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Cycloids, hipocycloids and epicycloids have an often forgotten commonproperty: they are homothetic to their evolutes. But what if use convexsymmetric polygons as unit balls, can we define evolutes and cycloids which aregenuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectorsof a discrete double evolute transform which can be seen as a linear operatoron a vector space we call curvature radius space. We are also able to classifysuch cycloids according to the eigenvalues of that transform, and show that thenumber of cusps of each cycloid is well determined by the ordering of thoseeigenvalues. As an elegant application, we easily establish a version of thefour-vertex theorem for closed convex polygons. The whole theory is developedusing only linear algebra, and concrete examples are given.

Discrete Cycloids from Convex Symmetric Polygons