No acute tetrahedron is an 8-reptile

Abstract

An r-gentiling is a dissection of a shape into r ≥ 2 parts which are all similar to the original shape. An r-reptiling is an r-gentiling of which all parts are mutually congruent. The complete characterization of all reptile tetrahedra has been a long-standing open problem. This note concerns acute tetrahedra in particular. We find that no acute tetrahedron is an r-gentile or r-reptile for any r < 10. The proof is based on showing that no acute spherical diangle can be dissected into less than ten acute spherical triangles.

Computer Geometry: Rep-Tiles with a Hole

“…Hill already in 1895, and two others found in 1994 [13]. Recent results support the conjecture that there are no further 8rep-tile tetrahedra [11]. Figure 3 shows two other polyhedral examples found with the IFStile package.…”

Fractals: An Eclectic Survey, Part-I

“…The regular tetrahedron or octahedron is not a 8−rep tile. A special tetrahedra, which is an 8−rep tile, was found by Hill in 1895 and the conjecture that there are no further 8−rep tile tetrahedra [43,44] seems true till date. Sometime ago, it was not known whether three dimensional m−rep tiles can have holes.…”

“…Even for m = 8, not too many examples are known (see Figure 12 for some examples of polyhedral tiles). We refer to the papers [39,43,44] for important results and examples on the existence and non-existence of three-dimensional m−rep tiles and also to [45,46] for 3D tilings with large values of m. The rep tiles presented in these papers are produced using the similarity maps of the form g(x) = mx, and the IFS mappings have the form f j (x) = q j (x) + r j where r j denotes an integer translation and q j a symmetry map of the unit cube with center 0.…”

Three-dimensional Rep-tiles

“…For example, a cube is an 8-index 3D rep-tile homeomorphic to a 3-ball, see Figure 2. Additionally, various authors have investigated which tetrahedra are 3D rep-tiles [16], [18], [15], [13]. However, examples homeomorphic to other 3-manifolds have been more challenging to generate.…”