IJPAM: Volume 58, No. 1 (2010)



Kumamoto University
Kumamoto, 860-8555, JAPAN
e-mail: j-itoh@kumamoto-u.ac.jp

Aso Campus
Tokai University
Aso, Kumamoto, 869-1404, JAPAN
e-mail: cnara@ktmail.tokai-u.jp
Abstract.A polyhedron in 3-space is called a space-filler or a space-filling polyhedron if its infinitely many (directly or reflectively) congruent copies fill the space with no gaps and no (3-dimensional) overlaps. A space-filling polyhedron is called a reflective space-filler if a tiling by congruent copies of
satisfies the following three conditions: (1) the tiling is face-to-face, (2) if two tiles
and
in the tiling have a common face,
is the mirror-image of
in the plane containing
, and (3) the chromatic number of the tiling is two. H.S. Coxeter [2], [3] proved that there exist only seven types of reflective space-fillers. In this paper, we give an elementary proof of this fact.
Received: December 19, 2009
AMS Subject Classification: 52B10
Key Words and Phrases: space-filler, polyhedron, reflective, tiling
Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 58
Issue: 1