A characterization of regular tetrahedra in Z3Z3

TextIn this note we characterize all regular tetrahedra whose vertices in R3R3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates [R. Chandler, E.J. Ionascu, A characterization of all equilateral triangles in Z3Z3, Integers 8 (2008), Article A19]. Previous work on this topic begun in [E.J. Ionascu, A parametrization of equilateral triangles having integer coordinates, J. Integer Seq. 10 (2007), Article 07.6.7]. We will use this characterization to point out some corollaries. The number of such tetrahedra whose vertices are in the finite set {0,1,2,…,n}3{0,1,2,…,n}3, n∈Nn∈N, is related to the sequence A103158 in the Online Encyclopedia of Integer Sequences [Neil J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at: http://www.research.att.com/~njas/sequences/, 2005].VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=LT3aAUUFMFk.