
Unfortunately this will force us to shrink the black arrows, too, In other words, *to make our major thirds just, we need to shrink our fifths*. It turns out that we need fifths with a frequency ratio of
This is only a tiny bit less than the ideal fifth, namely 1.5. It’s not a nasty wolf fifth: it sounds pretty good. In fact it’s quite wonderful that the fourth root of 5 is so close to 3/2. So, using some fifths like this may count as an acceptable sacrifice if we want just major thirds.
Here’s what we get:
[](http://math.ucr.edu/home/baez/cultural/tuning/quarter-comma_circle_of_fifths_with_just_thirds.jpg )[](https://johncarlosbaez.wordpress.com/wp-content/uploads/2024/01/quarter-comma_circle_of_fifths_with_just_thirds-2.jpg )
This tuning system is called **[quarter-comma meantone](https://en.wikipedia.org/wiki/Quarter-comma_meantone )**.
You’ll note that by shrinking the blue and black arrows—that is, the thirds and fifths—we’ve now made the note F♯ *lower* than G♭, rather than higher, as it was in Pythagorean tuning. Their frequency ratio is now
which is yet another of those annoying little glitches: this one is called the **[lesser diesis](https://en.wikipedia.org/wiki/Diesis )**.
So that’s quarter-comma meantone tuning in a nutshell. But there’s a lot more to say about it. For example, I haven’t explained all the numbers in that last picture. Where do and the lesser diesis 128/125 come from??? I haven’t even explained why this system called ‘quarter-comma meantone’. These issues are related. I’ll explain them both next time, but I’ll give you a hint now. I told you that the Pythagorean major third
is a bit bigger than the just major third:
But how much bigger? Their ratio is
This number, yet *another* of those annoying glitches in harmony theory, is called the **[syntonic comma](https://en.wikipedia.org/wiki/Syntonic_comma )**. And this, not the Pythagorean comma, is the comma that gives ‘quarter-comma meantone’ its name! By taking the syntonic comma and dividing it into four equal parts—or more precisely, taking its fourth root—we are led to quarter-comma meantone. I’ll show you the details next time.
Quarter-comma meantone is dramatically different from the earlier tuning systems I’ve discussed, since it uses an irrational number: the fourth root of 5. I think this is why it took so long for quarter-comma meantone to be discovered. After all, irrational numbers were anathema in the old Pythagorean tradition relating harmony to mathematics.
It seems that quarter-comma meantone was discovered in a burst of more sophisticated mathematical music theory in Renaissance Italy—along with other meantone systems, but I’ll explain what that means later. References to tuning systems that could be meantone appeared as early as the 1496 text *Practicae musica* by [Franchinus Gaffurius](https://en.wikipedia.org/wiki/Franchinus_Gaffurius ). [Pietro Aron](https://en.wikipedia.org/wiki/Pietro_Aron ) unmistakably discussed quarter-comma meantone in *Toscanello in musica* in 1523. However, the first mathematically precise descriptions appeared in the late 16th century treatises by the great [Gioseffo Zarlino](https://en.wikipedia.org/wiki/Gioseffo_Zarlino ) (*Le istitutioni harmoniche*, 1558) and [Francisco de Salinas](https://en.wikipedia.org/wiki/Francisco_de_Salinas ) (*De musica libri septem*, 1577). Those two also talked about ‘third-comma’ and ‘two-sevenths-comma’ meantone systems.
For more on Pythagorean tuning, read this series:
• [Pythagorean tuning](https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/ ).
For more on just intonation, read this series:
• [Just intonation](https://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1/ ).
For more on quarter-comma meantone tuning, read these:
• [Part 1](https://johncarlosbaez.wordpress.com/2023/12/13/quarter-comma-meantone-part-1/ ). Review of Pythagorean tuning. How to obtain quarter-comma meantone by expanding the Pythagorean major thirds to just major thirds.
• [Part 2](https://johncarlosbaez.wordpress.com/2023/12/18/quarter-comma-meantone-part-2/ ). How Pythagorean tuning, quarter-comma meantone and equal temperament all fit into a single 1-parameter family of tuning systems.
• [Part 3](https://johncarlosbaez.wordpress.com/2023/12/21/quarter-comma-meantone-part-3/ ). What ‘quarter-comma’ means in the phrase ‘quarter-comma meantone’: most of the fifths are lowered by a quarter of the syntonic comma.
• [Part 4.](https://johncarlosbaez.wordpress.com/2023/12/25/quarter-comma-meantone-part-4/ ) Omitting the diminished fifth or augmented fourth from quarter-comma meantone. The relation between the Pythagorean comma, lesser diesis and syntonic comma.
• [Part 5.](https://johncarlosbaez.wordpress.com/2024/01/01/quarter-comma-meantone-part-5/ ) The sizes of the two kinds of semitone in quarter-comma meantone: the chromatic semitone and diatonic semitone. The size of the tone, and what the ‘meantone’ means in the phrase ‘quarter-comma meantone’.
• [Part 6.](https://johncarlosbaez.wordpress.com/2024/01/04/quarter-comma-meantone-part-6/ ) What happens to quarter-comma meantone when you change it from a 13-tone scale to a more useful 12-tone scale by removing the diminished fifth.
• [Part 7.](https://johncarlosbaez.wordpress.com/2024/01/08/quarter-comma-meantone-part-7/ ) Why it’s better to start the quarter-comma meantone scale on D rather than C.
