Abstract. We study the existence of regular polygons and regular solidswhose vertices have integer coordinates in the three dimensional space andstudy side lengths of such squares, cubes and tetrahedra. We show thatexcept for equilateral triangles, squares and regular hexagons there is noregular polygon whose vertices have integer coordinates. By using this,we show that there is no regular icosahedron and no regular dodecahedronwhose vertices have integer coordinates. We characterize side lengths ofsuch squares and cubes. In addition to these results, we prove Ionascu’sresult [4, Theorem2.2] that every equilateral triangle of side lengthp2mfor a positive integer m whose vertices have integer coordinate can be aface of a regular tetrahedron with vertices having integer coordinates ina di erent way. 1. IntroductionEugen J. Ionascu and his colleagues did much work on regular polygons andregular solids whose vertices have integer coordinates[1, 3, 4, 5]. Some of theirresults are as follows:Theorem 1.1. [3] For an equilateral triangle OPQ(Ois the origin) in thethree dimensionl space R