CONVEX POLYHEDRA WITH REGULAR FACES
NORMAN W. JOHNSON
1. Introduction. An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular. That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, "semi-regular" polyhedra, but his work on the subject has been lost. Kepler (12, pp. 114-127) showed that the convex uniform polyhedra consist of the Platonic and Archimedean solids together with two infinite families—the regular prisms and antiprisms. It was Kepler also who gave the Archimedean polyhedra their generally accepted names. It is fairly easy to show that there are only a finite number of non-uniform regular-faced polyhedra (11; 13), but it is no simple matter to establish the exact number. However, it appears that there are just ninety-two such solids. Some special cases were discussed by Freudenthal and van der Waerden (8), and a more general treatment was attempted by Zalgaller (13). Subsequently, Zalgaller et al. (14) determined all the regular-faced polyhedra having one or more trivalent vertices and all those having only pentavalent vertices. Griin- baum and Johnson (9) proved that the only kinds of faces that a regular-faced solid, other than a prism or an antiprism, may have are triangles, squares, pentagons, hexagons, octagons, and decagons. A regular n-gon is conveniently denoted by the Schldfli symbol {n}. Thus {3} is an equilateral triangle, {4} a square, {5} a regular pentagon, etc. An edge of a regular-faced polyhedron common to an \m\ and an \n) will be said to be of type (m-n). The sum of the face angles at a vertex of a convex poly hedron must be less than 360°. If the faces are regular, it follows that no more than five can meet at any vertex; in other words, each vertex of a convex poly hedron with regular faces must be trivalent, tetravalent, or pentavalent. Various combinations of faces give rise to many different types of vertices. For example : (4-6-8)—a square, a hexagon, and an octagon; (3-4-3-6)—a triangle, a square, a triangle, and a hexagon; (32-4-6)—two triangles, a square, and a hexagon; (34-5)—four triangles and a pentagon. Received October 22,1964. 169
Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 170 NORMAN W. JOHNSON
The symbol (4-6-8) could also be written (6 • 8 • 4), (4 • 8 • 6), etc. Note, however, that (3-4-3-6) and (32-4-6) are not equivalent, since the two triangles are separated in the one case but adjacent in the other.
2. Uniform polyhedra. Since a uniform polyhedron is completely charac terized by the faces that surround one of its vertices, the vertex-type symbol, without parentheses, may be used as a symbol for the polyhedron (2, pp. 107, 130ff.; 3, p. 394; 7, pp. 56-57). The triangular prism, for example, is 3-42. However, an extension of the Schlàfli symbol devised by Coxeter (3, pp. 394-395; 5, pp. 403-404) reveals more clearly the relationships between polyhedra. In this notation, \m,n} is the regular polyhedron whose faces are {w}'s, n surrounding each vertex, i.e., the polyhedron whose vertices are of type (mn)\
{3, 3} is the tetrahedron) {3, 4} is the octahedron-, {4, 3} is the cube; (3, 5) is the icosahedron; {5, 3} is the dodecahedron. If we let
4ra ,T 2mn l No = ~4A - (m-, - 7W72)0 - ^2 ) ,' Nx 4- (m - 2)0 - 2) '
AT 4^ 2 4 - (m - 2)0 - 2)
(4, p. 13), then {m} n) has N0 vertices, Ni edges, and iV^ faces. m I > is the quasi-regular polyhedron with N2 faces {w}, No faces j n) 2Ni edges (m-n), and iVi vertices (m-n-m-n):
