John Carlos Baez on Nostr: I proved a fun theorem about this here: Say you have any regular pentagon in ℝⁿ ...

I proved a fun theorem about this here:

https://johncarlosbaez.wordpress.com/2026/05/29/from-pentagons-to-pentagrams/

Say you have any regular pentagon in ℝⁿ whose vertices have coordinates of the form a + b√5 with a,b rational. If you replace √5 by -√5, you get a regular pentagram!

So, Galois transformations can be demonic.

Such pentagons are actually impossible in the plane (that is, n = 2). For n = 2, to draw a regular pentagon, we need coordinates lying in a larger field. But in higher dimensions, the so-called 'golden field' is enough: that's the field ℚ[√5], consisting of numbers a + b√5 with a,b rational.

(2/2)