Structure of the Truncated Icosahedron

Home , Truncated icosahedron

Proc. Nati. Acad. Sci. USA Vol. 91, pp. 11714-11717, November 1994 Mathematics Structure of the truncated icosahedron (such as fullerene or viral coatings) and a 60-element conjugacy class in PSl(2, 11) BERTRAM KOSTANT Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4301 Contributed by Bertram Kostant, June 29, 1994

ABSTRACT The proper symmetry group of a truncated gation) is A-invariant. In effect then, say for fullerene, one icosahedron P is the icosahedral group PS1(2, 5). However, may label each carbon atom by an element of order 11 in knowing the symmetry group is not enough to specify the graph PSl(2, 11) and express the carbon bonds in terms of group structore (e.g., the carbon bonds for fullerene, C6) of P. The multiplication. group PSl(2, 5) is a subgroup of the 660-element group PS1(2, There is apparently a classic precedent relating the icosa- 11). The latter contains a 60-ement coujugacy class, say M, hedral group to PS1(2, 11). It goes back to the discoverer of ofelements oforder 11. I show here thatMexhibits a model for group theory himself. On the night before his life-ending duel, P where the graph structure is expressed group-theoretically. Galois, in a letter to Chevalier, wrote that P51(2, p) cannot For example, the 12 pentagons are the maximal commuting operate transitively on a set of cardinality - p in casep > 11. subsets of M. Such a model creates the opportunity of using Implicit in this statement is certainly the knowledge that this group-based harmonic analysis (e.g., convolution calculus) to is not the case whenp = 11. The 660-element group PSl(2, 11) deal with problems concerning the truncated icosahedron. operates transitively on a set of cardinality 11, and an isotropy subgroup for this action is an icosahedral subgroup A truncated icosahedron is the polyhedron, P, obtained by (see ref. 4, p. 268). cutting offeach ofthe 12 vertices ofan icosahedron, H, when Chung et al. (2) introduced the action of PSl(2, 11) on the each cut is made sufficiently close to the vertex to be 12-element set of vertices of II and recognized that the edge removed. The faces ofthe resulting P are either pentagons (n structure for the icosahedron can be expressed in terms ofthe = 12) or hexagons (n = 20). Up to an isomorphism the cross ratio in the projective line PI1 over the field F11 of 11 structure of P is completely determined by the graph, A, of elements. We also considered an extended action of PSI(2, its vertices V (n = 60) and edges E (n = 90). The symmetry 11) on the 60-element set V of vertices of P. There are two group of II is a 120-element group whose commutator sub- kinds ofedges in E, pentagonal edges (n = 60) and hexagonal groupA is isomorphic to the 60-element alternating group A5. edges (n = 30). An edge is pentagonal if it bounds one of the A group isomorphic toA5 will be referred to as an icosahedral 12 