< > is the icosidodecahedron. 15/ t{m,n} has N0 faces {n}, N2 faces {2m}, and 2Ni vertices (n-2nt'2m): t{3, 3} is the truncated tetrahedron; t{3, 4} is the truncated octahedron; t{4, 3} is the truncated cube; t{3, 5} is the truncated icosahedron; t{5, 3} is the truncated dodecahedron. Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 171 r •ft is the rhombicuboctahedron; r< ** > is the rhombicosidodecahedron. t{ > has TVi square faces, iV2 faces {2m}, iV0 faces {2w}, and 4iVi vertices n> (4-2w-2n): t|^| = t{3,4}; Hi}'i s the truncated cuboctahedron] t\ > is the truncated icosidodecahedron. l5j s< > has 27Vi triangular faces, 7V2 faces {w}, 7V0 faces \n}, and 27Vi vertices w (32-ra-3-w): $-18.51; s< / is the snub cuboctahedron; I4j sv / is the snub icosidodecahedron. ! } X \n] has » square faces separating two jwj's, and 2M vertices (42-w): { } X {4} = }4,3}; { } X {n} (n = 3, 5, 6, . . .) is the n-gonal prism. h{ }s{n] has 2n triangular faces separating two {w}'s, and 2n vertices (33-w): h{ }s{3} = {3,4}; h{ }s{n} {n = 4, 5, 6, . . .) is the n-gonal antiprism. (s\ (3l The prefix "rhomb(i)-" in the names for H.? and r< > derives from the fact that the former has 12 square faces whose planes bound a rhombic dodecahedron, the solid dual to < . >, while the latter has 30 squares lying in the face-planes of a rhombic triacontahedron, the dual of < >. Some persons object to the names "truncated cuboctahedron" and "trun- Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 172 NORMAN W. JOHNSON cated icosidodecahedron" on the ground that actual truncations of i . \ or < > would have rectangular faces instead of squares. For this reason 4-6-8 is sometimes called the "great rhombicuboctahedron," with 3-43 then being known as the "small rhombicuboctahedron," and similarly for 4-6-10 and 3-4-5-4 (2, p. 138; 7, p. 94). But this nomenclature is subject to the more serious objection that the words "great" and "small" have an entirely different connotation in connection with star polyhedra, as in the names of the Kepler- Poinsot solids (2, pp. 143-145; 5, p. 410; 7, pp. 83-93). It should also be pointed out that s\ > and s< > are more commonly known as the "snub cube" and the "snub dodecahedron," respectively, the names given them by Kepler. But it is clear that they are related just as closely to the octahedron and the icosahedron, and I have renamed them accordingly (cf. 2, p. 138, or 3, p. 395). It is sometimes useful to consider the prisms and antiprisms as being derived from the fictitious polyhedra {2, n) and < >. Thus, for n > 3, 2 2 2 t{2,n} = r( \ = {} X \n\, t| l = {} X [2n\, s| l = h{js{W}, where, in the case of r< > and s< >, digonal "faces" are to be disregarded (cf. 5, p. 403). Also, t|2J= {4,3} and s^j = {3,3}. All the vertices of a uniform polyhedron are necessarily of the same type. However, the fact that a regular-faced solid has only one type of vertex does not guarantee that the solid is uniform, as is shown by the existence of a non-uniform polyhedron which, like the rhombicuboctahedron, has vertices all of type (3-43). This solid, depicted in Fig. 1, was discovered by J. C. P. FIGURE 1 Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 173 Miller sometime before 1930 (2, p. 137). More recently, Askinuze (1) claimed that it should be counted as a fourteenth Archimedean polyhedron. 3. Cut-and-paste polyhedra. If a uniform polyhedron has a set of non-adjacent edges that form a regular polygon, then it is separated by the plane of this polygon into two pieces, each of which is a convex polyhedron with regular faces. This can be done with the octahedron and the icosahedron, the cuboctahedron and the icosidodecahedron, the rhombicuboctahedron and the rhombicosidodecahedron. In this manner or by using uniform polyhedra and pieces of uniform polyhedra as building blocks, eighty-three non-uniform regular-faced solids can be constructed. An n-gonal pyramid Yn (n = 3, 4, 5) has n triangular faces and one {n}, n vertices (32- n) and one (3W). A triangular pyramid is, of course, a tetrahedron. A square pyramid is half of {3, 4}, and a pentagonal pyramid is part of {3, 5}. An n-gonal cupola Qn (n = 3, 4, 5), obtainable as a fraction of r< >, is a polyhedron having n triangular and n square faces separating an \n] and a {2n}, with 2n vertices (3 • 4 • 2n) and n vertices (3 • 4 • n • 4). A pentagonal rotunda R5, half of < _>, has 