For more on equal temperament, read this series:
• [Equal temperament](https://johncarlosbaez.wordpress.com/2023/10/13/perfect-fifths-in-equal-tempered-scales/ ).
https://johncarlosbaez.wordpress.com/wp-content/uploads/2023/12/quarter-comma_circle_of_fifths_with_just_thirds.jpg?w=450

2021-09-13 16:21:39 UTC

##
The Cyclic Identity for Partial Derivatives
As an undergrad I learned a lot about partial derivatives in physics classes. But they told us rules as needed, without proving them. This rule completely freaked me out. If derivatives are kinda like fractions, shouldn’t this equal 1?
Let me show you why it’s -1.
First, consider an example:
This example shows that the identity is not crazy. But in fact it
holds the key to the general proof! Since is a coordinate system we can assume without loss of generality that . At any point we can approximate to first order as for some constants . But for derivatives the constant doesn’t matter, so we can assume it’s zero.
Then just compute!
There’s also a proof using differential forms that you might like
better. You can see it here, along with an application to
thermodynamics:
But this still leaves us yearning for more intuition — and for me, at least, a more symmetrical, conceptual proof. Over on Twitter, someone named
[Postdoc/cake](https://twitter.com/postdocforever/status/1437413772168024071 ) provided some intuition using the same example from thermodynamics:
> Using physics intuition to get the minus sign:<li>increasing temperature at const volume = more pressure (gas pushes out more)</li><li>increasing temperature at const pressure = increasing volume (ditto)</li><li>increasing pressure at const temperature = decreasing volume (you push in more)</li>
[](https://twitter.com/postdocforever/status/1437413772168024071 )
[Jules Jacobs](https://twitter.com/JulesJacobs5/status/1437415867185344517 ) gave the symmetrical, conceptual proof that I was dreaming of:
[](https://twitter.com/JulesJacobs5/status/1437415867185344517 )
This proof is more sophisticated than my simple argument, but it’s very pretty, and it generalizes to higher dimensions in ways that’d be hard to guess otherwise.
He uses some tricks that deserve explanation. As I’d hoped, the minus signs come from the anticommutativity of the wedge product of 1-forms, e.g.
But what lets us divide quantities like this? Remember, are all functions on the plane, so are 1-forms on the plane. And since the space of 2-forms at a point in the plane is 1-dimensional, we can divide them! After all, given two vectors in a 1d vector space, the first is always some number times the second, as long as the second is nonzero. So we can define their ratio to be that number.
For Jacobs’ argument, we also need that these ratios obey rules like
But this is easy to check: whenever are vectors in a 1-dimensional vector space, division obeys the above rule. To put it in fancy language, nonzero vectors in any 1-dimensional real vector space form a ‘torsor’ for the multiplicative group of nonzero real numbers:
• John Baez, [Torsors made easy](http://math.ucr.edu/home/baez/torsors.html ).
Finally, Jules used this sort of fact:
Actually he *forgot* to write down this particular equation at the top of his short proof—but he wrote down three others of the same form, and they all work the same way. Why are they true?
I claim this equation is true at some point of the plane whenever are smooth functions on the plane and doesn’t vanish at that point. Let’s see why!
First of all, by the inverse function theorem, if at some point in the plane, the functions and serve as coordinates in some neighborhood of that point. In this case we can work out in terms of these coordinates, and we get
or more precisely
Thus
so
as desired!

2013-01-24 19:02:35 UTC

##
Civilizational Collapse (Part 2)
[](http://en.wikipedia.org/wiki/Basketmaker_III_Era )
[Last time](https://johncarlosbaez.wordpress.com/2013/01/20/anasazi-america-part-1/ ) I told you a story of the American Southwest, starting with the arrival of small bands of hunters around 10,000 BC. I focused on the Anasazi, or [‘ancient Pueblo people’](http://en.wikipedia.org/wiki/Ancient_Pueblo_People ), and I led up to the [Late Basketmaker III Era](http://en.wikipedia.org/wiki/Basketmaker_III_Era ), from 500 to 750 AD.
The big invention during this time was the bow and arrow. Before then, large animals were killed by darts thrown from slings, which required a lot more skill and luck. But even more important was the continuing growth of agriculture: the cultivation of corn, beans and squash. This was fueled a period of dramatic population growth.
But this was just the start!###
The Pueblo I and II ErasThe [Pueblo I Era](http://en.wikipedia.org/wiki/Pueblo_I_Era ) began around 750 AD. At this time people started living in ‘pueblos’: houses with flat roofs held up by wooden poles. Towns became bigger, holding up to 600 people. But these towns typically lasted only 30 years or so. It seems people needed to move when conditions changed.
Starting around 800 AD, the ancient Pueblo people started building ‘great houses’: multi-storied buildings with high ceilings, rooms much larger than those in domestic dwellings, and elaborate subterranean rooms called ‘kivas’. And around 900 AD, people started building houses with *stone* roofs. We call this the start of the [Pueblo II Era](http://en.wikipedia.org/wiki/Pueblo_II_Era ).
The center of these developments was the [Chaco Canyon](http://en.wikipedia.org/wiki/Chaco_Culture_National_Historical_Park ) area in New Mexico:
[](https://en.wikipedia.org/wiki/Chaco_Culture_National_Historical_Park )
Chaco Canyon is 125 kilometers east of Canyon de Chelly.
Unfortunately, I didn’t see it on my trip—I wanted to, but we didn’t have time.
By 950 AD, there were pueblos on every ridge and hilltop of the Chaco Canyon area. Due to the high population density and unpredictable rainfall, this area could no longer provide enough meat to sustain the needs of the local population. Apparently they couldn’t get enough fat, salt and minerals from a purely vegan diet—a shortcoming we have now overcome!
