Title: Discrete Cycloids from Convex Symmetric Polygons,
Abstract: Cycloids, hipocycloids and epicycloids have an often forgotten common
property: they are homothetic to their evolutes. But what if use convex
symmetric polygons as unit balls, can we define evolutes and cycloids which are
genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors
of a discrete double evolute transform which can be seen as a linear operator
on a vector space we call curvature radius space. We are also able to classify
such cycloids according to the eigenvalues of that transform, and show that the
number of cusps of each cycloid is well determined by the ordering of those
eigenvalues. As an elegant application, we easily establish a version of the
four-vertex theorem for closed convex polygons. The whole theory is developed
using only linear algebra, and concrete examples are given.