Wikipedia:Reference desk/Archives/Mathematics/2024 August 30

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August 30

Projections of hypercubes

Looking at Hypercube#Graphs, it looks like the projection of the n-cube into the B_n Coxeter plane has a no central vertex exactly when n can be written as 2^m for some positive integer m. The pictures confirm this is true for 1 ≤ n ≤ 15.

So:

1. Is this true in general?
2. What's the general term of the sequence (a_n), where a_n is the number of vertices projected to the centre (i.e. 0, 0, 2, 0, 2, 4, 2, 0, ...) ?

Double sharp (talk) 18:21, 30 August 2024 (UTC)

Did you intend to include a "not" in the question? I get no central vertex for n=2, 4, 8. For n=9 I get 8 points projected to the center. --RDBury (talk) 23:02, 30 August 2024 (UTC)

Um, yes. Oops. T_T Double sharp (talk) 03:54, 31 August 2024 (UTC)

For those who want to play along at home, I'm pretty sure the projection in question, translated to R², is given by the matrix with columns $(1,\cos {\tfrac {\pi }{n}},\cos {\tfrac {2\pi }{n}},\cdots \cos {\tfrac {(n-1\pi }{n}})$ and $(0,\sin {\tfrac {\pi }{n}},\sin {\tfrac {2\pi }{n}},\dots \sin {\tfrac {(n-1\pi }{n}}).$ There may be a scaling factor involved if you're picky about distance being preserved, but this is irrelevant for the question. It's not too hard to show that these vectors are orthogonal and have the same length. So the question becomes, given n, how many combinations of $\pm (1,0)\pm (\cos {\tfrac {\pi }{n}},\sin {\tfrac {\pi }{n}})\pm (\cos {\tfrac {2\pi }{n}},\sin {\tfrac {2\pi }{n}})\dots \pm (\cos {\tfrac {(n-1)\pi }{n}},\sin {\tfrac {(n-1)\pi }{n}})$ add to $(0,0)?$ These vectors form half of the points on a regular 2n-gon. It's not hard to see that there are at least two ways of doing it if n is odd; just alternate signs. A similar sign alternating idea shows that the number must be at least 2^mp if n=mp where p is odd. So if n has an odd factor then there are points which project to 0. Proving the converse seems tricky though. --RDBury (talk) 00:04, 31 August 2024 (UTC)

PS. I think I have an argument for the converse. The n points on the circle are all of the from ρ^k where ρ = e^πi/n. We need to find a combination of these powers of ρ, which amounts to a polynomial in p of degree n-1, where all coefficients are ±1. If n is odd then ρ satisfies (ρⁿ+1)/(ρ+1) = 1 - ρ + ρ² - ... + ρ^n-1 = 0, and this polynomial has the desired properties. If n has an odd factor, say n=pq with p odd, then p satisfies (ρⁿ+1)/(ρ^q+1) = 1 - ρ^q + ρ^2q - ... + ρ^n-q = 0. Multiply by any polynomial of the form 1 ± ρ ± ρ² - ... + ρ^q-1 to get a polynomial with the desired properties. But if n is a power of 2 then the minimum polynomial for ρ is ρⁿ+1=0. The degree n is greater than n-1, so no integer combination of the powers of ρ from 1 to n-1 can add to 0 except when all the coefficients are 0. In other words, the condition that the coefficients are all ±1 isn't needed; we only need that they are not all 0, FWIW, it appears that the number of vertices projecting to the center is given by OEIS: A182256. It's a lower bound in any case. --RDBury (talk) 00:44, 31 August 2024 (UTC)

That's nice indeed! So it was really a question about roots of unity, after all. :) Double sharp (talk) 04:00, 31 August 2024 (UTC)