Oscar Cunningham

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OscarCunningham@mathstodon-xyz.mostr.pub

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nprofile1qqsdj0xlrr5g27jav3lmmezme40c93zspkmun2mm7t5qzmqv6p8atdqpz4mhxue69uhhyetvv9ujuerfw36x7tnsw43qz9thwden5te0wfjkccte9ekk7um5wgh8qatz25l32m

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npub1my7d7x8gs4a96erlhhj9hn2lstz9qrdhex4hhuhgq9kqe5z06k6qldtj8l

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Last Notes

2026-07-13 18:13:57 UTC

Do any image formats allow you to have negatives in the alpha channel? You could use this to make writing that's black on a white background and white on a black background.

2026-07-05 22:00:57 UTC

Kinda interesting that the betting odds have the England-Mexico game as 50/50, but then England is about 3 times more likely to go on to win the tournament. Why would England be better than Mexico at facing other teams, and yet not better against Mexico themselves?

2026-05-21 12:37:44 UTC

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@nprofile…pfuj The most common "pter" word is "helico-pter", but people don't see it because they tend to break it down as "heli-copter".

2026-05-20 16:48:43 UTC

how are loading and hiding not antonyms?

2026-05-08 17:08:42 UTC

The Planck momentum is so sane compared to the other units. A fast bowl in top level cricket has about 1 momentum. #Physics

2026-05-04 13:08:02 UTC

https://media.mathstodon.xyz/media_attachments/files/116/516/454/589/754/626/original/7c99bd9684500dbf.jpeg

2026-05-01 17:43:11 UTC

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@nprofile…pm95 I think you can do it as often as you want. The times I've used it is when I've said something cool in a reply to someone. Boosting that reply shows it to all my followers. Maybe that's less necessary now because you could quote-post.

2026-04-22 19:33:26 UTC

> show about post-scarcity future
> watch episode
> scarcity

2026-04-05 09:20:43 UTC

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@nprofile…pm95 @nprofile…g0zc Yes, I've also been sceptical of this advice. I think the correct strategy is to increase risk if you're doing well and decrease it if you're doing badly, regardless of how old you are.

2026-02-17 03:44:18 UTC

Quiz! What links Bill Gates with David 'X' Cohen?

2026-02-12 22:35:42 UTC

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@nprofile…vhdw @nprofile…f38c When I wanted to make this https://mathoverflow.net/questions/354327/atlas-like-websites-on-specific-areas-of-mathematics/354396#354396 comment I discovered the trick of adding several copies of the 'zero width space' character.

2026-02-11 13:17:53 UTC

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@nprofile…dn5h @nprofile…qjyt 5D Billiards With Multiverse Time Travel

2026-02-11 10:43:15 UTC

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@nprofile…dn5h So what happens when a ball goes straight at the portal?

2026-02-10 17:45:02 UTC

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@nprofile…n4rp There's some suggestion that A000000 should be used for the empty sequence: https://oeis.org/wiki/The_empty_sequence.

2026-02-07 11:38:21 UTC

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@nprofile…3n8d (https://oeis.org/A121273)

2026-02-06 16:52:39 UTC

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@nprofile…sqfw I think there's an interesting story you can tell with Barr's Theorem that sheds some light on constructive mathematics.
In particular I'm thinking of oddities regarding cardinality when you don't have the Axiom of Choice. For example you can have sets A and B such that there exist surjections A → B and B → A, and yet there's no bijection between A and B. This disagrees with our intuition that two sets should either be the same size or one should be larger than the other.
Barr's Theorem says that we can find some topos obeying Choice such that our original sets are coalgebras of some lex comonad on this topos. Applying the left adjoint to the surjections between A and B yields surjections between f*(A) and f*(B). Then because we have Choice we get that f*(A) and f*(B) have the same cardinality.
So the story we could tell is that the reals sets are the ones that obey Choice. What constructivists have been studying aren't actually sets but are instead sets equipped with the addional structure of a coalgebra for some comonad. The objects A and B have coalgebra structures that aren't isomorphic, but their underlying sets are indeed the same size.
I like this story because it replaces the deranged statement 'some sets have incomparable sizes' with the sane statement 'sets can have the same size while being equipped with different structures'.

