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1
4
b
c
a
c
b
a
3
4
1
2
4
2
1
3
a
c
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Fig. 1.
Dudeney's dissection.
W
in the 3-dimensional Euclidean space
R
3
is dissected into a finite number of
pieces which can be rearranged to form a convex polyhedron
X
, in such a way
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
∂W
(the surface of
W
) is transformed to the interior of
X
except some edges of pieces, we say
W
is
reversible
to
X
.J. Akiyama et al. [
3
] investigated reversibility of the family of
canonical parallelohedra.
In this paper, we study the family of convex unfoldings of doubly-covered
polyhedra. For a polyhedron
P
, the doubly-covered
P
(denoted by
d
(
P
)) is
the degenerate 4-dimensional polytope consisting of
P
and its congruent copy
(denoted by
P
∗
) whose corresponding faces are identified, which means the sur-
face of
d
(
P
) is identical to that of
P
and the volume of an unfolding of
d
(
P
)is
twice that of
P
.
If
P
in
R
3
is a reflective space-filling polyhedron (there are seven types of
such polyhedra with no obtuse dihedral angle up to congruence and similarity
[
4
,
7
]), any convex unfolding
W
of
d
(
P
)isa
space-filler
, which means that its
infinitely many congruent copies tile the space with no gaps and no 3-dimensional
overlaps [
8
].
We call two figures in
R
2
or
R
3
strictly congruent
if they can be mapped to
each other by rotation and translation only (with no reflection).
We show that if
P
is a reflective-space filler in
R
3
whose mirror image is
strictly congruent to
P
, any two convex unfoldings
W
and
X
of
d
(
P
) are trans-
formable to each other; moreover, if
W
is dissected into
n
non-empty pieces
(where
n
is the number of faces of
P
),
W
is reversible to
X
,whereweassume
both
W
and
X
include
P
as a subset. If
P
is a reflective-space filler in
R
3
whose
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