On Regular Polygons and Regular Solids Having Integer Coordinates for Their Vertices

Abstract

We study the existence of regular polygons and regular solids whose verticeshave integer coordinates in the three dimensional space andstudy side lengths of such squares, cubes and tetrahedra. We show that except for equilateral triangles, squares and regular hexagonsthere is no regular polygon whose vertices have integer coordinates. By using this, we show that there is no regular icosahedron and no regular dodecahedron whose vertices have integer coordinates. We characterize side lengths of such squares and cubes. In addition to these results, we prove Ionascu's result \cite[Theorem2.2]{E2} thatevery equilateral triangle of side length $\sqrt{2} m$ for a positive integer $m$ whose vertices have integer coordinate can be a face of a regular tetrahedron with vertices having integer coordinatesin a different way.