Transformability and Reversibility of Unfoldings of Doubly-Covered Polyhedra - Discrete and Computational Geometry and Graphs - page 79

mirror image (denoted by P m ) is not strictly congruent to P , any unfolding W
of d ( P ) is transformable to any unfolding X of d ( P m ); and moreover, if W is
dissected into n non-empty pieces (where n is the number of faces of P ), W is
reversible to X , where we assume P

W and P m

ↂ ↂ X .

2 Definitions and Preliminaries

Definition 1. If a convex polyhedron W is dissected into a finite number of
pieces which can be rearranged to form a convex polyhedron X ,insuchaway
that there is a dissection tree whose vertex set is the set of pieces and whose edge
set is the set of hinges each of which corresponds to a common edge of two pieces,
we say that W is transformable to X . Moreover if the surface ∂W is transformed
to the interior of X except some edges of pieces, we say W is reversible to X .
The pair of W and X is called transformable or reversible , respectively.

Note that the transformability defined in [ 3 ], is not allowed to cut any side (face)
of polyhedra, but in this paper we allow to do so, because Dudeney's dissection
cuts sides. Figure 2 shows the pair of a cube and a rectangular parallelepiped
which is transformable, and Fig. 3 shows the pair of a rhombic dodecahedron
and a rectangular parallelepiped which is reversible.

Fig. 2. The pair of a cube and a rectangular parallelepiped which is transformable.

Definition 2. A convex polyhedron W is called a reflective space-fil ler if its infi-
nitely many congruent copies tile space (without no gaps and no 3-dim. overlaps)
such that

(1) the tiling is face-to-face,
(2) if two copies have a face in common, one is obtained from the other by a
reflection in the common face, and
(3) any dihedral angle of W is ˀ/k (an integer k ≥ 2).

The third condition is used for a simple polyhedral unfolding X of the doubly-
covered W (see Definition 3) to be convex since dihedral angles of X are less
than or equal twice dihedral angles of W . H. S. M. Coxeter [ 4 ] showed that
there are seven types of 3-dimensional reflective space-fillers (up to congruence
and similarity): four of them are prisms (whose bases are a square, an equilateral
triangle, a right triangle with an angle ˀ/ 3, or an isosceles right triangle), and
three of them are tetrahedra one of which has congruent triangular faces with