Acta Math. Univ. Comenianae Vol. LXXX, 2 (2011), pp. 161–170 161 REGULAR TETRAHEDRA WHOSE VERTICES HAVE INTEGER COORDINATES E. J. IONASCU Abstract. In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0, 1, ..., n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences [16]. These results lead to the consideration of an infinite graph having fractal nature which is tightly connected to the set of orthogonal 3-by-3 matrices with rational coefficients. The vertices of this graph are the primitive integer solutions of the Diophantine equation a 2 + b 2 + c 2 =3d 2 . Our aim here is to lay down the basis of finding good estimates, if not exact formulae, for the sequence A103158. 1. Introduction The story of regular tetrahedra having vertices with integer coordinates starts with the parametrization of some equilateral triangles in Z 3 that began in [9]. There was an additional hypothesis that did not cover all the generality in the result obtained in [9] but it was removed successfully in [2]. In this note we are interested in the following problem How many regular tetrahedra, T (n), can be found if the coordinates of its vertices must be in the set {0, 1, ..., n}? We observe that A103158 = 1 2 T (n) (see [16]). This sequence starts as in the following table. n 1 2 3 4 5 6 7 8 9 10 11 A103158 1 9 36 104 257 549 1058 1896 3199 5154 7926 n 12 13 14 15 16 17 18 A103158 11768 16967 23859 32846 44378 58977 77215 These values were computed by Hugo Pfoertner in 2005, using a brute force program. Our method of counting is based on several theoretical facts. Roughly, it is an extension of the technique described in [10] using the results from [12] about the existence of regular tetrahedrons in Z 3 . The program can be used to cover values of T (n) for n quite bigger than 100, but we included here only Received December 5, 2009; revised July 1, 2011. 2010 Mathematics Subject Classification. Primary 11D09. Key words and phrases. diophantine equations; integers; infinite graph; fractal.