2026-02-05 06:45:30 UTC

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@nprofile…t6k2 Maybe it's not that driving is particularly bad, but just that people have different politeness levels for each activity. So the bad driver might be polite in the bakery, but likewise the guy listening to music without headphones in the bakery might not be a bad driver.

2026-02-01 03:00:54 UTC

The adoption of early Christianity really began to snowball after it was adopted by Paul. This is known as the Katamari Damascene Conversion.

2026-01-30 16:06:40 UTC

We're getting close to someone using the phrase 'Leibniz monad' and not even knowing why it's funny.

2026-01-28 09:59:48 UTC

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@nprofile…kwrt Rearrange to get x = y + 1 + 1/(y - 1). So y - 1 must be a factor of 1, i.e. 1 or -1. But y can't be 0 since it originally appeared in a denominator. So y = 2 and x = 4 is the only solution.

2026-01-26 12:45:41 UTC

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@nprofile…q0jd We don't know the secret prefix to make '-eans' refer to a pair of trousers.

2026-01-18 17:31:11 UTC

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@nprofile…f3gx I think fiddles normally have four strings. Hope that helps!

2026-01-18 12:36:56 UTC

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@nprofile…vhyf I can prove you need at least five checks. This is because you have six variable, and you already know obey one equation (that the probabilities sum to one). Each check gives you the additional equation that three of the variables sum to the same as the other three. So with fewer than five checks we would have an underdetermined linear system. And since we already have one solution (all probabilities equal) we would therefore have a family of solutions with dimension at least one. Any solution near the fair solution would also be a valid set of probabilities.
So we need at least five checks.
Furthermore, it's clear that some number of checks will eventually work. Any alternative set of probabilities will be ruled out by the check that compares the three largest probabilities to the three smallest. From this I think we can conclude that it can be done in exactly five. Just keep picking a check that's linearly independent from the information you already have.

2026-01-18 12:25:47 UTC

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@nprofile…vhyf This feels like the same genre as the questions where you have to find a counterfeit coin with a balance scale.

2026-01-16 18:06:32 UTC

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@nprofile…lhdz The unit m⁻¹ is also called the 'dioptre', so you can measure speed in hertz per dioptre.

2026-01-16 10:08:44 UTC

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@nprofile…tfwq Could you retrace the symbols and extend them to make an art piece?

2026-01-13 23:16:47 UTC

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@nprofile…vtpe @nprofile…f38c @nprofile…dxrd What if a polyhedron has some vertices of degree 3 and some of degree > 3, and likewise for faces? I don't see why we could find a perturbation that works in that case.

2026-01-13 19:54:12 UTC

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@nprofile…f38c @nprofile…dxrd If you perturb the vertices of a quadrilateral, how do you guarantee that they remain coplanar? In fact I'm not surprised that this doesn't hold in 4 dimensions so much as I'm surprised that it does hold in 3 dimensions.

2026-01-10 22:58:36 UTC

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@nprofile…r4wp This reminds me of some posts by @nprofile…c3q9 like this one https://mstdn.social/@Phracker2Art/113998758622520120.

2025-12-28 07:17:16 UTC

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@nprofile…w448 It's suspicious that the Fibonacci numbers have 56%, while the one-away numbers only have 19%. You can't do this with a decreasing distribution because the number before each Fibonacci number would always have more probability than it. And you can't do it with a broad distribution, because the two numbers either side of each Fibonacci number would always add to one in the middle. I think the only way to do it with a natural looking distribution is to have a narrow distribution focused on a single Fibonacci number. Like if 8 has 50%, 7 and 9 each have 5%, etc.

