IJPAM: Volume 58, No. 1 (2010)

REFLECTIVE SPACE-FILLING POLYHEDRA

Jin-Ichi Itoh$^1$, Chie Nara$^2$
$^1$Faculty of Education
Kumamoto University
Kumamoto, 860-8555, JAPAN
e-mail: j-itoh@kumamoto-u.ac.jp
$^2$Liberal Arts Education Center
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 $P$ is called a reflective space-filler if a tiling by congruent copies of $P$ satisfies the following three conditions: (1) the tiling is face-to-face, (2) if two tiles $P_1$ and $P_2$ in the tiling have a common face, $P_1$ is the mirror-image of $P_2$ in the plane containing $P_1 \cap P_2$, 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