10 triangular and 5 pentagonal faces separating a {5} and a {10}, 10 vertices (3-5-10) and 10 vertices (3-5-3-5). A pyramid, cupola, or rotunda is elongated if it is adjoined to an appropriate prism (a pentagonal pyramid to a pentagonal prism, a pentagonal cupola or rotunda to a decagonal prism, etc.) or gyroelongated if it is adjoined to an antiprism. Two pyramids can be put together to form a dipyramid; two cupolas, a bicupola; a cupola and a rotunda, a cupolarotunda; and two rotundas, a birotunda. In the last three cases, the prefix ortho- is used to indicate that one of the two bases is the orthogonal projection of the other (as in a prism) ; the prefix gyro- indicates that one base is turned relative to the other (as in an antiprism). One of these polyhedra is elongated or gyroelongated when the two parts are separated by a prism or an antiprism. Two triangular prisms can be joined to form a gyrobifastigium. Certain uniform polyhedra can be augmented by adjoining other solids to them: square pyramids may be added to an w-gonal prism (n = 3, 5, 6), pentagonal pyramids to {5, 3}, and w-gonal cupolas to t{n, 3} (n = 3, 4, 5). Other uniform polyhedra can be diminished by removing pieces of them— pentagonal pyramids from {3,5}, pentagonal cupolas from r< >. And pentagonal cupolas in r< > can be rotated 36° to produce a gyrate solid. Prefixes bi- and tri- are used where more than one piece is added, subtracted, or twisted. Where there are two different ways of adjoining, removing, or Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 174 NORMAN W. JOHNSON turning a pair of pieces, these are distinguished by the further prefixes para- if the pieces are opposite each other and meta- if they are not. To facilitate the description of regular-faced solids obtained from uniform polyhedra, certain of the latter are given abbreviated symbols: S2= Y3= {3,3} T3 = t{3,3} Q2 p3 == {} X {3} 2 S3 = Y4 ={3,4} T4 = t{4,3} PB == { i X {«} 0»>5 ) P4= {4, 3} T6 = t{5,3} on == hj }s{n] (n >4) i* = 13,5} h 5 r D5={5,3} * \5j Each of the non-uniform regular-faced polyhedra can then be given a concise symbol indicative of its structure. For example, the Miller-Askinuze solid, 2 the elongated square gyrobicupola, is g Q4 Ps- A convex polyhedron with regular faces is elementary if it contains no set of non-adjacent edges forming a regular polygon, i.e., if it cannot be separated by a plane into two smaller convex polyhedra with regular faces. The regular polyhedra {n, 3} (n = 3, 4, 5), nine of the Archimedean polyhedra, all the prisms, and all the antiprisms except h{ }s{3} are elementary. Of the eighty- three non-uniform regular-faced solids obtained from uniform polyhedra, nine are elementary: 3 2 3 Y4, Y5, Q3, Q4, QB, RB, Y5- I5, £-Q5- E5, Q5- E5. The last three solids result from the respective removal of three pentagonal pyramids from an icosahedron, of two opposite pentagonal cupolas from a rhombicosidodecahedron, and of three pentagonal cupolas from a rhomb- 3 icosidodecahedron. The tridiminished icosahedron Y5~ I5 is the vertex figure of a uniform four-dimensional polytope, the snub 24-ce/Z s{3,4, 3} (4, p. 163). All of these elementary polyhedra were listed by Zalgaller (13, pp. 7-8). 4. Other non-uniform polyhedra. Freudenthal and van der Waerden (8) enumerated all the convex polyhedra with congruent regular faces. In 2 addition to the five Platonic solids, these are the triangular dipyramid Y3 , 2 2 the pentagonal dipyramid Y5 , the gyroelongated square dipyramid Y4 S4, the 3 triaugmented triangular prism Y4 P3, and one other figure, which they call a "Siamese dodecahedron.'' Unlike all the polyhedra discussed so far, this last solid cannot be obtained by taking apart or putting together pieces of uniform polyhedra. It is, however, related to a disphenoid—a tetrahedron regarded as a belt of four triangles between two opposite edges—in the same way that s< . > and s< r > are related to < . > and < - f • Consequently, it will be called the snub disphenoid and denoted by the symbol s S2. From the square antiprism there can be derived in a similar Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 175 manner the snub square antiprism s S4, whose faces consist of 24 triangles and 2 squares. (Since S3 = {3,4} = < >,sS3 = s< > = {3, 5}.) Snub disphenoid Snub square antiprism FIGURE 2 Four other convex polyhedra all of whose faces are {3}'s and {4}'s are the sphenocorona V2 N2, the sphenomegacorona V2M2, the hebesphenornegacorona 2 U2 M2, and the disphenocingulum V2 G2. If we define a lune as a complex consisting of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lines. The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune. The suffix -corona refers to a crownlike complex of 8 triangles, and -megacorona, to a larger such complex of 12 triangles. The suffix -cingulum indicates a belt of 12 triangles. Two polyhedra whose faces include pentagons are the bilunabirotunda 2 2 L2 R2 and the triangular hebesphenorotunda U3 R3. The former is regarded as being composed of two lunes and two rotundas—a rotunda here being the complex of faces surrounding a vertex of type (3-5-3-5). The latter is the union of a complex consisting of three lunes separated by a hexagon with Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 176 NORMAN W. JOHNSON Sphenocorona Sphenomegacorona Hebesphenomegacorona Disphenocingulum FIGURE 3 Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 177 triangles attached to its alternate sides and a complex of three triangles and three pentagons surrounding another triangle. Top and bottom, or front and back, views of each of the polyhedra 2 2 2 sS2, sS4, V2N2, V2M2, U2M2, V2 G2, L2 R2 , U3 R3 are shown in Figures 2, 3, and 4. These solids are all elementary, bringing the number of such non-uniform regular-faced polyhedra up to seventeen. In addition, a non-elementary solid, the augmented sphenocorona Y4V2N2, can be formed by placing a pyramid on one of the square faces of V2 N2. Bilunabirotunda Triangular hebesphenorotunda FIGURE 4 5. Symmetry groups. Coxeter and Moser (6, pp. 33-40, 135) have given abstract definitions for each of the finite groups of isometries in Ez. The follow ing is a description of the individual groups, especially as they relate to polyhedra. The bilateral group [ ], of order 2, is generated by a single reflection R, satisfying the relation R2 = E. Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 178 NORMAN W. JOHNSON The kaleidoscopic group [n] ~ Sn (n > 2), of order 2n, is generated by two reflections, Ri and R2l satisfying the relations JRI2 = R22 = (Ri R*Y = E. The cyclic group [n]+ = En (w > 2), of order n, is a subgroup of index 2 in [»]. It is generated by the rotation S = Ri R2} satisfying the relation S*1 = E. When n = 1, the above relations imply that i?i = R2 and that 5 = E; thus the kaleidoscopic group of order 2 is the bilateral group [ ] = 3)i, and the cyclic subgroup is the identity group [ ]+ = (Si. When n = 2, the kaleidoscopic group is the direct product of two bilateral groups: [2] S [ ] X [ ]. For n > 3, [w]+ is the rotation group of a right regular w-gonal pyramid (other than a regular tetrahedron), and [n] is the complete symmetry group (including reflections). For integers m and n, 2 < m < w, (w — 2)(w — 2) < 4, the group [w, w], of order ^x~ 4- (m- 2)(n- 2)' is generated by three reflections, j?i, i^2, Rz, satisfying the relations m 2 Rf = R2* = RS = (R1R2) = (R2R,r = (RiR*) = E. For m = 2, Ni = n; for m = 3, JVi = 6w/(6 — w). The group [m, w]+, of order 2iVi, is a subgroup of index 2 in [ra, w]. It is generated by the rotations S12 = Ri R2 and S23 = ^2 Rz, satisfying the relations Sl2m = 5*23W = (5*12 »523)2 = E. The group [2, 2] ^ [ ] X [ ] X [ ], of order 8, is generated by reflections Ri, R2, R3 in three mutually perpendicular planes. Each