## 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 2*n* points, *n* on the unit circle and *n* on an inner circle with:

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

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 (6*i*–2)/*i* and (18*i*–6)/(6*i*–2) have the same inner radius, as do (6*i*–4)/*i* and (18*i*–12)/(6*i*–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 andn₂/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?

SIAM J. Discrete Math11(1998), 135-156 (www-math.mit.edu/~poonen/papers/ngon.pdf). $\endgroup$3more comments