Rational Area Profiles of Regular Polygons Cut by Two Congruent Diagonals
Abstract
Regular-polygon geometry is tightly linked to cyclotomic arithmetic: Poonen and Rubinstein’s treatment of three-diagonal concurrence, for example, turns a geometric incidence condition into a short vanishing sum of roots of unity.
We prove an analogous rigidity result for areas. Two congruent crossing diagonals divide a regular n-gon into four regions. For the four regions cut out by the two congruent crossing diagonals V0Vm and VkVn−m+k of a regular n-gon, we completely classify, for all parameters (n, k, m), which sums of the normalized areas a0, . . . , a3 are rational. The classification has a sharp finite–infinite contrast: a0 is rational in only five configurations, whereas the rational cases for a2 and adjacent two-region sums form infinite families. Rationality is delicately sensitive to the parameters: for the configuration (14, 3, 5), no nontrivial subset sum is rational, while the neighboring cut (14, 4, 5) gives a2 = 5/7.
The proof reduces each rationality condition to trigonometric relations at rational multiples of π and combines cyclotomic norm arguments with the classification theorems of Conway–Jones and Poonen–Rubinstein.
We prove an analogous rigidity result for areas. Two congruent crossing diagonals divide a regular n-gon into four regions. For the four regions cut out by the two congruent crossing diagonals V0Vm and VkVn−m+k of a regular n-gon, we completely classify, for all parameters (n, k, m), which sums of the normalized areas a0, . . . , a3 are rational. The classification has a sharp finite–infinite contrast: a0 is rational in only five configurations, whereas the rational cases for a2 and adjacent two-region sums form infinite families. Rationality is delicately sensitive to the parameters: for the configuration (14, 3, 5), no nontrivial subset sum is rational, while the neighboring cut (14, 4, 5) gives a2 = 5/7.
The proof reduces each rationality condition to trigonometric relations at rational multiples of π and combines cyclotomic norm arguments with the classification theorems of Conway–Jones and Poonen–Rubinstein.
Research Areas