Yet the population continued to grow until 1000 AD. In his book *Anasazi America*, David Stuart wrote:
>
Millions of us buy mutual funds, believing the risk is spread among millions of investors and a large “basket” of fund stocks. Millions divert a portion of each hard-earned paycheck to purchase such funds for retirement. “Get in! Get in!” hawk the TV ads. “The market is going up. Historically, it always goes up in the long haul. The average rate of return this century is 9 percent per year!” Every one of us who does that is a Californian at heart, believing in growth, risk, > *power*> . It works—until an episode of too-rapid expansion in the market, combined with brutal business competition, threatens to undo it.
> That is about what it was like, economically, at Chaco Canyon in the year 1000—rapid agricultural expansion, no more land to be gotten, and deepening competition. Don’t think of it as “romantic” or “primitive”. Think of it as just like 1999 in the United States, when the Dow Jones Industrial Average hit 11,000 and 30 million investors held their breath to see what would happen next.
###
The Chaco phenomenon In 1020 the rainfall became more predictable. There wasn’t *more* rain, it was simply less erratic. This was good for the ancient Pueblo people. At this point the ‘Chaco phenomenon’ began: an amazing flowering of civilization.
We see this in places like [Pueblo Bonito](http://en.wikipedia.org/wiki/Pueblo_Bonito ), the largest great house in Chaco Canyon:
[](http://en.wikipedia.org/wiki/Pueblo_Bonito )
[](http://en.wikipedia.org/wiki/Pueblo_Bonito )
Pueblo Bonito was founded in the 800s. But starting in 1020 it grew immensely, and it kept growing until 1120. By this time it had 700 rooms, nearly half devoted to grain storage. It also had 33 kivas, which are the round structures you see here.
But Pueblo Bonito is just one of a dozen great houses built in Chaco Canyon by 1120. About 215 thousand ponderosa pine trees were cut down in this building spree! Stuart estimates that building these houses took over 2 million man-hours of work. They also built about 650 kilometers of roads! Most of these connect one great house to another… but some mysteriously seem to go to ‘nowhere’.
By 1080, however, the summer rainfall had started to decline. And by 1090 there were serious summer drought lasting for five years. We know this sort of thing from tree rings: there are enough ponderosa logs and the like that archaeologists have built up a detailed year-by-year record.
Thanks to overpopulation and these droughts, Chaco Canyon civilization was in serious trouble at this point, but it charged ahead:
>
Part of Chacoan society were already in deep trouble after AD 1050 as health and living conditions progressively eroded in the southern districts’ open farming communities. The small farmers in the south had first created reliable surpluses to be stored in the great houses. Ultimately, it was the increasingly terrible conditions of those farmers, the people who grew the corn, that had made Chacoan society so fatally vulnerable. They simply got back too little from their efforts to carry on.
> [….]
> Still, the great-house dwellers didn’t merely sit on their hands. As some farms failed, they used farm labor to expand roads, rituals, and great houses. This prehistoric version of a Keynesian growth model apparently alleviated enough of the stresses and strains to sustain growth through the 1070s. Then came the waning rainfall of the 1080s, followed by drought in the 1090s.
> Circumstances in farming communities worsened quickly and dramatically with this drought; the very survival of many was at stake. The great-house elites at Chaco Canyon apparently responded with even more roads, rituals, and great houses. This was actually a period of great-house and road infrastructure “in-fill”, both in and near established open communities. In a few years, the rains returned. This could not help but powerfully reinforce the elites’ now well-established, formulaic response to problems.
> But roads, rituals, and great houses simply did not do enough for the hungry farmers who produced corn and pottery. As the eleventh century drew to a close, even though the rains had come again, they walked away, further eroding the surpluses that had fueled the system. Imagine it: the elites must have believe the situation was saved, even as more farmers gave up in despair. Inexplicably, they never “exported” the modest irrigation system that had caught and diverted midsummer runoff from the mesa tops at Chaco Canyon and made local fields more productive. Instead, once again the elites responded with the sacred formula—more roads, more rituals, more great houses.
So, Stuart argues that the last of the Chaco Canyon building projects were “the desperate economic reactions of a fragile and frightened society”.
Regardless of whether this is true, we know that starting around 1100 AD, many of the ancient Pueblo people left the Chaco Canyon area. Many moved upland, to places with more rain and snow. Instead of great houses, many returned to building the simpler pit houses of old.
Tribes descending from the ancient Pueblo people still have myths about the decline of the Chaco civilization. While such tales should be taken with a huge grain of salt, these are too fascinating not to repeat. Here are two quotes:
>
In our history we talk of things that occurred a long time ago, of people who had enormous amounts of power, spiritual power and power over people. I think that those kinds of people lived here in Chaco…. Here at Chaco there were very powerful people who had a lot of spiritual power, and these people probably used their power in ways that caused things to change, and that may have been one of the reasons why the migrations were set to start again, because these these people were causing changes that were never meant to occur.
>
My response to the canyon was that some sensibility other than my Pueblo ancestors had worked on the Chaco great houses. There were the familiar elements such as the > *nansipu*> (the symbolic opening into the underworld), kivas, plazas and earth materials, but they were overlain by a strictness and precision of design that was unfamiliar…. It was clear that the purpose of these great villages was not to restate their oneness with the earth but to show the power and specialness of humans… a desire to control human and natural resources… These were men who embraced a social-political-religious hierarchy and envisioned control and power over places, resources and people.
These quotes are from an excellent book on the changing techniques and theories of archaeologists of the American Southwest:
• Stephen H. Lekson, *A History of the Ancient Southwest*, School for Advanced Research, Santa Fe, New Mexico, 2008.
