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Geometry, algebra, and examples

Let n and m be integers, with 2 ≤ m < n/2. Consider the bounding polygon of an n/m star (that is, a star with n points each of which connects to the two points ±m away) inscribed in the unit circle. Such a bounding polygon has 2n points, n on the unit circle and n on an inner circle with:

$Inner Radius = \frac{\cos{\left(\frac{\pi m}{n}\right)}}{\cos{\left(\frac{\pi (m-1)}{n}\right)}}$

5/2 star 8/2 star 8/3 star

E.g., inner radius of 5/2 star is ½(3–√5) ≈ 0.381966, that of 8/2 star is √(2–√2) ≈ 0.765367, and that of 8/3 star is √(1–√½) ≈ 0.541196.

I wish to characterise the different stars that have the same inner radius. Two series of pairs are known: ∀ integer i ≥ 2, stars (6i–2)/i and (18i–6)/(6i–2) have the same inner radius, as do (6i–4)/i and (18i–12)/(6i–3). Proof that these pairs do match is boringly elementary, given the identity Cos[θ] Cos[φ] = ½Cos[θ+φ] + ½Cos[θ–φ], and at jdawiseman.com.

But are there any other non-trivial matches, perhaps another series, perhaps sporadic?

(I know that any other matches must have inner radius > 0.998122, and of course strictly < 1. So any other matches shown as a graphic smaller than about 2k pixels across must look like a circle — practical rather than proper progress.)

The Question

Characterise all pairs of stars, n₁/m₁ and n₂/m₂ (all integer), such that the stars have the same inner radius. It is known that there are two series of such pairs of stars, {(6i–2)/i, (18i–6)/(6i–2)} and {(6i–4)/i, (18i–12)/(6i–3)}. Are there any other series? Are there any sporadic matches?

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In short, you are asking the following question. If $m_1,m_2,n_1,n_2$ are integers with $2\leq m_1<n_1/2$ and $2\leq m_2<n_2/2$ and $n_1\leq n_2$, and if $f(n_1,m_1)=f(n_2,m_2)$, with $f(n,m):=\cos(\pi*m/n)/\cos(\pi*(m-1)/n)$, then is it the case that either $n_2=n_1$ or $n_2=6n_1-2$ or $n_2=6n_1-3$? –  user30035 Mar 29 '13 at 8:44
    
Did you mean to write that “either $n_2=n_1$ or $n_2=3n_1$”? Even so, I think that is necessary, but not sufficient. Might there be multiple $m$s for any particular $n_1$ and $n_2$? –  jdaw1 Mar 29 '13 at 15:17
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This is a special case of a problem that was solved by Gerrit Bol in 1936; that nearly-forgotten result was rediscovered, using slightly different methods, by Bjorn Poonen and Michael Rubinstein: see their paper "The Number of Intersection Points Made by the Diagonals of a Regular Polygon", SIAM J. Discrete Math 11 (1998), 135-156 (www-math.mit.edu/~poonen/papers/ngon.pdf). –  Noam D. Elkies Apr 1 '13 at 15:43
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Ooops: offered bounty without noticing comment that is an answer. Noam: please grab prize by posting as a answer. –  jdaw1 Apr 1 '13 at 20:30
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If the inner radii coincide then we can rotate the two stars until they share an inner vertex. The outer vertices are then contained among the vertices of a single regular $N$-gon where $N$ is at most something like the product of the two different $n$'s. Now apply Bol / Poonen-Rubinstein. –  Noam D. Elkies Apr 2 '13 at 0:47

1 Answer 1

up vote 6 down vote accepted

[Expanding some on my comment of a few days ago]

This is a special case of a problem that was solved by Gerrit Bol in 1936 [B]; that nearly-forgotten result was rediscovered, using slightly different methods, by Bjorn Poonen and Michael Rubinstein [PR]. (As it happens I used such coincidences in my own work a few years ago [E].) They find all ways that three diagonals of a regular polygon can meet at a point: there are several infinite families (comprising algebraically "trivial" solutions that are not always geometrically obvious, plus the four "nontrivial" families of Table 3 on page 12), and 65 sporadic solutions (Table 4 on page 13). If two stars have the same inner and outer radii then we can rotate them so they share an inner vertex; then the outer vertices are contained among the vertices of a regular polygon, and the shared inner vertex is on at least four diagonals $-$ indeed at least five if we include the line of symmetry (and double the order of the regular polygon if necessary). Four infinite families (all symmetrical) of such quintuple intersections are listed in Table 6 (page 16), and a finite computation limits further sporadic solutions to denominators 18, 24, and 30 (pages 15-16). If you've already computed far enough to find any sporadic solutions then the infinite families must account for everything else.

References

[B] Gerrit Bol: Beantwoording van prijsvraag no. 17, Nieuw Archief voor Wiskunde 18 (1936), 14-66.

[PR] Bjorn Poonen and Michael Rubinstein: The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math 11 (1998), 135-156 (http://www-math.mit.edu/~poonen/papers/ngon.pdf).

[E] Noam D. Elkies: On some points-and-lines problems and configurations, Periodica Mathematica Hungarica 53 #1-2 (2006), 133-148 (arXiv:MG/0612749).

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Splendid, thank you. Posted just too late for me to approve it, but the machine awarded you the bounty anyway. –  jdaw1 Apr 9 '13 at 7:03

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