2025-12-26 07:06:07 UTC

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@nprofile…vpth Got this one exactly

2025-12-24 15:57:28 UTC

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@nprofile…5ug2 @nprofile…yw5f Maybe it would have been clearer with symbols. Suppose we have a halting oracle A(-,-), so that A(B,d) halts with success if B(d) halts, and halts with failure if B(d) loops. Then we modify A(-,-) to make a new program A'(-) such that A'(B) halts if A(B,B) halts with failure, and loops if A(B,B) halts with success. The A'(A') halts if and only if A(A',A') halts with failure, which happens if and only if A'(A') loops.

2025-12-24 15:32:48 UTC

@nprofile…yw5f Hi Cody. I was listening to your Church of Logic podcast, and the episode 'Computability III: ourobouros' https://youtube.com/watch?v=_pgHGRJtWlw reminded me of a question I've had for a while about Turing's proof. Why is the Universal Turing Machine necessary? In particular I'm thinking of the following proof that seems not to need a UTM:
Assume for contradiction that there's a program that takes as input another program and some data, and tells you if that program halts when given that data. Alter that program in the following two ways:
1. Prepend a program that simply repeats its input twice.
2. Create a new state that loops infinitely regardless of the tape, and replace each 'halt successfully' instruction with an instruction to jump to this state.
Now we have a program that takes a program as input and halts if and only if that program doesn't halt when given itself as input. Now give this program itself as input, it halts if and only if it doesn't. Contradiction. □

2025-12-23 18:07:25 UTC

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@nprofile…vqu7 These days a hacker is a much more likely threat than a burglar. Sticky notes are pretty secure, especially if you're working remotely.

2025-12-23 12:38:46 UTC

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@nprofile…rrqe I think there's a 'Nixon to China' aspect to Lurie's writing. Because ∞-categories are the most category-theoretic thing possible, it helps if the explanation of them isn't too category-theoretic.

2025-12-23 12:24:45 UTC

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@nprofile…ngug You can also do that. You get a different answer, called the 'mean absolute deviation' (or MAD). But the standard deviation is usually easier to work with. Thus is because, from the point of view of doing algebra, squaring is a simpler operation than taking the absolute value.

2025-12-22 07:00:59 UTC

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@nprofile…sekg I think it's just interpreting the % as a percentage operator. The number -7 % is equal to -0.07, and then it subtracts 3 to get -3.07. I'd be interested to see what it thinks (-7) % (-3) is.

2025-12-17 18:38:14 UTC

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@nprofile…hkxc This reminds me of this bug https://bugzilla.mozilla.org/show_bug.cgi?id=1758975 where a cursed* GIF made part of the Firefox window transparent.
*Missing its global colour table.

2025-12-16 16:07:37 UTC

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@nprofile…vkca I think I have a solution that works for all the rows and columns. Did you notice this part of the description? 'The sums should be read left to right and top to bottom ignoring the usual order of operations.'

2025-12-15 16:14:11 UTC

It happened again. My tumble dryer was broken. Wouldn't even turn on. I check the fuse because that's the most likely explanation (UK electrical plugs have built-in fuses). The fuse seems fine. Reassemble the plug. Tumble dryer works fine.

2025-12-13 11:36:59 UTC

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@nprofile…22u8 Aren't they just transferring the stereotypes about those accents/places to the fantasy characters. Like in the LotR movies the elves have a 'standard' English accent to suggest high class and orthodoxy, the orcs are cockneys to suggest low class and criminality, Gimli is Scottish to suggest bravery and self-sufficiency, etc.
I think the accents from cities outside of London are mostly associated with the industrial history of those cities, which would explain why they're absent from fantasy. And the sterotype of the Welsh isn't quite as strong as for Ireland and Scotland, mostly it's just sheep jokes.

2025-12-11 16:53:24 UTC

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@nprofile…26qu Up to rotations I think the number will eventually be constant. Any n hypercubes lie in some n-1 dimensional space.

2025-12-07 10:02:34 UTC

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@nprofile…8eua Sometimes I like to wonder about how much of the standard model we'd have been able to figure out if humanity were only allowed to do chemistry experiments. This is an analogy to trying to go beyond the standard model without doing further high-energy experiments.