reflection generates a subgroup [ ], and each pair of reflections generates a subgroup [2]. Each half-turn about a line of intersection of two planes generates a subgroup [2]+. The rotational subgroup [2, 2]+ ^ [2]+ X [2]+ is generated by any two of the three half-turns. A subgroup [2, 2+] = [ ] X [2]+ is generated by each reflection together with the orthogonal half-turn, e.g., the reflection Ri and the half-turn S23 = ^2 Rz, satisfying the relations 2 2 2 Ri = 523 = (RiS2z) = E. The subgroups [2], [2, 2+], and [2, 2]+ are isomorphic, and the whole group can be obtained by adjoining to any of them one of the missing reflections: [2, 2] ^ [ ] X [2] ^ [ ] X [2, 2+] ^ [ ] X [2, 2]+. Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 179 The rotatory reflection Z = Ri R2 Rz is the central inversion, i.e., the ''reflection" in the point of intersection of the three planes. The subgroup of [2, 2] generated + by Z is the central group [2 , 2+] = S2, defined by Z2 = E. This gives two other ways of expressing [2, 2] as a direct product: [2, 2] ^ [2+, 2+] X [2] ^ [2+, 2+] X [2, 2]+. + Since i?i 523 = Ri ^2 Rz, the group [2, 2 ] also contains the central inversion, and [2, 2+] ^ [2+, 2+] X [ ] ^ [2+, 2+] X [2]+. Note that the three groups [ ], [2]+, and [2+, 2+], respectively generated by a reflection, a half-turn, and the central inversion, are merely three different geometric representations of the single abstract group of order 2, for which S)i and 62 are alternative symbols. The dihedral group [2, n)+ = 3\, of order 2n, and the extended dihedral group [2, n] ^ [ ] X [n] 9* [ ] X [2, n]+, of order 4w, are respectively the rotation group and the complete symmetry group of a regular w-gonal prism (other than a cube) for n > 3. When n is even, the group [2, n] contains the central inversion in the form Z = RxiRiRz)"'2, so that [2, n] ^ [2+, 2+] X M ÊË [2+, 2+] X [2, »]+ (» even). The group [2, n] contains a subgroup of index 2 generated by the reflection Ri and the rotation S2z = R2 Rz, satisfying the relations 2 n R1 = Sn = E, ^i523 = SnRi. This is the extended cyclic group [2, n+) == [ ] X [n]+, of order 2n. The rotational subgroup, generated by S23, is the cyclic group [n}+. When n is + w/2 even, the group [2, n ] contains the central inversion in the form Z = Ri 523 , and [2, n+] ^ [2+, 2+] X [»]+ in even). The group [2, 2w] contains a subgroup of index 2 generated by the half- turn 5i2 = R\ R2 and the reflection Rz, satisfying the relations 2n 5122 = Rz* = (S12Rz) = E. + This is the group [2 , 2n] ~ £>2n, of order 4n, the complete symmetry group of a regular n-gonal antiprism (other than a regular octahedron) for n > 3. + The rotational subgroup, generated by S12 and 7?3 «S^ Rz, is [2, w] . When n is w odd, (5i2 i^3) is the central inversion Z, and Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 180 NORMAN W. JOHNSON [2+, 2w] S [2+, 2+] X [n] S [2+, 2+] X [2, »]+ (w odd). The groups [2+, 2w] and [2, 2n+] have a common subgroup of index 2, a subgroup of index 4 in [2, 2n], generated by the rotatory reflection T = S12R3 = RiS2z = Ri R2 R%, satisfying the relation This is the group [2+, 2n+j = Ê2n» of order 2w. The rotational subgroup, generated by T2, is [w]+. When « is odd, Tn is the central inversion Z, and [2+ 2w+] ^ [2+, 2+] X [n]+ (n odd). The groups [3, 3]+, [3, 4]+, and [3, 5]+, of orders 12, 24, and 60, being the rotation groups of the regular polyhedra, are known as the tetrahedral, octahedral, and icosahedral groups. They are isomorphic to symmetric or alter nating groups of degree 4 or 5 (4, pp. 48-50): [3, 3]+ S 2Ï4, [3, 4]+ s @4, [3, 5]+ ^ SI5. The extended polyhedral groups [3, 3], [3, 4], and [3, 5], of orders 24, 48, and 120, are the complete symmetry groups of the regular polyhedra. The extended tetrahedral group is the symmetric group of degree 4: [3, 3] £* ©4. The extended octahedral group contains the central inversion in the form Z = (i?ii?2^3)3 and is obtained by adjoining this operation to either of the isomorphic groups [3, 3] and [3, 4]+: [3, 4] ££ [2+, 2+] X [3, 3] S [2+, 2+] X [3, 4]+. In the extended icosahedral group the central inversion occurs as Z = (R1R2RZ)\ and [3, 5] ^ [2+, 2+] X [3, 5]+. The group [3, 4] contains a subgroup of index 2 generated by the rotation Su = Ri R2 and the reflection i?3, satisfying the relations 3 2 2 5i2 = i?3 = (Su-'RsSnRz) = E. This is the pyritohedral group [3+, 4], of order 24, the complete symmetry group of the crystallographic pyritohedron or of the figure consisting of a cube inscribed in a regular dodecahedron. The central inversion occurs in the form Z = (S^Rz)*, while the rotational subgroup, generated by S12 and Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. CONVEX POLYHEDRA 181 RzSuRzy is [3, 3]+, so that [3+, 4] ^ [2+, 2+] X [3, 3]+. The pyritohedral group is also a subgroup of index 5 in [3, 5], generated by the rotation Rz Ri R2 Rz and the reflection R2. All the finite three-dimensional symmetry groups are listed in Table I. TABLE I FINITE GROUPS OF ÏSOMETRIES IN E3 Rotation groups Extended groups Group Structure Order Group Structure Order + / [] S)i 2 [] Si 1 §2 2 I [2+,2+] 2w + + W , n>2 £n w [2 ,W 2»+ ] @2n 2w [ [2, K+] 2)1 X + / [2 , 2»] 2>2n 4n [2, »]+, n > 2 £>n 2w I [2, n] 2)1 X S)n 4w / [3,3] ©4 24 [3, 3]+ «4 12 + I [3 , 4] e2 X 2Ï4 24 [3, 4]+ ©4 24 [3,4] S2 X ©4 48 [3, 5]+ 2Ï5 60 [3,5] (Ê2 X 2Ï5 120 Every rotation, other than the identity, that transforms a solid into itself leaves invariant a unique line, called an axis of symmetry of the solid. If the greatest period of any rotation about a given axis is n, the axis is said to be an n-fold one. A solid whose rotation group is the identity has no axes. Other wise, the number of axes of symmetry for each finite rotation group is as follows : [n]+: 1 w-fold; [2, n]+: n twofold, 1 n-fold; [3, 3]+: 3 twofold, 4 threefold; [3, 4]+: 6 twofold, 4 threefold, 3 fourfold; [3, 5]+: 15 twofold, 10 threefold, 6 fivefold. Since these are the only finite rotation groups, it follows that a polyhedron that has more than one threefold, fourfold, or fivefold axis must have tetra- hedral, octahedral, or icosahedral symmetry and that no polyhedron can have more than one w-fold axis for n > 6. Of the convex polyhedra with regular faces, the only ones that have tetra- hedral, octahedral, or icosahedral symmetry are the Platonic and Archimedean Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 182 NORMAN W. JOHNSON solids. The only ones having an axis other than twofold, threefold, fourfold, or fivefold are the n-gonal prisms and antiprisms (n > 6). Thus the rotation group of a non-uniform regular-faced polyhedron is either the identity group or one of the groups [n]+ or [2, n]+ (n = 2, 3, 4, 5). If any of the symmetry operations of a solid are reflections or rotatory reflections, then exactly half of them are, and the rotation group of the solid is a subgroup of index 2 in its complete symmetry group. If not, the rotation group is the whole group, and the solid occurs in two enantiomorphous forms, mirror images of each other. There are seven regular-faced polyhedra of this kind: the Archimedean snub cuboctahedron and snub icosidodecahedron and the non-uniform figures QZ2SQ, QA2SS, Q52SIO, Q5 R5 S10, R.52Sio. On the other hand, four polyhedra have only bilateral symmetry: 2 1 2 m-g Qr'Ei, g Q5- E5, gQ5- E5, Y4V2N2. The complete symmetry group of each of the remaining non-uniform solids is one of the groups [n], [2, w], or [2+, 2n] (n = 2, 3, 4, 5). It is remarkable that none of the known convex polyhedra with regular faces is completely asym metric, i.e., has a symmetry group consisting of the identity alone. The vertices, edges, and faces of any symmetric polyhedron fall into various equivalence classes. Two vertices, edges, or faces belong to the same equivalence class if there is a