Transformability and Reversibility of Unfoldings of Doubly-Covered Polyhedra - Discrete and Computational Geometry and Graphs - page 77

Transformability and Reversibility of Unfoldings
of Doubly-Covered Polyhedra

B

Jin-ichi Itoh 1 andChieNara 2(

)

1 Faculty of Education, Kumamoto University, Kumamoto 860-8555, Japan
[e-mail]
2 Liberal Arts Education Center, Aso Campus, Tokai University,
Aso, Kumamoto 869-1404, Japan
[e-mail]

Abstract. Let W and X be convex polyhedra in the 3-dimensional
Euclidean space. If W is dissected into a finite number of pieces which can
be rearranged to form X with hinges (which compose a dissection tree),
W is called transformable to X , and if the surface of W is transformed
to the interior of X except some edges of pieces, W is called reversible to
X .Let P be a reflective space-filler in the 3-space and let P m be a mir-
ror image of P . In this paper, we show that any convex unfolding W of
the doubly covered polyhedron d ( P )of P is transformable to any convex
unfolding X of the doubly covered polyhedron d ( P m )of P m ,wherewe
assume that W (resp. X ) includes P (resp. P m ) as a subset. Moreover if
W is dissected into n non-empty pieces (where n isthenumberoffaces
of P ), W is reversible to X .

1 Introduction

The famous hinged dissection problem asked if an equilateral triangle W can be
dissected into a finite number of pieces that can be rearranged to form a square
X with hinges. H.E. Dudeney [ 5 ] gave an answer by giving a dissection by four
pieces (see Fig. 1 ). There is a related topic for the n -dimensional case, the so-
called Hilbert's third problem: Given any two polyhedra of equal volume, is it
always possible to cut the first into finitely many polyhedral pieces which can
be reassembled to yield the second? If n = 2, the answer is armative, and is
known as the Bolyai-Gerwein Theorem [ 11 ]; moreover, they can be reassembled
with hinges [ 1 ]. On the other hand, if n = 3, M. Dehn [ 6 ] gave a counterexample;
the pair of a cube and a regular tetrahedron of equal volume.
We study the problem of finding a family of convex polyhedra such that any
pair in the family has the above-mentioned property. If a convex polyhedron

Jin-ichi Itoh—Supported by Grant-in Aid for Scientific Research (No. 23540098),
JSPS.
Chie Nara—Supported by Grant-in Aid for Scientific Research (No. 23540160),
JSPS.