2025-12-06 15:07:40 UTC

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@nprofile…ral2 keep it as long as possible

2025-12-05 08:46:01 UTC

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@nprofile…hkxc
Catching a bug 🖥️😀
vs
Catching a bug 🦠☹️

2025-12-04 19:05:58 UTC

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@nprofile…d9sd The spherical AAS and SSA cases could also be ∞. I'm imagining one point at the North Pole, one point fixed on the equator and one point sliding along the equator.

2025-12-04 18:13:29 UTC

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@nprofile…ral2 Most people don't care about ads as much as we do. They wouldn't pay slightly more for a fridge without ads. That's why it's already impossible to buy a TV without ads on the homescreen. It's not worth it for the company to make a more expensive TV without ads for our market segment. Soon it will be the same for fridges.

2025-12-03 17:11:38 UTC

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@nprofile…rpwp @nprofile…rlzs I'm assuming that 1-ε was the largest representable number below 1, so 1-ε/2 isn't representable.

2025-12-03 13:16:00 UTC

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@nprofile…rlzs Suppose the largest representable number below 1 is 1 - ε. Then the Taylor expansion of √(1 - ε) is 1 - ε/2 - ε²/8 + …, which is closer to 1 - ε than it is to 1. So it's correctly rounding the correct result.

2025-12-02 10:29:51 UTC

Can someone give me a (spoiler tagged) hint for this one?

2025-12-01 07:12:47 UTC

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@nprofile…u2p5 I think the solution is a form of insurance where the farmers pay the bank half their profits during the good years, and the bank pays them slightly less than that amount during the bad years. That way the farmers can keep operating with a constant income, and the bank makes a profit on average.
Note that this solution will be adopted only if it actually is economically efficient to keep the farms open. It might be better for everyone to just eat some other vegetable and 'free up' the farmers for other work. Or for the farmers to migrate back and forth every few decades.

2025-11-30 18:50:46 UTC

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@nprofile…es2w I think 91 is old enough that they're noticeably more likely to be on the younger end of the 12 months.

2025-11-27 10:03:49 UTC

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@nprofile…pklh the thing is that they pack them in cubic boxes, so you're mostly paying for the cardboard

2025-11-20 06:40:14 UTC

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@nprofile…5l32 I think the vpte happened after he was convicted but before he was sentenced.

2025-11-19 20:18:57 UTC

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@nprofile…l4xy There's at least one octagon on the list: https://commons.wikimedia.org/wiki/Category:Greenwich_Park_Bandstand

2025-11-19 15:00:33 UTC

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@nprofile…5l32 This came up when Trump voted in the 2024 election. https://apnews.com/article/trump-felony-conviction-can-he-vote-b95e7b4c9158d999e8bc89b00fbda911 He was convicted in New York, so he could vote in Florida even though Florida wouldn't normally allow someone with a felony conviction to vote.

2025-11-18 13:48:24 UTC

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@nprofile…j6np It would be y,x↦f(x,y). The Y-combinator is f↦(x↦f(xx))(x↦f(xx)). You just delete the lambdas and replace the dot by the arrow.

2025-11-18 12:20:19 UTC

The notation 'x↦f(x)' is so much better than 'λx.f(x)'. I'm wondering if this is one of the cases where there world could switch over without too many transition costs. Just start teaching with '↦'. Anyone who was taught the old notation would immediately understand the new notation. Anyone who was taught the new notation would only need a one-sentence explanation to understand the old notation. We could still call it 'lambda calculus'.

2025-11-17 06:25:55 UTC

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@nprofile…v64a Lungfish are more closely related to giraffes than they are to trout: https://en.wikipedia.org/wiki/Fish#Phylogeny.
Elm trees are more closely related to strawberries than they are to oak: https://www.lesswrong.com/posts/fRwdkop6tyhi3d22L/there-s-no-such-thing-as-a-tree-phylogenetically.

2025-11-16 17:03:50 UTC

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@nprofile…ral2 Once I was trying to pick up a takeaway in Edinburgh. I was at the right place on the map, but I eventually realised my food was about 20m above me.

2025-11-15 15:29:25 UTC

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@nprofile…xk2w This is why picking a basis is ugly; it means working directly with numbers.

