Unfoldings of doubly covered polyhedra and applications to space-fillers
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UNFOLDINGS OF DOUBLY COVERED POLYHEDRA AND APPLICATIONS TO SPACE-FILLERS Jin-ichi Itoh1 and Chie Nara2 1
2
Faculty of Education, Kumamoto University Kumamoto, 860-8555, Japan E-mail: [email protected]
Liberal Arts Education Center, Aso Campus, Tokai University Aso, Kumamoto, 869-1404, Japan E-mail: [email protected] (Received September 6, 2009; Accepted May 13, 2010) [Communicated by Imre B´ ar´ any]
Abstract We study unfoldings (developments) of doubly covered polyhedra, which are space-fillers in the case of cuboids and some others. All five types of parallelohedra are examples of unfoldings of doubly covered cuboids (Proposition 1). We give geometric properties of convex unfoldings of doubly covered cuboids and determine all convex unfoldings (Theorem 1). We prove that every unfolding of doubly covered cuboids has a space-filling (consisting of its congruent copies) generated by three specified translates and three specified rotations, and that all such space-fillers are derived from unfoldings of doubly covered cuboids (Theorem 2). Finally, we extend these results from cuboids to polyhedra which are fundamental regions of the Coxeter groups generated by reflections in the 3-space and which have no obtuse dihedral angles (Theorem 3).
1. Introduction A doubly covered square is a degenerated polyhedron consisting of two congruent squares whose corresponding edges are identified. The geometric properties of convex unfoldings (developments) of a doubly covered square were studied by J. Mathematics subject classification numbers: 05B45, 51M20, 52C22. Key words and phrases: unfolding, parallelohedron, tiling, space-filler. 1
Supported by Grant-in-Aid for Scientific Research No. 23540098, JSPS.
2
Supported by Grant-in-Aid for Scientific Research No. 23540160, JSPS.
0031-5303/2011/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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J.-I. ITOH and C. NARA
Akiyama, K. Hirata, M. P. Ruiz and J. Urrutia, all such unfoldings were determined by them, and it was showed that such unfoldings are plane-fillers [1, 2, 3, 4]. In this paper, we extend these results from the plane to the 3-space. We denote by ∂W the boundary of a set W in R2 or R3 . Definition 1. Let P be a polyhedron. The doubly covered P (denoted by D(P )) is the degenerated polytope in the 4-space consisting of P and its congruent copy (denoted by P ∗ ) whose corresponding faces are identified. Definition 2. We call a compact set W ⊂ R3 a body if W is homeomorphic to a closed unit ball in R3 . Let P be a polyhedron. A body W is called an unfolding of D(P ) if there is a continuous map (denoted by fW,D(P ) ) from W onto D(W ) such that (i) fW,D(P ) is locally isometric on the interior of W , and (ii) fW,D(P ) has no 3-dimensional overlaps (that is, for disjoint open sets of W the images have no common interior point). We call fW,D(P ) a folding map of W onto D(P ), and the image of ∂W is called a cut 2-complex of D(P ) for W . Then the map fW,D(P ) is one-to-one in the interior of W , but not one-to-one on ∂W (see
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