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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