What these quotes show, I think, is that the sensibility of current-day Pueblo people is very different from that of the people who built the great houses of Chaco Canyon. According to David Stuart, the Chaco civilization was a ‘powerful’ culture, while their descendants became an ‘efficient’ culture:
>
… a powerful society (or organism) captures more energy and expends (metabolizes) it more rapidly than an efficient one. Such societies tend to be structurally more complex, more wasteful of energy, more competitive, and faster paced than an efficient one. Think of modern urban America as powerful, and you will get the picture. In contrast, an efficient society “metabolizes” its energy more slowly, and so it is structurally less complex, less wasteful, less competitive, and slower. Think of Amish farmers in Pennsylvania or contemporary Pueblo farms in the American Southwest.
> In competitive terms, the powerful society has an enormous short-term advantage over the efficient one > *if*> enough energy is naturally available to “feed” it, or if its technology and trade can bring in energy rapidly enough to sustain it. But when energy (food, fuel and resources) becomes scarce, or when trade and technology fail, an efficient society is advantageous because it simpler, less wasteful structure is more easily sustained in times of scarcity.
###
The Pueblo III Era, and collapse
[](http://commons.wikimedia.org/wiki/File:Mesa_Verde_National_Park_Cliff_Palace_2006_09_12.jpg )
By 1150 AD, some of the ancient Pueblo people began building cliff dwellings at higher elevations—like [Mesa Verde](http://en.wikipedia.org/wiki/Mesa_Verde_National_Park ) in Colorado, shown above. This marks the start of the [Pueblo III Era](http://en.wikipedia.org/wiki/Pueblo_III_Era ). But this era lasted a short time. By 1280, Mesa Verde was deserted!
Some of the ruins in Canyon de Chelly also date to the Pueblo III Era. For example, the White House Ruins were built around 1200. Here are some of my pictures of this marvelous place. Click to enlarge:
[](http://math.ucr.edu/home/baez/diary/arizona_2012/white_house_ruin_5.jpg )
[](http://math.ucr.edu/home/baez/diary/arizona_2012/white_house_ruin_9.jpg )
[](http://math.ucr.edu/home/baez/diary/arizona_2012/white_house_ruin_14.jpg )
But again, they were deserted by the end of the Pueblo III Era.
Why did the ancient Pueblo people move to cliff dwellings? And why did they move out so soon?
Nobody is sure. Cliff dwellings are easy to defend against attack. Built into the south face of a cliff, they catch the sun in winter to stay warm—it gets cold here in winter!—but they stay cool when the sun is straight overhead in summer. These are good reasons to build cliff dwellings. But these reasons don’t explain why cliff dwellings were so popular from 1150 to 1280, and then were abandoned!
One important factor seems to be this: there was a series of severe droughts starting around 1275. There were also raids from other tribes: speakers of [Na-Dené languages](http://en.wikipedia.org/wiki/Na-Dene_languages ), who eventually became the current-day Navajo inhabitants of this area.
But drought alone may be unable to explain what happened. There have been some fascinating attempts to model the collapse of the Anasazi culture. One is called the [Artificial Anasazi Project](http://jasss.soc.surrey.ac.uk/12/4/13.html ). It used ‘agent-based modeling’ to study what the ancient Pueblo people did in Long House Valley, Arizona, from 200 to 1300. The [Villages Project](http://village.anth.wsu.edu/ ), a collaboration of Washington State University and the Crow Canyon Archaeological Center, focused on the region near Mesa Verde.
Quoting Stephen Lekson’s book:
>
Both projects mirrored actual settlement patterns from 800 to 1250 with admirable accuracy. Problems rose, however, with the abandonments of the regions, in both cases after 1250. There were unexplained exceptions, misfits between the models and reality.
> Those misfits were not minor. Neither model predicted complete abandonment. Yet it happened. That’s perplexing. In the > *Scientific American*> summary of the Long House Valley model, Kohler, Gummerman, and Reynolds write, “We can only conclude that sociopolitical, ideological or environmental factors > *not*> included in our model must have contributed to the total depopulation of the valley.” Similar conundrums best the Villages Project: “None of our simulations terminated with a population decline as dramatic as what actually happened in the Mesa Verde region in the late 1200.”
These simulation projects look interesting! Of course they leave out many factors, but that’s okay: it suggests that one of *those* factors could be important in understanding the collapse.
For more info, click on the links. Also try this short review by the author of a famous book on why civilizations collapse:
• Jared Diamond, [Life with the artificial Anasazi](http://faculty.bennington.edu/~kwoods/classes/agric_hist/readings_11/Diamond%20-%202002%20-%20Archaeology%20Life%20with%20the%20artificial%20Anasazi.pdf ), *Nature* **419** (2002), 567–569.
From this article, here are the simulated versus ‘actual’ populations of the ancient Pueblo people in Long House Valley, Arizona, from 800 to 1350 AD:
[](http://math.ucr.edu/home/baez/cultural/anasazi_long_house_valley_population.jpg )
The so-called ‘actual’ population is estimated using the number of house sites that were active at a given time, assuming five people per house.
This graph gives a shocking and dramatic ending to our tale! Let’s hope our current-day tale doesn’t end so abruptly, because in abrupt transitions much gets lost. But of course the ancient Pueblo people didn’t disappear. They didn’t all die. They became an ‘efficient’ society: they learned to make do with diminished resources.