symmetry operation of the polyhedron that takes one into the other. The order of the equivalence class to which a particular vertex, edge, or face belongs is equal to the index of the subgroup of the complete symmetry group of the polyhedron that leaves the particular vertex, edge, or face invariant. The uniform polyhedra are just those regular-faced solids whose vertices all belong to a single equivalence class. Besides the Platonic solids, the only convex polyhedra with regular faces that have all their edges equivalent are the cuboctahedron and the icosidodecahedron, and the only ones whose faces are all equivalent are the triangular and pentagonal dipyramids. Tables II and III list the different types of faces, edges, and vertices to be found in each convex polyhedron with regular faces. Those edges or vertices that are locally congruent, i.e., edges or vertices of the same type that form equal dihedral angles, are grouped together. In each case the number of faces, edges, or vertices in each equivalence class is indicated in roman type and the number of equivalence classes of the same order in italic type. Dihedral angles are given to the nearest second. Where minutes and seconds are not shown, the given value is exact. Some of the following references were supplied by Branko Griinbaum, for whose interest and encouragement I am most grateful. I am also indebted to the referee for several helpful suggestions. Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 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CO CO CO CO CO CO CO CO, CO, ^ COMOÔWNÔOO ÔCOQOMOÔINONNO ibûOOOOÔNO^M CO ^ CO CO CO (N (N COCO CO CO CO CO CO TtH r^ r*H CO CO CO CO CO CO CO CO TH ^ r^ CO CO CO CO CO CO rfH Tt< ^ COCOCOCOCOCOCO'^1 COCOCOCOCOCO.COCO^CO^COCO COCOCO,COCOCO'^3-S3-^- (M^^H^^(NH Cq co^^ CO + + + > > h o Ou - Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. (M ci *o to 10 »o ^ ^ CO CO ^CO CO 9 ^ ^ îb -^ 10 *? ^ *ô 1* S2- i-s & ss CO, CO, CO CO CO CO CO, £0, ÇO< $S ^ COWCOOONWiC H c^ Q0N 00 Q 00 H © H T)( îO CO H Q H TH CO Q lO iO cococococo^Tt ^HTtt^WN^lN "tf 00 GO ^ "** 00 CQ ^ "tf 00 00 CO ^ »0 CO^^ ^ CO^ 00 ' "^ CM -^ CO CO CO i-l SSI + CM + C+O £ > J T3 S o c o u Je G £ a 42 a cd en a J2 •Ë-S P3 Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use. 200 NORMAN W. JOHNSON REFERENCES 1. V. G. Askinuze, Oïisle polupraviVnyh mnogogrannikov, Mat. Prosvesc, 1 (1957), 107-118. 2. W. W. R. Ball, Mathematical recreations and essays, 11th éd., revised by H. S. M. Coxeter (London, 1939). 3. H. S. M. Coxeter, Regular and semi-regular polytopes, Math. Z., 46 (1940), 380-407. 4. Regular polytopes, 2nd ed. (New York, 1963). 5. H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform polyhedra, Philos. Trans. Roy. Soc. London, Ser. A, 246 (1954), 401-450. 6. H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 14), 2nd ed. (Berlin, 1963). 7. H. M. Cundy and A. P. Rollett, Mathematical models, 2nd printing (London, 1954). 8. H. Freudenthal and B. L. van der Waerden, Over een bewering van Euclides, Simon Stevin, 25 (1947), 115-121. 9. B. Griinbaum and N. W. Johnson, The faces of a regular-faced polyhedron, J. London Math. Soc, 40 (1965), 577-586. 10. T. L. Heath, The thirteen books of Euclid's elements, vol. 3 (London, 1908; New York, 1956). 11. N. W. Johnson, Convex polyhedra with regular faces (preliminary report), Abstract 576-157, Notices Amer. Math. Soc, 7 (1960), 952. 12. J. Kepler, Harmonice Mundi, Opera Omnia, vol. 5 (Frankfurt, 1864), 75-334. 13. V. A. Zalgaller, PraviVnogrannye mnogogranniki, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astron., 18 (1963), No. 7, 5-8. 14. V. A. Zalgaller and others, O praviVnogrannyh mnogogrannikah, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astron., 20 (1965), No. 1, 150-152. Michigan State University Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 21:29:54, subject to the Cambridge Core terms of use.