2025-11-14 11:31:02 UTC

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@nprofile…9kud You can even use the plural with one if you say something like '1.0 seconds'.

2025-11-13 10:36:22 UTC

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@nprofile…8eua For the usual sphere S² is the 'reflection about p' actually a rotation of π about the axis through p? If it was an actual reflection then it would fix some vectors in the tangent space rather than negating them.

2025-11-12 12:34:07 UTC

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@nprofile…m7tv @nprofile…wuts @nprofile…8eua A way to see close to the true olo is to stare at the complementary colour and then switch to staring at (a desaturated version of) olo. Here are the colours for your convenience: (presumably the pink colour is called 'lol')
https://media.mathstodon.xyz/media_attachments/files/115/536/742/367/065/633/original/801ee665d2b19949.jpeg

2025-11-12 06:16:46 UTC

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@nprofile…5e2g There are some good notes about random permutations by @nprofile…8eua here: https://math.ucr.edu/home/baez/permutations/. Looks like he answers your first question, and probably provides some information that will be useful for your second question.

2025-11-11 13:44:46 UTC

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@nprofile…4c0l I've mostly heard it pronounced the same as 'soup'.

2025-11-10 15:30:48 UTC

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@nprofile…qlkf you divide 146000/55068 to get 2.65, which as a percentage is 265%.

2025-11-09 07:14:06 UTC

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@nprofile…pwph @nprofile…gvpc Sometimes I notice my own posts often disappear from my timeline after a short amount of time. When I first post, the post appears in my own timeline. But if I go back to check later, it's not there. I think this is what you're observing and the boost is just an added layer of complication.

2025-11-08 09:41:45 UTC

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@nprofile…v64a I have thought about this. At the moment the code is capable of finding max-wall solutions, but it's much slower. The min-wall solver takes seconds whereas for max-walls it can take hours. Because I run the bot in the cloud, it would be a bit expensive for it to run that long every day. I have some ideas for how to optimise this, but I don't have time to implement them at the moment.
Because I do #Celtix daily I do always mentally check whether there might be a 104 wall solution. If I can't convince myself that it's impossible then I often run the solver to confirm either way. As far as I know there's only ever been one day when 104 walls was possible (day 49) and I did put that solution on the bot account.
If I do optimise the max-wall solver I will run it on all the historical puzzles to double-check that I didn't miss any 104 wall solutions.

2025-11-06 21:26:00 UTC

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@nprofile…8pq6 Apparently the column-sums are also integers, but I don't know a proof.

2025-11-06 15:35:11 UTC

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@nprofile…ral2 @nprofile…lp7j I don't see why this is always silly. If I have a proof that starts with not-A and proves A, is there always a way to unwind it to get a direct proof of A?

2025-11-02 11:34:56 UTC

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@nprofile…nq39 when it gets bought by venture capital and sold off in parts

2025-11-02 11:08:37 UTC

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@nprofile…es2w You can set the visibility to 'Private mention' and then not mention anyone.

2025-10-30 09:42:39 UTC

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@nprofile…es2w There is a bflat programming language: https://flattened.net/

2025-10-30 09:32:13 UTC

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@nprofile…p3nn Because of bra-ket notation I thought bracket was spelt that way. I only noticed the c a few months ago.

2025-10-13 15:53:07 UTC

#Mathober #Mathober2025
The prompt for day 5 was 'Digraph Sink'. In mathematics 'digraph' is an abreviation for 'directed graph' which is a kind of network made by joining nodes with arrows. A 'digraph sink' is any node which has no arrows that point out of it. If you walk around a digraph by following the arrows then the sinks are the places where you can get stuck.
In linguistics a 'digraph' is instead a pair of letters that make a different sound when written together than you would expect from their individual sounds. Let's draw a digraph digraph where the nodes are the letters of the alphabet, and the arrows represent which pairs of letters form digraphs.
The sinks in this digraph are B, D, F, H, J, K, L, M, V, X, Y and Z.
#Math #Maths #Mathematics #GraphTheory #Linguistics #Orthography #EnglishOrthography
https://media.mathstodon.xyz/media_attachments/files/115/367/641/466/024/877/original/99a02769418944fa.png

2025-10-13 14:53:23 UTC

#Mathober #Mathober2025
The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.
The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.
The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.
But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.
The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!
So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!
#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

2025-10-11 14:59:12 UTC

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@nprofile…m2ru @nprofile…yu7x Yeah, if I remember correctly there are two different theorems about the error term? It took me a while but I think I remember both of them. He had all night so I think I would have been fine.