For more about our current-day tale, read this:
• [Civilizational collapse (part 3)](https://johncarlosbaez.wordpress.com/2014/03/25/civilizational-collapse-part-3/ ).
https://johncarlosbaez.wordpress.com/wp-content/uploads/2013/01/white_house_ruin_5_small.webp?w=450

2013-01-20 23:38:23 UTC

##
Civilizational Collapse (Part 1)[](http://math.ucr.edu/home/baez/diary/arizona_2012/first_ruin_5.jpg )
A few weeks ago I visited Canyon de Chelly, which is home to some amazing cliff dwellings. I took a bunch of photos, like this picture of the so-called ‘First Ruin’. You can see them and read about my adventures starting here:
• John Baez, [Diary, 21 December 2012](http://math.ucr.edu/home/baez/diary/december_2012.html#december21_12 ).
Here I’d like to talk about what happened to the civilization that built these cliff dwellings! It’s a fascinating tale full of mystery… and it’s full of lessons for the problems we face today, involving climate change, agriculture, energy production, and advances in technology.
First let me set the stage! Canyon de Chelly is in the [Navajo Nation](http://en.wikipedia.org/wiki/Navajo_nation ), a huge region with its own laws and government, not exactly part of the United States, located at the corners of Arizona, New Mexico, and Utah:
[](http://en.wikipedia.org/wiki/Navajo_nation )
The hole in the middle is the [Hopi Reservation](http://en.wikipedia.org/wiki/Hopi_Reservation ). The [Hopi](http://en.wikipedia.org/wiki/Hopi_people ) are descended from,the people who built the cliff dwellings in Canyon de Chelly. Those people are often called the Anasazi, but these days the favored term is [ancient Pueblo peoples](http://en.wikipedia.org/wiki/Ancient_Pueblo_Peoples ).
The Hopi speak a [Uto-Aztecan language](http://en.wikipedia.org/wiki/Uto-Aztecan_languages ), and so presumably did the Anasazi. Uto-Aztecan speakers were spread out like this shortly before the Europeans invaded:
[](http://en.wikipedia.org/wiki/Uto-Aztecan_languages )
with a bunch more down in what’s now Mexico. The [Navajo](http://en.wikipedia.org/wiki/Navajo_people ) are part of a different group, the [Na-Dené language](http://en.wikipedia.org/wiki/Na-Dene_languages ) group:
[](http://en.wikipedia.org/wiki/Na-Den%C3%A9_languages )
So, the Navajo aren’t a big part of the story in this fascinating book:
• David E. Stuart, *Anasazi America*, U. of New Mexico Press, Albuquerque, New Mexico, 2000.
Let me summarize this story here!###
After the ice The last Ice Age, called the [Wisconsin glaciation](http://en.wikipedia.org/wiki/Wisconsin_glaciation ), began around 70,000 BC. The glaciers reached their [ maximum extent](http://en.wikipedia.org/wiki/Last_Glacial_Maximum ) about 18,000 BC, with ice sheets down to what are now the Great Lakes. In places the ice was over 1.6 kilometers thick!
Then it started warming up. By 16,000 BC people started cultivating plants and herding animals. Around 12,000 BC, before the land bridge connecting Siberia and Canada melted, people from the so-called [Clovis culture](http://en.wikipedia.org/wiki/Clovis_culture ) came to the Americas.
It [seems likely](http://en.wikipedia.org/wiki/Settlement_of_the_Americas#Problems_with_Clovis_migration_models ) that other people got to America earlier, moving down the Pacific coast before the inland glaciers melted. But even if the Clovis culture didn’t get there first, their arrival was a big deal. They be traced by their distinctive and elegant spear tips, called [Clovis points](http://en.wikipedia.org/wiki/Clovis_point ):
[](http://en.wikipedia.org/wiki/Clovis_point )
After they arrived, the Clovis people broke into several local cultures, roughly around the time of the [Younger Dryas](http://en.wikipedia.org/wiki/Younger_Dryas ) cold spell beginning around 10,800 BC. By 10,000 BC, small bands of hunters roamed the Southwest, first hunting mammoths, huge bison, camels, horses and elk, and later—perhaps because they killed off the really big animals—the more familiar bison, deer, elk and antelopes we see today.
For about 5000 years the population of current-day New Mexico probably fluctuated between 2 and 6 thousand people—a density of just one person per 50 to 150 square kilometers! Changes in culture and climate were slow.###
The Altithermal Around 5,000 BC, the climate near Canyon de Chelly began to warm up, dry out, and become more strongly seasonal. This epoch is called the ‘Altithermal’. The lush grasslands that once supported huge herds of bison began to disappear in New Mexico, and those bison moved north. By 4,000 BC, the area near Canyon de Chelly became very hot, with summers often reaching 45°C, and sometimes 57° at the ground’s surface.
The people in this area responded in an interesting way: by focusing much more on gathering, and less on hunting. We know this from their improved tools for processing plants, especially yucca roots. The yucca is now the state flower of New Mexico. Here’s a picture taken by Stan Shebs:
[](http://en.wikipedia.org/wiki/Yucca_schidigera )
David Stuart writes:
> At first this might seem an unlikely response to unremitting heat and aridity. One could argue that the deteriorating climate might first have forced people to reduce their numbers by restricting sex, marriage, and child-bearing so that survivors would have enough game. That might well have been the short-term solution [….] When once-plentiful game becomes scarce, hunter-gatherers typically become extremely conservative about sex and reproduction. […] But by early Archaic times, the change in focus to plant resources—undoubtedly by necessity—had actually produced a marginally growing population in the San Juan Basin and its margins in spite of climatic adversity.
> [….]
> Ecologically, these Archaic hunters and gatherers had moved one entire link > *down*> the food chain, thereby eliminating the approximately 90-percent loss in food value that occurs when one feeds on an animal that is a plant-eater.
