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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
ↂ
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
ↂ
ↂ
T
and
P
m
S
.If
T
divides
P
∗
into three pieces by its corresponding cut 2-complex,
T
is reversible to
S
.
ↂ
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
1. Abbott, T.G., Abel, Z., Charlton, D., Demaine, E.D., Demaine, M.L., Kominers,
S.D.: Hinged dissections exist. Discrete Comput. Geom.
47
(1), 150-186 (2012)
2. Akiyama, J., Kobayashi, M., Nakagawa, H., Sato, I.: Atoms for parallelohedra.
Geometry intuitive, discrete and convex. Bolyai Soc. Math. Stud.
24
, 23-43 (2013).
Springer
3. Akiyama, J., Sato, I., Seong, H.: On Reversibility among parallelohedra. In:
Marquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 14-28.
Springer, Heidelberg (2012)
4. Coxeter, H.S.: Discrete groups generated by reflections. Ann. Math.
35
(3), 588-621
(1934)
5. Dudeney, H.E.: The Canterbury Puzzles and Other Curious Problems.
W. Heinemann, London (2010)
6. Dehn, M.: Uber den Rauminhalt. Math. Ann.
55
, 465-478 (1902)
7. Itoh, J., Nara, C.: Reflective space-filling polyhedra. Int. J. Pure Appl. Math.
58
(1), 87-98 (2010)
8. Itoh, J., Nara, C.: Unfoldings of doubly covered polyhedra and applications to
space-fillers. Periodica Math. Hung.
63
(1), 47-64 (2011)
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