2025-10-11 13:47:29 UTC

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@nprofile…yu7x Something like this happened in real life to Igor Tamm: https://damtson.wordpress.com/2017/08/12/igor-tamm-and-the-taylor-expansion/
> […] The two were soon detained by a Red Army detachment. Neither of them had any documents with them. To the people who arrested them, it appeared obvious that they were White scouts, deserved to be shot.
> Luckily for Tamm, the commander of the detachment was a drop-out student. He grinned grimly while listening to Tamm’s explanation that he had graduated from the Physics and Mathematics Department of the Moscow University.
> “So, you’re a mathematician? You’re lying, aren’t you? No problem, we will give you a test now. Here! Derive for me the formula for the expansion of a function in Taylor series. Including the remainder term! If you can do it, you will be freed. If not – you and your friend will face the firing squad.” […]

2025-10-09 14:17:15 UTC

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@nprofile…uts4 oh christ I thought I was a late arrival to QC but I've now been reading it for more than half its existence

2025-10-09 13:04:08 UTC

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@nprofile…yu7x @nprofile…yajh I would have expected the smallest eigenvectors to be a combination of positive shells and negative shells with 0 overall mass. Then the field outside them will be very nearly 0 regardless of how the sample points are positioned.

2025-10-08 21:16:30 UTC

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@nprofile…uwy3 No, it's a valid section of that page: https://en.wikipedia.org/wiki/Lovin%27_You.

2025-10-08 17:59:56 UTC

Well that took a turn
https://media.mathstodon.xyz/media_attachments/files/115/339/834/251/598/303/original/0e2af340eb1ce62b.png

2025-10-07 18:30:19 UTC

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@nprofile…2k88
https://media.mathstodon.xyz/media_attachments/files/115/334/299/686/210/902/original/d6ffce3640f35ca7.jpg

2025-10-06 16:25:17 UTC

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@nprofile…vl5l Was the school electricity on an AC generator with a slightly inaccurate frequency?

2025-10-06 16:19:05 UTC

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@nprofile…q8c7
https://media.mathstodon.xyz/media_attachments/files/115/328/117/072/114/097/original/7d09710640ddb34c.jpg

2025-10-06 15:00:18 UTC

#Mathober #Mathober2025
The prompt for day 3 was 'Polyhedron'. In mathematics a 'convex polyhedron' is any subset of ℝ³ which can be obtained as the convex hull of finitely many noncoplanar points. However, when convexity is not required the definition of 'polyhedron' is in some dispute.
Certainly every convex polyhedron is a polyhedron, however there are multip-
On second thoughts, this post would be better expressed as an alignment chart meme.
#Math #Maths #Mathematics #Geometry #Polyhedra
https://media.mathstodon.xyz/media_attachments/files/115/327/792/523/383/162/original/1b2f1a9b23766898.png

2025-10-06 13:22:45 UTC

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@nprofile…ufp2 @nprofile…tca7 I calculated the coordinates of the 0-centre for the reference triangle https://faculty.evansville.edu/ck6/encyclopedia/Search_13_6_9.html and got 1.43335(8), which is indeed not on the list.

2025-10-03 20:11:16 UTC

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@nprofile…tca7 Good question. You can also think of this point as minimising the average logarithm of the distance to a point in the triangle. When I get the chance I'll calculate it's position for ETC's reference triangle. But my guess would be that it won't be listed, because it's an even more obscure property than the geometric median which also wasn't in there (see the link above).