> [….]
> This is sound ecological behavior—they could not have found a better basic strategy even if they had the advantage of a contemporary university education. Do I attribute this to their genius? No. It is simply that those who stubbornly clung to the traditional big game hunting of their Paleo-Indian ancestors could not prosper, so they left fewer descendents. Those more willing to experiment, or more desperate, fared better, so their behavior eventually became traditional among their more numerous descendents.###
The San Jose PeriodBy 3,000 BC the Altithermal was ending, big game was returning to the Southwest, yet the people retained their new-found agricultural skills. They also developed a new kind of dart for hunting, the ‘San Jose point’. So, this epoch is called the ‘San Jose period’. Populations rose to maybe about 15 to 30 thousand people in New Mexico, a vast increase over the earlier level of 2-6 thousand. But still, that’s just one person per 10 or 20 square kilometers!
The population increased until around 2,000 BC. At this point population pressures became acute… but two lucky things happened. First, the weather got wetter. Second, corn was introduced from Mexico. The first varieties had very small cobs, but gradually they were improved.
The wet weather lasted until around 500 BC. And at just about this time, *beans* were introduced, also from Mexico.
> Their addition was critical. Corn alone is a costly food to metabolize. Its proteins are incomplete and hard to synthesize. Beans contain large amounts of lysine, the amino acid missing from corn and squash. In reasonable balance, corn, beans and squash together provide complimentary amino acids and form the basis of a nearly complete diet. This diet lacks only the salt, fat and mineral nutrients found in most meats to be healthy and complete.
> By 500 BC, nearly all the elements for accelerating cultural and economic changes were finally in place—a fairly complete diet that could, if rainfall cooperated, largely replace the traditional foraging one; several additional, modestly larger-cobbed varieties of corn that not only prospered under varying growing conditions but also provided a bigger harvest; a population large enough to invest the labor necessary to plant and harvest; nearly 10 centuries of increasing familiarity with > [> cultigens](http://en.wikipedia.org/wiki/Cultigen )> ; and enhanced food-processing and storage techniques. Lacking were compelling reasons to transform an Archaic society accustomed to earning a living with approximately 500 hours of labor a year into one willing to invest the 1,000 to 2,000 yours coming to contemporary hand-tool horticulturalists.
> Nature then stepped in with one persuasive, though not compelling, reason for people to make the shift.
Namely, *droughts!* Precipitation became very erratic for about 500 years. People responded in various ways. Some went back to the old foraging techniques. Others improved their agricultural skills, developing better breeds of corn, and tricks for storing water. The latter are the ones whose populations grew.###
The Basketmakers This led to the [Basketmaker culture](http://en.wikipedia.org/wiki/Basketmaker_culture ), where people started living in dugout ‘pit houses’ in small villages. More precisely, the [Late Basketmaker II Era](http://en.wikipedia.org/wiki/Late_Basketmaker_II_Era ) lasted from about 50 AD to 500 AD. New technologies included the baskets that gave this culture its name:
[](http://en.wikipedia.org/wiki/Basketmaker_culture#Basketmaker_eras )
Pottery entered the scene around 300 AD. Have you ever thought about how important this is? Before pots, people had to cook corn and beans by putting rocks in fires and then transferring them to holes containing water!
> Now, porridge and stews could be put to boil in a pot set directly into a central fire pit. The amount of heat lost and fuel used in the old cooking process—an endless cycle of collecting, heating, transferring, removing and replacing hot stones just to boil a few quarts of water—had always been enormous. By comparison, cooking with pots became quick, easy, and far more efficient. In a world more densely populated, firewood had to be gathered from greater distances. Now, less of it was needed. And there was newer fuel to supplement it—dried corncobs.
Not all the changes were good. Most adult skeletons from this period show damage from long periods spend stooping—either using a stone hoe to tend garden plots, or grinding corn while kneeling. And as they ate more corn and beans and fewer other vegetables, mineral deficiencies became common. Extreme osteoporosis afflicted many of these people: we find skulls that are porous, and broken bones. It reminds me a little of the plague of obesity, with its many side-affects, afflicting modern Americans as we move to a culture where most people work sitting down.
On the other hand, there was a massive growth in population. The number of pit-house villages grew nine-fold from 200 AD to 700 AD!
> It must have been an exciting time. In only some 25 generations, these folks had transformed themselves from forager and hunters with a small economic sideline in corn, beans and squash into semisedentary villagers who farmed and kept up their foraging to fill in the economic gaps.
But this was just the beginning. By 1020, the ancient Pueblo people would begin to build housing complexes that would remain the biggest in North America until the 1880s! This happened in Chaco Canyon, 125 kilometers east of Canyon de Chelly.
Next time I’ll tell you the story of how that happened, and how later, around 1200, these people left Chaco Canyon and started to build cliff dwellings.
For now, I’ll leave you with some pictures I took of the most famous cliff dwelling in Canyon de Chelly: the ‘White House Ruins’. Click to enlarge:
[](http://math.ucr.edu/home/baez/diary/arizona_2012/white_house_ruin_6.jpg )
[](http://math.ucr.edu/home/baez/diary/arizona_2012/white_house_ruin_7.jpg )

2012-01-19 05:54:33 UTC

##
Classical Mechanics versus Thermodynamics (Part 1)It came as a bit of a shock last week when I realized that some of the equations I’d learned in thermodynamics were *just the same as equations I’d learned in classical mechanics*—with only the names of the variables changed, to protect the innocent.
Why didn’t anyone tell me?
For example: everybody loves Hamilton’s equations: there are just two, and they summarize the entire essence of classical mechanics. Most people hate the Maxwell relations in thermodynamics: there are lots, and they’re hard to remember.
