Download Convex polyhedra whose faces are equiangular or composed of such

CONVEX POLYHEDRA WHOSE FACES ARE EQUIANGULAR OR COMPOSED
OF SUCH
Yu. A. Pryakhin
UDC 513(02)
Depending on the type, considering only the topological structure of the network of
faces, and the angles of corresponding faces at corresponding vertices, convex polyhedra in
, each face of which is equiangular or composed of such, constitute four
infinite series (prism, antiprism, and two types of truncated antiprisms); outside
of this series, there are only a finite number of types.
i.
Parquet Polygon s
A convex polygon is said to be equiangular if all of its angles are equal.
A convex
polygon is called a parquet polygon if it can be formed from a finite number of equiangular
polygons.
The alternation of the angles obtained during a circuit of the vertices of a par-
quet polygon is called the type of the polygon and is designated by a special symbol.
example, the symbol
(3, 30, ~ ,
30)
For
refers to a polygon of 6 vertices, the angles for which
are successively equal to the angles of a regular triangle, a regular tridecagon, a regular
pentagon, a regular pentagon, a regular pentagon, and a regular tridecagon.
We have the fol-
lowing
LEMMA i.
polygon,
then
If a parquet polygon admits a decomposition into more than one equiangular
it has
one of the following 23
types: (3, 30, 53 , 30), (3, 30, 5, 30, 3,
30~),
(62, 30, 53 , 30), (3, 302 , 62 , 30, 5, 30), (6 ~, 30, 5, 30, 6~,302), (44), (3, 6, 3, 6), (3~), (32 ,
62), (3, 6"), (3, 12, 42 , 12), (6r 12, 42 , 12), (4, 12~, 4, 12, 6, 12), (4, 12 =, 4, 124),(3, 12 ~,
3, 122), (3, I22, 62 , 12~), (62 , 122,
6=
,
T~, 2
122), (6, ~z ,6, 12 ~, 6, 122), (6, 12 = , 6, i2+), (6+),
(12+2 ), (6, i2 ~ ), (6, I2", 6, i2~).
2.
~-Antiprisms and 6-Antiprisms
We consider an antiprism ~ , the parallel bases of which are regular
lateral faces of which are i~ pieces of regular triangles.
parallel to the bases, into three parts.
antiprisms;
n-gons and the
We section A , by two planes
The parts adjacent to the bases we refer to as B -
the middle part we refer to as a ~ -antiprism.
All the faces of these polyhedra
are parquet polygons.
3.
The Main Result
We say that two convex polyhedra have one and the same type if they have a common net-
work of faces and if at corresponding faces the corresponding angles are equal.
THEOREM.
For convex polydedra in ~ ,
all faces of which are parquet polygons, there
exists, except for the types associated with prisms, antiprisms,
~-antiprisms,
and Z-anti-
prisms, only a finite number of types.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo
Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 111-112, 1974.
486
0096-4104/78/1003-0486507.50 9
1978 Plenum Publishing Corporation
Remark.
M,
It is well known that in ~
there exists a finite number of convex polyhedra
distinct from prisms and antiprisms, whose faces are regular polygons (see [i]).
assume that the faces of M
ing at the vertices of M ,
If we
are formed from regular polygons, the vertices of the latter lythen the number of polyhedra M
increases but stays finite (see
[1-31).
If we allow the vertices of the regular polygons, which constitute the faces of M , to
be permuted arbitrarily, then the number of polyhedra M , distinct from prisms and antiprisms,
becomes infinite.
(One need only imagine rods of different proportion formed from cubes.)
The theorem stated above shows that if we disregard lengths of edges and concentrate only on
the type of the polyhedron, then in this case, except for the series of prisms, antiprisms,
B-antiprisms, and G-antiprisms,
there are a finite number of such types visible for the
type of ~.
4.
Plan of the Proof
To prove the theorem it is sufficient to proceed, similarly to what was done in [I, w
by eliminating from the table in [i, pp. 9-10] three infinite series of types of vertices.
The steps in the proof are similar, in many ways, to those in [i, w
Moreover, instead of
using Lemmas 2, 3, and 4 of [I], we prove, respectively, the following lemmas for a convex
polyhedron ~ with parquet faces.
LEMMA 2.
M
cannot have a vertex of the type (3.6.~)
for ~>60
if its type differs
from that of a ~-antiprism or a C-antiprism.
LEMMA 3.
~
cannot have a vertex of the type (4~. r~ ) for ~>30
if its type differs from
that of a prism.
LEMMA 4.
M
cannot have a vertex of the type
(3~.rL) for ~>60
if its type differs
from that of an antiprism or a ~-antiprism.
LITERATURE CITED
i.
2.
3.
V. A. Zalgaller, "Convex polyhedra with regular faces," in: Zap. Nauchn. Sem~ Leningr.
Otd. Mat. Inst., 2 (1967).
B. A. Ivanov, "Po~yhedra with faces composed of regular polygons," Ukr. Geomo Sb., I0,
20-34 (1971).
Yu. A. Pryakhin, "On convex polyhedra with regular faces," Ukr. Geomo Sb., 14, 83-88
(1973).
487