The Number of Intersection Points Made by the Diagonals of a Regular Polygon

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Bjorn Poonen, Michael Rubinstein, "The Number of Intersection Points Made by the Diagonals of a Regular Polygon," SIAM Journal on Discrete Mathematics 11 (1998), no. 1, 135-156 (1995).

Summary

The paper determines the number (I(n)) of distinct interior intersection points formed by the diagonals of a regular (n)-gon and the number (R(n)) of resulting regions. For a generic convex polygon one would have (I(n)=\binom n4), but regular polygons have multiple concurrences. The explicit quasipolynomial formula for (I(n)) is Theorem 1; its correction terms are supported on divisibility conditions through modulus (2520). Theorem 2 derives the corresponding formula for (R(n)) using Euler's formula. The introduction also extracts a geometric classification: away from the center, at most seven diagonals can meet, and the maximum multiplicity is determined by the divisibility of (n) by (2,6,) and (30), with the exceptions (n=6,12).

Section 2 converts concurrence of three diagonals into the trigonometric Diophantine equation (2), an equality of two products of three sines of rational multiples of (pi). Expanding the sines turns it into a positive vanishing sum of twelve roots of unity. Section 3 therefore studies relations

$$S=\sum_{i=1}^k a_i\eta_i=0, \qquad a_i\in\mathbb Z_{>0},$$

with distinct roots (eta_i), weight (w(S)=\sum_i a_i), and no proper vanishing subrelation. Theorem 3 and Table 1 classify all minimal relations of weight at most (12), up to rotation; there are (107) rotational classes. The notation ((R_p:T_1,\ldots,T_j)) records the basic recursive operation: start with the prime polygon relation (R_p), subtract smaller relations at selected terms, and cancel the common terms.

Three lemmas give the structural core of that classification. Lemma 1, using Mann's theorem, says that a minimal relation with (k) distinct roots can be rotated so that every term has squarefree order whose prime factors are at most (k). Lemma 2 shows that the only minimal relations supported on (2p)-th roots are the prime polygons (R_2) and (R_p). Lemma 3 gives the recursive step. Let the primes (p_1<\cdots<p_s) of Lemma 1 be chosen with (p_1=2) and (p_s) minimal. If (w(S)<2p_s), then, after rotation, (S=(R_{p_s}:T_1,\ldots,T_j)), where the (T_i) are minimal relations other than (R_2) that use only the smaller conductor primes, (j<p_s), and

$$\sum_{i=1}^j\bigl(w(T_i)-2\bigr)=w(S)-p_s.$$

The proof groups the relation by its (p_s)-coordinate. Cyclotomic degree forces all layer sums to be equal, while the weight bound makes one layer a singleton; subtracting that singleton from the remaining layers produces the smaller minimal relations. The proof of Theorem 3 then handles the finitely many possible largest primes (2,3,5,7,11) and the exceptional cases at the boundary of the (2p_s) inequality.

Section 4 imposes complex-conjugation symmetry and uses the classified relations to obtain Theorem 4, the explicit list of positive rational solutions to the trigonometric equation governing triple concurrence. Lemmas 4 and 5 control conjugation-stable decompositions. Section 5 treats higher multiplicity: Lemma 6 bounds the denominator of every further diagonal through a point once one concurrence denominator is known. Sections 6 and 7 package the enumeration into tame arithmetic functions; Lemma 7 gives a finite set of arguments determining such a function, after which the computer calculations in the appendix determine the formulas of Theorems 1 and 2.