But what I’d like to show you now is that Hamilton’s equations *are* Maxwell relations! They’re a special case, and you can derive them the same way. I hope this will make you like the Maxwell relations more, instead of liking Hamilton’s equations less.
First, let’s see what these equations look like. Then let’s see why Hamilton’s equations are a special case of the Maxwell relations. And then let’s talk about how this might help us unify different aspects of physics.####
Hamilton’s equations Suppose you have a particle on the line whose position and momentum are functions of time, If the energy is a function of position and momentum, [**Hamilton’s equations**](http://en.wikipedia.org/wiki/Hamiltonian_mechanics#Simplified_overview_of_uses ) say:####
Maxwell’s relations There are lots of Maxwell relations, and that’s one reason people hate them. But let’s just talk about two; most of the others work the same way.
Suppose you have a physical system like a box of gas that has some volume pressure temperature and entropy Then the first and second [**Maxwell relations**](http://en.wikipedia.org/wiki/Maxwell_relations ) say:####
Comparison Clearly Hamilton’s equations resemble the Maxwell relations. Please check for yourself that the patterns of variables are exactly the same: only the names have been changed! So, apart from a key subtlety, Hamilton’s equations *become* the first and second Maxwell relations if we make these replacements:
What’s the key subtlety? One reason people hate the Maxwell’s relations is they have lots of little symbols like saying what to hold constant when we take our partial derivatives. Hamilton’s equations don’t have those.
So, you probably won’t like this, but let’s see what we get if we write Hamilton’s equations so they *exactly* match the pattern of the Maxwell relations:
This looks a bit weird, and it set me back a day. What does it mean to take the partial derivative of in the direction while holding constant, for example?
I still think it’s weird. But I think it’s correct. To see this, let’s derive the Maxwell relations, and then derive Hamilton’s equations using *the exact same reasoning*, with only the names of variables changed.####
Deriving the Maxwell relations The Maxwell relations are extremely general, so let’s derive them in a way that makes that painfully clear. Suppose we have any smooth function on the plane. Just for laughs, let’s call the coordinates of this plane and . Then we have
for some functions and This equation is just a concise way of saying that
and
The minus sign here is unimportant: you can think of it as a whimsical joke. All the math would work just as well if we left it out.
(In reality, physicists call as the [**internal energy**](http://en.wikipedia.org/wiki/Internal_energy ) of a system, regarded as a function of its [**entropy**](http://en.wikipedia.org/wiki/Entropy ) and [**volume**](http://en.wikipedia.org/wiki/Volume_%28thermodynamics%29 ) They then call the [**temperature**](http://en.wikipedia.org/wiki/Temperature ) and the [**pressure**](http://en.wikipedia.org/wiki/Pressure ). It just so happens that for lots of systems, their internal energy goes down as you increase their volume, so works out to be positive if we stick in this minus sign, so that’s what people did. But you don’t need to know any of this physics to follow the derivation of the Maxwell relations!)
Now, mixed partial derivatives commute, so we have:
Plugging in our definitions of and this says
And that’s the first Maxwell relation! So, there’s nothing to it: it’s just a sneaky way of saying that the mixed partial derivatives of the function commute.
The second Maxwell relation works the same way. But seeing this takes a bit of thought, since we need to cook up a suitable function whose mixed partial derivatives are the two sides of this equation:
There are different ways to do this, but for now let me use the time-honored method of ‘pulling the rabbit from the hat’.
Here’s the function we want:
(In thermodynamics this function is called the [**Helmholtz free energy**](http://en.wikipedia.org/wiki/Helmholtz_free_energy ). It’s sometimes denoted but the International Union of Pure and Applied Chemistry recommends calling it which stands for the German word ‘Arbeit’, meaning ‘work’.)
Let’s check that this function does the trick:
If we restrict ourselves to any subset of the plane where and serve as coordinates, the above equation is just a concise way of saying
and
Then since mixed partial derivatives commute, we get:
or in other words:
which is the second Maxwell relation.
We can keep playing this game using various pairs of the four functions as coordinates, and get more Maxwell relations: enough to give ourselves a headache! But we have more better things to do today.####
Hamilton’s equations as Maxwell relations For example: let’s see how Hamilton’s equations fit into this game. Suppose we have a particle on the line. Consider smooth paths where it starts at some fixed position at some fixed time and ends at the point at the time Nature will choose a path with least action—or at least one that’s a stationary point of the action. Let’s assume there’s a unique such path, and that it depends smoothly on and . For this to be true, we may need to restrict and to a subset of the plane, but that’s okay: go ahead and pick such a subset.
Given and in this set, nature will pick the path that’s a stationary point of action; the action of this path is called [**Hamilton’s principal function**](http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation ) and denoted (Beware: this is not the same as entropy!)
Let’s assume is smooth. Then we can copy our derivation of the Maxwell equations line for line and get Hamilton’s equations! Let’s do it, skipping some steps but writing down the key results.
For starters we have
for some functions and called the **momentum** and **energy**, which obey
and
As far as I can tell it’s just a cute coincidence that we see a minus sign in the same place as before! Anyway, the fact that mixed partials commute gives us
which is the first of Hamilton’s equations. And now we see that all the funny and things are actually correct!
Next, we pull a rabbit out of our hat. We define this function:
and check that
This function probably has a standard name, but I don’t know it. Do you?
Then, considering any subset of the plane where and serve as coordinates, we see that because mixed partials commute:
we get
So, we’re done!