pentagons; otherwise it is hexagonal. While it is true that group. The action of A on H1 induces an action of A on P the action ofPSl(2, 11) no longer preserves all the edges inE, making A a symmetry group of A. As such, V is a principal it does, however, preserve the pentagonal edges. homogeneous space forA so that all the structures above are Returning to the present paper, one knows that a presen- determined as soon as one knows which pairs of elements of tation for an icosahedral group A' is given by a pair of V define E. elements 1 # 4, r E A' such that An encounter with P or A is a commonplace experience. P and A appear, for example, on the surface of a soccer ball. s = 1, 2 = 1, and (4T)3 = 1. More importantly, these structures materialize in the coatings of a large number of viruses (see, e.g., ref. 1). More We will refer to such a presentation as a standard presenta- recently, A has been encountered in the structure of the tion. The following result characterizes truncated icosahedral (relatively) newly recognized and celebrated carbon mole- structures in terms of standard presentations. cule fullerene (C6o). In this case the elements of V are carbon THEOREM 1. Assume that X is a principal homogeneous atoms and the elements of E are carbon-carbon bonds. space for an icosahedral group A'. Let {4, r} C A' be a Given the interesting symmetry group of A, one should standard presentation for A'. Then there is a unique trun- expect that physical or eventually biological applications cated icosahedral structure R on X such thatfor any x E X would flow from the harmonic analysis associated with A. To the twopentagonal edges having x as an endpoint are {x, 4x} a certain extent this has already been the case (see refs. 2 and and{x, 4-1x}, and the unique hexagonal edge having x as an 3). However, to bring to bear the full power of group and endpoint is {x, Tx}. Moreover, any truncated icosahedral representation theory and to do more subtle harmonic anal- structure on any 60-element set X is uniquely (up to inversion ysis, it seems very desirable to have a model of A where the for 4) of this form. In addition, if A is the centralizer elements of V are themselves group elements and where the (necessarily an icosahedral group) ofA' in Perm X, then A edge structure can be expressed group-theoretically. Some- is the icosahedral symmetry group of R. thing like x, y E V defines an edge if x-ly has order 2. It is If we started with A instead of its centralizer A', it follows the purpose of this paper to present such a model. In this easily from Theorem I that there are exactly 60 truncated model V is replaced by one of the two conjugacy classes, icosahedral structures on X for which A is the icosahedral denoted by M, of elements of order 11 in PSl(2, 11). Once an symmetry group. icosahedral subgroup A C PSI(2, 11) is chosen, then one Now let A be an icosahedral subgroup ofPS1(2, 11). (There conjugacy class M is preferred and I will