Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations Gunnar Brinkmann1, Pieter Goetschalckx1, Stan Schein2 1Applied Mathematics, Computer Science and Statistics Ghent University Krijgslaan 281-S9 9000 Ghent, Belgium 2California Nanosystems Institute and Department of Psychology University of California Los Angeles, CA 90095 [email protected], [email protected], [email protected]. (Received ...) Abstract Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug – not knowing about Goldberg’s work – in 1962 in a arXiv:1705.02848v1 [math.CO] 8 May 2017 biological context. In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. In 1971 Coxeter published a survey article that refers to these constructions. Subsequently, the literature often refers to the Goldberg-Coxeter construction. This construction is actually that of Caspar and Klug. Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We will sketch the different approaches and generalize Goldberg’s approach to a systematic one encompassing all local symmetry- preserving operations on polyhedra. Keywords: polyhedra, symmetry preserving, chamber system, Goldberg-Coxeter operation Author’s contributions: All authors contributed critically in ideas and writing. 1 Data accessibility: No data was involved in this research. Competing interests: None. Acknowledgements: The authors want to thank Nico Van Cleemput (Ghent Uni- versity) for helpful discussions. Funding statement: There was no external funding for this project. Introduction In mathematics and chemistry the article by Caspar and Klug [1] is less popular than the ones by Coxeter [2] and Goldberg [3]. A reason might be that the former has a strong biological focus and is vague with regard to mathematical details. Indeed, for details it refers to another paper [4] that is listed as to be submitted in the article but was never completed, as Caspar writes in a later comment [5] on [1]. The most commonly used source for information about the Caspar-Klug construction is the article by Coxeter [2] in which the construction is described in a more formal way and without referring much to the biological context. In this article Coxeter mentions the works of Caspar and Klug, of Buckminster Fuller [6] and of Goldberg, but when describing the method of covering an icosahedron with identical equilateral triangles cut out of the triangular lattice – the construction often cited as the Goldberg-Coxeter operation (see e.g. [7] or [8]) – he explicitly refers to Caspar and Klug. The misunderstanding that this operation is identical to the much earlier proposed operation by Goldberg might have been caused by the sentence in [2]: “Independently of Michael Goldberg, Caspar and Klug proposed the following rule for making a suitable pattern.” But, although the 2-parameter description of the operation is the same, the equilateral triangle used by Caspar and Klug is not even an intermediate step in Goldberg’s approach and is not what Goldberg – who in fact works in the duals, the hexagonal lattice and the dodecahedron – glues onto the dodecahedron. The results are the same (or, to be exact, duals of each other), but the methods to obtain these results differ in essential points. We will now describe the methods of Goldberg [3], of Fuller [6], and of Caspar and Klug [1] in the order in which they were developed. Finally we will show how Goldberg’s approach can be modified to form a general approach to local operations preserving sym- metry. The approach of Goldberg We will describe the approach of Goldberg in more modern language than the one in his article from 1937. This language allows a more formal description and will make generalization easier. In this article we discuss graphs that are finite or infinite, have a finite degree of the vertices, are embedded in the sphere or the Euclidean plane so that all faces have an interior that is homeomorphic to an open disc, and have a boundary for each face that is a simple cycle. We will refer to a plane graph (or polyhedron if the graph 2 0 1 0 0 1 0 1 1 1 1 0 1 0 2 0 1 0 2 0 1 0 1 1 1 1 1 1 2 2 2 0 0 1 0 0 0 0 1 1 1 1 1 1 1 2 1 2 1 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 1 0 Figure 1: The chamber system of the hexagonal tiling. Here and in the remainder of this article dotted lines correspond to edges of the original tiling (not containing the vertex labelled 2), full lines to connections between face centers and edge centers (not containing vertex 0) and dashed lines to connections between face centers and vertices (not containing vertex 1). is 3-connected) if a finite graph is embedded in the Euclidean plane or on the sphere. We will refer to a tiling if it is an infinite tiling of the Euclidean plane. (Other tilings will not be discussed in detail in this article.) As shown in Figure 1 for the example of the hexagonal tiling of the Euclidean plane, we can obtain a barycentric subdivision of such a tiling by labeling each vertex with a 0, by inserting one vertex within each edge and labelling it with a 1, and by placing one vertex in the interior of each face and labelling it with a 2. Then we connect the vertices with label 2 to all of the vertices (0’s and 1’s) in the boundary of the face in which it lies in the cyclic order around the boundary. This procedure can also be used when the boundary of each face is not a simple cycle and vertices occur more than once in the walk around the boundary. In this case the result would be a multigraph. By this method we can obtain a tiling where all faces are triangles with vertices labeled (in clockwise order around the boundary) either 0, 1, 2 (we can call them black) or 2, 1, 0 (white), and each triangle shares only edges with triangles of another colour. We call such 2 2 2 labelled triangles chambers. LetΣ= σ0, σ1, σ2 σ0 = σ1 = σ2 =1 denote the free Coxeter h | i ′ group with generators σ0, σ1, σ2. Now, if we define for 0 i 2 and two chambers t, t ′ ≤ ≤ ′ such that t and t share an edge but do not share the vertex labelled i that σi(t) = t , we have a chamber system. In an abstract way, these chambers can also be defined as elements of the flag space of the polyhedron. The flags are given by triples (v, e, f), so that vertex v is contained in edge e and edge e is contained in face f. See [9] and [10] for more detailed descriptions. The chamber system encodes the combinatorial structure of the polyhedron completely, and the symmetry group of the polyhedron induces an operation on the chamber system. While Caspar and Klug cut out triangular parts that fill complete faces of the trian- gulation (e.g. of the icosahedron) onto which they are glued, Goldberg proposes to cut out smaller triangles and to fill an f-gon with 2f copies of small triangles. Although he applies the method only to pentagons, different from the Caspar-Klug method, his method can be applied to faces of any size. Unfortunately Goldberg’s article [3] has a mistake. The construction only works for 3 v=(3.5,3.5) 1 (0,1) v=(0,0) 2 (1,0) v=(7,0) 0 Figure 2: The triangle used by Goldberg for the parameters (7, 0). achiral icosahedral fullerenes, which have the full icosahedral group of order 120, and not – as he claims – also for chiral fullerenes with only the rotational subgroup of order 60. He argues that for fullerenes (or medial polyhedra as he calls them) that are assembled from 12 congruent patches that are pentagonal in shape we have that “Each pentagonal patch may be divided into ten equivalent (congruent or symmetric) triangular patches.” This statement is true for achiral icosahedral fullerenes but not for pentagonal patches of chiral icosahedral fullerenes. The latter can be divided into five congruent triangular patches or ten triangular patches consisting of two sets of five congruent triangles. The error is surprising because in his article he gives a picture (Figure 3 in [3]) of the triangular patch he claims to be sufficient for the chiral parameters (5, 3), and copies of this triangular patch can obviously not be identified along edges to form a tiling. Goldberg’s method for cutting the patches out of the hexagonal lattice can thus (in a more formal and corrected way) be described as follows: Assume that the regular hexagonal lattice is equipped with a coordinate system so that (0, 0) is in the center of a hexagonal face f, (1, 0) is a vertex v of f and (0, 1) is the vertex of f that is obtained by a 60 degrees rotation of v in counterclockwise direction around the origin.