But you might be wondering how we pulled this rabbit out of the hat. More precisely, why did we suspect it was there in the first place? There’s a nice answer if you’re comfortable with [differential forms](http://en.wikipedia.org/wiki/Differential_form ). We start with what we know:
Next, we use this fundamental equation:
to note that:
See? We’ve managed to switch the roles of and at the cost of an extra minus sign!
Then, if we restrict attention to any contractible open subset of the plane, the [**Poincaré Lemma**](http://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar.C3.A9_lemma ) says
Since
it follows that there’s a function with
This is our rabbit. And if you ponder the difference between and , you’ll see it’s So, it’s no surprise that####
The big picture
Now let’s step back and think about what’s going on.
Lately I’ve been trying to unify a bunch of ‘extremal principles’, including:
1) the [principle of least action](http://en.wikipedia.org/wiki/Principle_of_least_action )
2) the [principle of least energy](http://en.wikipedia.org/wiki/Principle_of_minimum_energy )
3) the [principle of maximum entropy](http://en.wikipedia.org/wiki/Principle_of_maximum_entropy )
4) the principle of maximum simplicity, or [Occam’s razor](http://en.wikipedia.org/wiki/Occam%27s_razor )
In my post on [quantropy](https://johncarlosbaez.wordpress.com/2011/12/22/quantropy/ ) I explained how the first three principles fit into a single framework if we treat Planck’s constant as an imaginary temperature. The guiding principle of this framework is
***maximize entropy
subject to the constraints imposed by what you believe***
And that’s nice, because E. T. Jaynes has made a [powerful case](http://bayes.wustl.edu/etj/articles/stand.on.entropy.pdf ) for this principle.
However, when the temperature is imaginary, entropy is so different that it may deserves a new name: say, ‘quantropy’. In particular, it’s complex-valued, so instead of maximizing it we have to look for stationary points: places where its first derivative is zero. But this isn’t so bad. Indeed, a lot of minimum and maximum principles are really ‘stationary principles’ if you examine them carefully.
What about the fourth principle: Occam’s razor? We can formalize this using [algorithmic probability theory](http://www.scholarpedia.org/article/Algorithmic_probability ). Occam’s razor then becomes yet another special case of
***maximize entropy
subject the constraints imposed by what you believe***
once we realize that [algorithmic entropy is a special case of ordinary entropy](https://johncarlosbaez.wordpress.com/2011/01/06/algorithmic-thermodynamics-part-2/ ).
All of this deserves plenty of further thought and discussion—but not today!
Today I just want to point out that once we’ve formally unified classical mechanics and thermal statics (often misleadingly called ‘thermodynamics’), as sketched in the article on [quantropy](https://johncarlosbaez.wordpress.com/2011/12/22/quantropy/ ), we should be able to take any idea from one subject and transpose it to the other. And it’s true. I just showed you an example, but there are lots of others!
I guessed this should be possible after pondering three famous facts:
• In classical mechanics, if we fix the initial position of a particle, we can pick any position and time at which the particle’s path ends, and nature will seek the path to this endpoint that minimizes the *action*. This minimal action is Hamilton’s principal function which obeys
In thermodynamics, if we fix the entropy and volume of a box of gas, nature will seek the probability distribution of microstates the minimizes the *energy*. This minimal energy is the internal energy , which obeys
• In classical mechanics we have [canonically conjugate quantities](http://en.wikipedia.org/wiki/Conjugate_variables ), while in statistical mechanics we have [conjugate variables](http://en.wikipedia.org/wiki/Conjugate_variables_%28thermodynamics%29 ). In classical mechanics the canonical conjugate of the position is the momentum , while the canonical conjugate of time is energy In thermodynamics, the conjugate of entropy is temperature while the conjugate of volume is pressure All this is fits in perfectly with the analogy we’ve been using today:
• Something called the [Legendre transformation](http://en.wikipedia.org/wiki/Legendre_transformation ) plays a big role both in classical mechanics and thermodynamics. This transformation takes a function of some variable and turns it into a function of the conjugate variable. In our proof of the Maxwell relations, we secretly used a Legendre transformation to pass from the internal energy to the Helmholtz free energy
where we must solve for the entropy in terms of and to think of as a function of these two variables.
Similarly, in our proof of Hamilton’s equations, we passed from Hamilton’s principal function to the function
where we must solve for the position in terms of and to think of as a function of these two variables.
I hope you see that all this stuff fits together in a nice picture, and I hope to say a bit more about it soon. The most exciting thing for me will be to see how [symplectic geometry](http://en.wikipedia.org/wiki/Symplectic_manifold ), so important in classical mechanics, can be carried over to thermodynamics. Why? Because I’ve never seen anyone use symplectic geometry in thermodynamics. But maybe I just haven’t looked hard enough!
Indeed, it’s perfectly possible that some people already know what I’ve been saying today. Have you seen someone point out that Hamilton’s equations are a special case of the Maxwell relations? This would seem to be the first step towards importing all of symplectic geometry to thermodynamics.
• [Part 1](https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ ): Hamilton’s equations versus the Maxwell relations.
• [Part 2](https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ ): the role of symplectic geometry.
• [Part 3](https://johncarlosbaez.wordpress.com/2021/09/23/classical-mechanics-versus-thermodynamics-part-3/ ): a detailed analogy between classical mechanics and thermodynamics.
• [Part 4](https://johncarlosbaez.wordpress.com/2021/09/26/classical-mechanics-versus-thermodynamics-part-4/ ): what is the analogue of quantization for thermodynamics?
https://johncarlosbaez.wordpress.com/wp-content/uploads/2021/09/classical_vs_thermo-e1723982721645.jpg?w=450