show that there is are two conjugacy classes of such subgroups.) There is then, a distinguished truncated icosahedral structure on Mwhich is when standard presentations for A arising from elements of characterized in the main theorem and which (under conju- order 11 in PS1(2, 11) are used, a preference of one, say M, of the two 60-element conjugacy classes of elements oforder The publication costs ofthis article were defrayed in part by page charge 11. Under conjugation, M is a principal homogeneous space payment. This article must therefore be hereby marked "advertisement" for A. Of the 60 possible A-invariant truncated icosahedral in accordance with 18 U.S.C. §1734 solely to indicate this fact. structures on M, one ofthem is distinguished and is described

11714 Downloaded by guest on September 25, 2021 Mathematics: Kostant Proc. Natl. Acad. Sci. USA 91 (1994) 11715

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3] FIG. 1. Placement of some of the elements of M at the vertices of the truncated icosahedron. in the following and main result. The proofof Theorems I and rather than overburden the diagram (Fig. 1) where this is 2 will be published elsewhere. partially done, I exhibit only 30 of them. More complete THEOREM 2. Let A be any icosahedral subgroup ofPS1(2, information is contained in Table 1. 11). Let A(2) be the set ofall elements oforder 2 in A. Then The two 60-element sets A and M (Tables 2 and 3) are of for any x E PSl(2, 11) of order 11 there exists a unique course subsets of PSl(2, 11). For notational convenience, element ax E A(2) such that the commutator px = x-laxxolx rather than using matrices, I have (going back to Galois' is in A(2). Moreover, there exists a unique choice, CA, of a letter) represented PS1(2, 11) as permutations (in cycle form) (12-element) coqjugacy class ofelements oforder 5 in A and of an 11-element set. The latter we take to be the finite field a unique choice of a (60-element) conjugacy class, M, of F1l. A subgroup of the symmetric group Si, which we can elements of order 11 in PS1(2, 11) such that if x E M and identify with PS1(2, 11) consists ofthose permutations which U E CA normalizes the cyclic groups generated by x and stabilize the set of eleven 5-element subsets of F11 obtained oxxax, then u,px defines a standardpresentation ofA. (That by all 11 additive translations of the set S = {1, 3, 4, 5, 9} of is, upx has order 3.) Furthermore, if x E M, then pxx E M, nonzero squares in F11. (See ref. 5, p. 7, biplane.) Theorem and a truncated icosahedral structure R is defined on M 2 depends upon the choice of an icosahedral subgroup A of where the two pentagonal edges containing x are {x, x3} and PS1(2, 11). For Fig. 1 and Table 1, I have taken (and can take) {X, X-3} and the unique hexagonal edge containing x is {x, A to be the subset of those elements of PSI(2, 11) which PXX}. stabilize S itself. The numbercodes forA (in square brackets) Next, given x E M, an