2025-10-03 14:00:41 UTC

#Mathober #Mathober2025
The prompt for day 2 was 'Deviation'. In statistics the 'average absolute deviation' of a distribution is the average distance from a central point to a point chosen according to that distribution. For a one-dimensional distribution the point that minimises the average absolute deviation is the median. If instead we square each distance before averaging, and then square root the result, we have defined the 'standard deviation'. The centre point that minimises the standard deviation is the mean.
We can generalise this by defining the p-deviation to be the pth root of the average distance to the power of p. We'll define the p-centre to be the point that minimises the p-deviation (or maximises it if p is negative). We can define the p-centre when p is -∞, 0 or ∞ by taking a limit. So the 2-centre is the mean, the 1-centre is the median, and in fact the -∞-centre is the mode.
What if we apply this to a two dimensional distribution? Consider the distribution with probability 1/3 on each of the vertices of an acute triangle. Then the ∞-centre is the circumcentre, the 2-centre is the centroid and the 1-centre is the Fermat point. For some p between 1 and 0 the p-centre jumps to whichever vertex has the largest angle and stays there until p=-∞.
If instead we let the probability distribution be uniform on the interior of the triangle we find that the ∞-centre is again the circumcentre, the 2-centre is again the centroid, but the 1-centre is nowhere in particular (https://mathoverflow.net/questions/240186/the-geometric-median-of-a-triangle), and the p-centre continues to change smoothly until we reach -∞ at the incentre.
#Math #Maths #Mathematics #Geometry #Statistics #Probability
https://media.mathstodon.xyz/media_attachments/files/115/310/564/441/435/694/original/55b97c8e48d9eaf1.png

2025-10-01 14:35:32 UTC

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@nprofile…sp4m I'm also assuming the balls stay in one plane while being juggled, which is more than can be said for my juggling!

2025-10-01 14:22:18 UTC

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@nprofile…sp4m Right exactly. Once the balls all return to their starting positions we imagine we've looped (just like in the mp4). The way I first thought of it was that if you stand in the middle of the three rings and rotate a radial half-plane then its intersection with the three rings traces out the juggling pattern.

2025-10-01 14:10:10 UTC

#Mathober #Mathober2025
The prompt for day 1 is 'Link'. In mathematics, a link is any arrangement of some number of loops in space. Mathematicians are interested in how the loops can be tangled together. A simple example of a link is a single loop with any knot tied in it. Another example is the Borromean rings. These three loops cannot be separated from each other even though no two of the loops are linked without the third.
When you juggle three balls, the balls trace out the Borromean rings in spacetime.
#Math #Maths #Mathematics #Juggling #BorromeanRings
https://media.mathstodon.xyz/media_attachments/files/115/299/279/229/432/345/original/472e2397ddd83cb4.mp4

2025-09-29 23:48:29 UTC

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@nprofile…sast category implies the existence of catidry

2025-09-29 20:25:36 UTC

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@nprofile…caat This gives 0 mod 10 = 5, the problem being that at a jump discontinuity the fourier transform converges to the midpoint of the limits from either side. If you just want this to work when a is an integer then maybe you could just shift everything over by 1/2?

2025-08-18 11:18:23 UTC

A working QR code in the style of Piet Mondrian. Inspired @nprofile…r7u3 and @nprofile…ucsf.
#Art #PietMondrian #QRCode
https://media.mathstodon.xyz/media_attachments/files/115/049/472/461/955/344/original/a8149e2f0aad79f4.png

2025-06-18 21:04:51 UTC

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@nprofile…me4e Melée? Wasn't she in the Spice Girls?

2025-06-18 13:10:55 UTC

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@nprofile…6kk0 I would be setting up a hidden camera to watch the bike overnight

2025-06-18 12:55:03 UTC

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@nprofile…6kk0 Um... this reminds me of the posts you made about someone disconnecting your bike chain overnight, and the time your brakes were disconnected. Are you sure someone isn't trying to kill you?

2025-06-14 21:44:49 UTC

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@nprofile…fa4k @nprofile…hszg @nprofile…mlph (We need 0⁰ = 1, not 0)