Rational tetrahedra with edges in arithmetic progression

Abstract

This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families are found which have rational volume, a face can have rational area only if its edges are themselves in arithmetic progression, and a tetrahedron can have at most one such rational face area.

Rational tetrahedra with edges in geometric progression

“…The discriminant of the curve is −2 4 3 12 23, so we can apply the 'Reduction mod p' map with p = 5. We find y 2 ≡ x 3 + 2 (mod 5) which has solutions (2, 0), (3,2), (3,3), (4, 1), (4,4). So |Ẽ(F 5 )| = 6 and since there are no points of order 2, E(Q) t ∼ = Z 3 .…”

“…, there are two non-trivial rational points of order 3: (3,27) and (3, −27). The discriminant of the curve is −2 4 3 12 23, so we can apply the 'Reduction mod p' map with p = 5.…”

“…2 1 r r 2 r 4 r 5 r 3 G1, G9, G14, G19 G1. 3 1 r r 2 r 3 r 5 r 4 G1, G10, G14, G18 G1. 4 1 r r 3 r 5 r 2 r 4 G2, G6, G16, G18 G1.…”

“…Face Edges (4A) 2 = Partner(s) G1 1, r, r 2 (r 2 + r + 1)(−r 2 + r + 1) G11, 17, 20 × (r 2 − r + 1)(r 2 + r − 1) G2 1, r, r 3 (r 3 + r + 1)(−r 3 + r + 1) G5, 12, 14, 18, 19 × (r 3 − r + 1)(r 3 + r − 1) G3 1, r, r 4 (r 4 + r + 1)(−r 4 + r + 1) G8, 13, 16 × (r 4 − r + 1)(r 4 + r − 1) G4 1, r, r 5 (r 3 − r 2 + 1)(r 3 + r 2 − 1)(r 5 − r + 1) G10 × (−r 5 + r + 1)(r 2 + r + 1)(r 2 − r + 1) G5 1, r 2 , r 3 (r 3 + r 2 + 1)(−r 3 + r 2 + 1) G2, 12, 14, 18, 19…”

“…In [3] the authors conducted a search for Heron tetrahedra with edges in arithmetic progression, inspired by the tetrahedron shown in Fig. 1 [4, p. 287] and the discovery by Buchholz and MacDougall [2] of a simple relation giving Heron triangles whose edges form an arithmetic progression.…”