element u E CA satisfying the and M follow Fig. 1 and Table 1. condition above is unique up to inversion and may be fixed Table 1 lists, for each element x E M, its three neighbors by the condition that uxu-1 = x5. As such, write u = u(x). The in M; xr, its unique hexagonal neighbor; and x3 and x-3, its space A of12pentagons is thenparameterized by CA so that, two pentagonal neighbors. There is also an antipodal struc- for u E CA, one has P. E A, where P. = {x E MIu = u(x)}. ture, defined by R, in M. In the table the antipode of x is In addition, the icosahedral graph structure Rt induced on A denoted by x*. Table 1 also lists the elements cr, px E A(2). by R is such that {Pu,, Pj is an edge ifand only ifuv E CA. As stated in Theorem 2 one has that xi = pxx. In addition, one Remark. It is to be noted that the 12 pentagons in M are, can prove that, in reverse order, x* = xpx. Finally, to locate rather elegantly, the 12 sets of maximally commutative pentagons which contain x and its hexagonal neighborxt, the subsets of M. table lists u(x) and u(xr). Forfuture computational purposes it is useful to exhibit, by Fig. 1 exhibits the bottom 6 pentagons ofP. The remaining a diagram, the placement ofthe elements of M at the various 30 elements of M are antipodal to those shown. Also dis- vertices ofthe truncated icosahedron defined by R. Actually, played are the 15 elements ofA(2) and the 12 elements ofCA. Downloaded by guest on September 25, 2021 11716 Mathematics: Kostant Proc. Natl. Acad. Sci. USA 91 (1994)

Table 1. Relevant elements in PSI(2, 11) x x* ax Px U(X) u(xl) XT x3 x-3 x x * ax px u(x) u(xT) XT x3 x-3 1 57 [401 [21] [3] [9] 18 2 5 31 26 [10] [35] [22] [17] 37 32 35 2 56 [39] [28] [3] [41] 24 3 1 32 30 [38] [58] [22] [6] 60 33 31 3 60 [38] [58] [3] [4] 30 4 2 33 29 [50] [40] [22] [20] 52 34 32 4 59 [37] [56] [3] [7] 6 5 3 34 28 [28] [54] [22] [9] 20 35 33 5 58 [35] [11] [3] [8] 12 1 4 35 27 [56] [25] [22] [8] 15 31 34 6 53 [37] [56] [7] [3] 4 7 10 36 24 [39] [28] [17] [6] 56 37 40 7 52 [50] [40] [7] [4] 29 8 6 37 23 [10] [35] [17] [22] 31 38 36 8 51 [11] [10] [7] [15] 42 9 7 38 22 [21] [50] [17] [8] 14 39 37 9 55 [58] [44] [7] [17] 39 10 8 39 21 [58] [44] [17] [7] 9 40 38 10 54 [54] [39] [7] [8] 13 6 9 40 25 [25] [37] [17] [15] 41 36 39 11 50 [44] [38] [8] [9] 19 12 15 41 17 [25] [37] [15] [17] 40 42 45 12 49 [35] [11] [8] [3] 5 13 11 42 16 [11] [10] [15] [7] 8 43 41 13 48 [54] [39] [8] [7] 10 14 12 43 20 [28] [54] [15] [4] 28 44 42 14 47 [21] [50] [8] [17] 38 15 13 44 19 [44] [38] [15] [12] 50 45 43 15 46 [56] [25] [8] [22] 35 11 14 45 18 [40] [21] [15] [6] 57 41 44 16 42 [11] [10] [9] [20] 51 17 20 46 15 [56] [25] [12] [4] 27 47 50 17 41 [25] [37] [9] [41] 25 18 16 47 14 [21] [50] [12] [41] 22 48 46 18 45 [40] [21] [9] [3] 1 19 17 48 13 [54] [39] [12] [20] 54 49 47 19 44 [44] [38] [9] [8] 11 20 18 49 12 [35] [11] [12] [6] 58 50 48 20 43 [28] [54] [9] [22] 34 16 19 50 11 [44] [38] [12] [15] 44 46 49 21 39 [58] [44] [41] [20] 55 22 25 51 8 [11] [10] [20] [9] 16 52 55 22 38 [21] [50] [41] [12] 47 23 21 52 7 [50] [40] [20] [22] 33 53 51 23 37 [10] [35] [41] [4] 26 24 22 53 6 [37] [56] [20] [6] 59 54 52 24 36 [39] [28] [41] [3] 2 25 23 54 10 [54] [39] [20] [12] 48 55 53 25 40 [25] [37] [41] [9] 17 21 24 55 9 [58] [44] [20] [41] 21 51 54 26 31 [10] [35] [4] [41] 23 27 30 56 2 [39] [28] [6] [17] 36 57 60 27 35 [56] [25] [4] [12] 46 28 26 57 1 [40] [21] [6] [15] 45 58 56 28 34 [28] [54] [4] [15] 43 29 27 58 5 [35] [11] [6] [12] 49 59 57 29 33 [50] [40] [4] [7] 7 30 28 59 4 [37] [56] [6] [20] 53 60 58 30 32 [38] [58] [4] [3] 3 26 29 60 3 [38] [58] [6] [22] 32 56 59

Table 2. Icosahedral group A

[1] () [21] (1, 4) (5, 9) (7, 8) (10, 11) [41] (1, 9, 5, 4, 3) (6, 8, 10, 11, 7) [2] (1, 3, 9, 5, 4)(2, 6, 7, 10, 8) [22] (1, 3, 9, 4, 5) (2, 8, 7, 6, 11) [42] (2, 11, 10) (3, 9, 5) (6, 8, 7) [3] (1, 4, 9, 5, 3)(2, 6, 10, 8, 11) [23] (2, 11, 6) (3, 5, 4) (7, 10, 8), [43] (1, 4, 5, 9, 3) (2, 8, 10, 7, 6) [4] (1, 5, 4, 9, 3) (2, 11, 6, 7, 8) [24] (1, 5, 3) (2, 8, 6) (7, 10, 11) [44] (1, 9) (2, 11) (3, 4) (7, 10) [5] (2, 7, 6) (3, 4, 9) (8, 11, 10) [25] (3, 9) (4, 5) (6, 10) (7, 11) [45] (1, 3, 5) (2, 6, 8) (7, 11, 10) [6] (1, 3, 5, 9, 4) (2, 11, 8, 10, 6) [26] (1, 9, 4, 5, 3) (2, 11, 10, 6, 7) [46] (2, 10, 7) (4, 9, 5) (6, 11, 8) [7] (1, 5, 3, 4, 9) (2, 10, 6, 8, 7) [27] (1, 4, 3, 5, 9) (2, 6, 8, 11, 7) [47] (1, 5, 9, 4, 3) (2, 10, 8, 7, 11) [8] (1, 4, 5, 3, 9) (2, 7, 10, 11, 8) [28] (1, 3) (2, 7) (5, 9) (6, 11) [48] (1, 3, 4) (2, 8, 7) (6, 10, 11) [9] (1, 4, 3, 9, 5) (2, 10, 7, 11, 6) [29] (1, 5, 4, 3, 9) (2, 8, 6, 11, 10) [49] (1, 4, 9, 3, 5) (6, 11, 8, 7, 10) [10] (1, 5) (3, 4) (6, 7) (8, 11) [30] (1, 9, 3) (2, 6, 10) (7, 11, 8) [50] (1, 9) (2, 6) (4, 5) (7, 8) [11] (1, 4) (2, 10) (3, 5) (6, 7) [31] (1, 4, 5)(2, 11, 7)(6, 10, 8) [51] (1, 3, 9) (2, 10, 6) (7, 8, 11) [12] (1, 9, 3, 5, 4) (2, 8, 11, 10, 7) [32] (1, 9, 3, 4, 5)(2, 10, 11, 6, 8) [52] (1, 4, 9) (2, 8, 10) (6, 7, 11) [13] (1, 3, 4, 9, 5) (2, 11, 7, 8, 10) [33] (1, 5, 3, 9, 4)(6, 10, 7, 8, 11) [53] (1, 9, 5, 3, 4) (2, 7, 11, 8, 6) [14] (1, 9, 5) (2, 8, 11) (6, 7, 10) [34] (1, 5, 4) (2, 7, 11) (6, 8, 10) [54] (1, 5) (2, 7) (3, 9) (8, 10) [15] (1, 5, 9, 3, 4) (2, 6, 11, 7, 10) [35] (1, 3) (2, 10) (4, 5) (8, 11) [55] (1, 4, 3) (2, 7, 8) (6, 11, 10) [16] (2, 6, 7) (3, 9, 4) (8, 10, 11) [36] (1, 5, 9) (2, 11, 8) (6, 10, 7) [56] (2, 8) (3, 5) (4, 9) (7, 11) [17] (1, 3, 4, 5, 9) (6, 7, 11, 10, 8) [37] (2, 8) (3, 4) (5, 9) (6, 10) [57] (2, 6, 11) (3, 4, 5) (7, 8, 10) [18] (1, 3, 5, 4, 9) (2, 7, 6, 10, 11) [38] (1, 4) (2, 11) (3, 9) (6, 8) [58] (1, 3) (4, 9) (6, 8) (7, 10) [19] (2, 7, 10) (4, 5, 9) (6, 8, 11) [39] (1, 9) (3, 5) (6, 11) (8, 10) [59] (1, 9, 4) (2, 10, 8) (6, 11, 7) [20] (1, 9, 4, 3, 5) (2, 7, 8, 6, 10) [40] (1, 5) (2, 6) (4, 9) (10, 11) [60] (2, 10, 11) (3, 5, 9) (6, 7, 8) Downloaded by guest on September 25, 2021 Mathematics: Kostant Proc. NatL. Acad. Sci. USA 91 (1994) 11717 Table 3. Conjugacy class M in PS1(2, 11) 1 (1,6,8,10,5,2,7,3,11,4,9) 21 (1,4,8,11,10,9,6,2,5,7,3) 41 (1, 6, 9, 3, 5, 7, 2, 10, 4, 11, 8) 2 (1, 10,7,4,6,5,3,9,8,2,11) 22 (1, 11,6,7,4,10,2,3,8,9,5) 42 (1, 3, 2, 11,6,5,10,8,9, 7, 4) 3 (1,4,3,2, 10,6,9,11,7,5,8) 23 (1,7,2,9,11,4,3,5,6,10,8) 43 (1, 11, 10, 7, 3, 6, 8, 4, 2, 5, 9) 4 (1,2,9,5,4,10, 11,8,3,6,7) 24 (1,9,3,10,7,11,5,8,2,4,6) 44 (1, 7, 8, 5, 11, 3, 4, 9, 10, 6, 2) 5 (1,5,11,6,2,4,8,7,9,10,3) 25 (1, 10,5,4,9,7,8,6,3,11,2) 45 (1, 5, 4, 6, 7, 11, 9, 2, 8, 3, 10) 6 (1,2,3,4,5,6,7,8,9,10,11) 26 (1,5,3,7,2,8,4,6,10,9,11) 46 (1, 7, 3, 5, 9, 8, 10, 11, 4, 2, 6) 7 (1,4,7,10,2,5,8,11,3,6,9) 27 (1,7,4,9,5,2,6,11,3,8,10) 47 (1, 5, 10, 2, 7, 9, 11, 6, 3, 8, 4) 8 (1, 10,8,6,4,2, 11,9,7,5,3) 28 (1,9,6,8,7,5, 11, 10,4,2,3) 48 (1, 2, 11, 8, 5, 7, 6, 4, 10, 9, 3) 9 (1,6, 11,5, 10,4,9,3,8,2,7) 29 (1,8, 11,2,9,7, 10,3,6,5,4) 49 (1, 8, 6, 9, 2, 5, 4, 3, 11, 7, 10) 10 (1,5,9,2,6,10,3,7,11,4,8) 30 (1,2,10,5,8,9,3,4,11,7,6) 50 (1, 9, 4, 7, 8, 2, 3, 10, 6, 5, 11) 11 (1,3,10,8,11,4,7,6,5,2,9) 31 (1,4,6,2,11,3,7,10,9,8,5) 51 (1, 10, 11, 9, 6, 3, 5, 4, 2, 8, 7) 12 (1,8,7,2,3,11,6,9,10,4,5) 32 (1,2,7,8,4,11,10,5,6,3,9) 52 (1, 9, 5, 8, 10, 6, 4, 7, 11, 3, 2) 13 (1,2,6,4,8,3,9,5,7,11,10) 33 (1,8,10,3,2,4,5,9,7,11,6) 53 (1, 8, 4, 3, 9, 10, 7, 2, 5, 6, 11) 14 (1,4,9, 11,2,8,5, 10,6,3,7) 34 (1,3,5, 11,8,2,9,6, 10,4,7) 54 (1, 3, 7, 6, 8, 9, 2, 11, 4, 10, 5) 15 (1, 11,5,3,4,2, 10,7,9,8,6) 35 (1, 11,9,4,3,8,6,7,5,2, 10) 55 (1, 6, 2, 10, 3, 8, 11, 5, 7, 9, 4) 16 (1,4,5,10,11,7,3,2,8,9,6) 36 (1,5,8,7,6,3, 10,2,4,11,9) 56 (1, 10, 2, 6, 9, 8, 7, 4, 11, 3, 5) 17 (1, 10,3,9,4,11,2,6,5,7,8) 37 (1,7,10,11,5,6,2,9,8,3,4) 57 (1, 6, 7, 3, 10, 9, 4, 5, 2, 8, 11) 18 (1,9,2,7,10,4,6,8,3,11,5) 38 (1,11,2,3,7,5,9,4, 10,6,8) 58 (1, 3, 4, 8, 6, 10, 5, 11, 7, 9, 2) 19 (1, 7,6, 11,9, 10,8,5,2,4,3) 39 (1,3,9,6, 11,7,4,8,2,5, 10) 59 (1, 8, 5, 9, 3, 6, 11, 2, 4, 10, 7) 20 (1, 11,8,4,7,9,5,3,6,10,2) 40 (1,6,4,5,3,11,8,10,9,7,2) 60 (1, 9, 11, 10, 8, 3, 2, 7, 5, 6, 4)

This display ofelements ofA is such that the number code (in 1. Home, R. W. (1974) Virus Structure (Academic, New York). square brackets) for the element points to the axis on the 2. Chung, F. & Sternberg, S. (1993) Am. Sci. 81, 56-71. 3. Chung, F., Kostant, B. & Sternberg, S. (1994) in Lie Theory and underlying icosahedron H about which the action of the Geometry, Progress in Mathematics, eds. Brylinski, J.-L., element is a rotation. For an element u in CA the arrow points, Brylinski, R., Guillemin, V. & Kac, V. (Birkhauser, Boston), more specifically, to the vertex of II, corresponding to the Vol. 123, pp. 97-126. pentagon 4. Conway, J. H. & Sloane, N. J. A. (1988) Grundlehr. Math. Pu. Wiss. 290. 5. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. & This work was supported in part by National Science Foundation Wilson, R. A. (1985) Atlas ofFinite Groups (Clarendon, Ox- Grant 9307460-DMS. ford). Downloaded by guest on September 25, 2021