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cut 2-complex divides P into six non-empty pieces. So the statement in Corol-
llary 1 follows from Theorem 1 .
If a polygon T in the 2-dimensional plane is reversible to itself, T is called self-
reversible . Note that the mirror image of a right triangle T with angle ˀ/ 3in
a given plane is not strictly congruent to T in the given plane. If we restrict
unfoldings of d ( P ) in Theorem 1 to prisms, we get the following.
Corollary 2. Let P be a polygon which is a rectangular parallelogram, an equi-
lateral triangle, or an isosceles right triangle. Any two convex unfoldings T and
S of d ( P ) are transformable to each other, where we assume that P
T and
S .If T divides P into n pieces by its corresponding cut 2-complex (where
n is the number of edges of P ), T is reversible to S .
If P is a right triangle with angles ˀ/ 3 , any convex unfoldings T of d ( P )
is transformable to any unfolding S of d ( P m ) , where we assume P
P
T and
P m
S .If T divides P into three pieces by its corresponding cut 2-complex, T
is reversible to S .
Proof. For rectangular parallelepipeds or triangular prisms P which are reflective
space-fillers, consider the subfamily of simple unfoldings of d ( P ) whose cut 2-
complex is included in faces of P (as well as P ), that is, which are orthogonal
prisms. Then their images by orthogonal projection to corresponding bases are
the family of unfoldings of d ( P ).
Remark. By observing the proof of Theorem 2, we notice that the result may
be extended to d ( P ) of any convex polyhedron P whose dihedral angles are less
than or equal to ˀ/ 2. To do so, we only need to give precise continuous motions
without self-intersection, which looks obvious, but we leave it for a future work.
References
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