Transformability and Reversibility of Unfoldings of Doubly-Covered Polyhedra - Discrete and Computational Geometry and Graphs - page 84

(1)

(2)

(3)

a'
b'

(4) (5) (6)

Fig. 8. (1) An unfolding W of d ( P )ofaprism P whose base is an equilateral triangle;
(2) P included in the unfolding; (3) dissection and hinges; (4) the resulting figure
transformed by dissection and hinges shown in (3); (5) another dissection of W by
the 2-complex strictly congruent to the mirror image of the cut 2-complex in Fig. 7 (4);
(6) the resulting figure by dissection and hinges shown in (5) which is strictly congruent
to the polyhedron shown in Fig. 7 (5).

If P ∗ is dissected into n (where n is the number of faces of P ) pieces to obtain
W , the cut 2-complex except its boundary is included in the interior of P ∗ ,and
hence ∂W is transformed to the interior of P m except some line segments. W is
reversible to X by P m ↂ X .

Theorem 2. Let P be a tetrahedron which is a half- ST or a quarter- ST .Any
convex unfoldings W of d ( P ) is transformable to any convex unfolding X of
d ( P m ) , where we assume P X .If W divides P ∗ into four pieces

by its corresponding cut 2-complex, W is reversible to X .

W and P m ↂ ↂ

Proof. The proof is similar to the one of Theorem 1 , so we omit it.

If a polyhedron W is reversible to itself, W is called self-reversible .

Corollary 1. Let P be a reflective space-filling polyhedron whose mirror image
is strictly congruent to P ,and n be the number of faces of P . If the cut 2-
complex of a convex unfolding W of d ( P ) divides P ∗ into n non-empty pieces,

W is self-reversible, where we assume P ↂ W .

Proof. If an unfolding W is combinatorially equivalent to a truncated octahe-
dron, rhombic dodecahedron, or an elongated dodecahedron, the corresponding