## Abstract

The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Archimedean solids, which can be constructed from the Platonic solids by local symmetry-preserving operations (e.g. truncation) on the solid. By contrast, the results of decorations of polyhedra, e.g. by Islamic artists and by Escher, have been interpreted as decorated polyhedra—and not as new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected. In this article, we first sketch and compare the operations of Goldberg, Fuller, Caspar & Klug and Coxeter to construct polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons and all vertices have three neighbours. We point out and correct an error in Goldberg’s construction. In addition, we transform the term *symmetry-preserving* into an exact requirement. This goal, symmetry-preserving, could also be obtained by taking global properties into account, e.g. the symmetry group itself, so we make precise the terms *local* and *operation*. As a result, we can generalize Goldberg’s approach to a systematic one that uses *chamber operations* to encompass all local symmetry-preserving operations on polyhedra.

## 1. Introduction

Polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons, and all vertices have three neighbours, have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene [1], a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of *spherical* viruses [2,3]. For an article discussing symmetry in both contexts—biology and chemistry—see [4]. Parametrized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context [5] and later by Caspar & Klug [3]—not knowing about Goldberg’s work—in 1962 in a biological context. (Some icosahedral viruses follow other rules [6,7].) In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes [8]. In 1971, Coxeter published a survey article [9] that refers to these constructions and describes the construction of Caspar and Klug in detail. Subsequently, the literature often refers to the *Goldberg-Coxeter construction*, although 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.

In mathematics and chemistry, the article by Caspar & Klug [3] is less popular than the ones by Coxeter [9] and Goldberg [5]. 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 that is listed as *to be submitted* but was never completed, as Caspar writes in a later comment [10] on [3].

The most commonly used source for information about the Caspar-Klug construction is the article by Coxeter [9], 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 Fuller [8] and of Goldberg, but when describing the method of covering an icosahedron with identical equilateral triangles cut out of the triangular lattice—the construction later often cited as the Goldberg-Coxeter operation (see e.g. [11] or [12])—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 [9]: *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 [5], of Fuller [8], and of Caspar & Klug [3] 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 that preserve symmetry.

## 2. The approach of Goldberg

We will describe the approach of Goldberg in more modern mathematical 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 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 labelling 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 labelled (in clockwise order around the boundary) either 0,1,2 (we can call these triangles black) or 2,1,0 (white), and each triangle shares only edges with triangles of another colour. We call such labelled triangles *chambers* [13]. Let *σ*_{0},*σ*_{1},*σ*_{2}. 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 *σ*_{i}(*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 [13,14] 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 triangulation (e.g. of the icosahedron) onto which they are glued, Goldberg proposes to cut out smaller triangles and to fill an *f*-gon with 2*f* 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 [5] has a mistake. The construction only works for 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 [5]) 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 centre 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^{°} rotation of *v* in counterclockwise direction around the origin. By induction it can be easily proved that a vertex with integer coordinates (*a*,*b*) is the centre of a face if *a*–*b*≡0(mod 3), and a vertex otherwise. Or, to be more precise, if *a*–*b*≢0 (mod 3), each vertex is either the left or the right vertex of an edge parallel to the vector (1,0). If *a*–*b*≡1 (mod 3), it is the left vertex. If *a*–*b*≡2 (mod 3), it is a right vertex. If *a* and *b* are not both integer but are both multiples of *a*,*b*) is the centre of an edge if 2(*a*–*b*)≡0 (mod 3).

For given parameters *l*≥*m*≥0, choose *v*_{2}=(0,0), *v*_{1}=((*l*−*m*)/2,(*l*+2*m*)/2) and *v*_{0}=(*l*,*m*). The point *v*_{1} is the midpoint between *v*_{0} and the image of *v*_{0} when rotating it by 60^{°} in the counterclockwise direction around the origin *v*_{2}. See figure 2 for an example. The resulting triangle is always a right triangle with a 30^{°} angle at *v*_{2}, a 90^{°} angle at *v*_{1} and a 60^{°} angle at *v*_{0}. While *v*_{2} is always the center of a face, *v*_{0} can be a vertex or the center of a face. For *v*_{1} we have that (*l*−*m*)/2−(*l*+2*m*)/2=−3*m*/2. Both coordinates of *v*_{1} are only integer if *l* and *m* are both even. In this case, *v*_{1} is the centre of a face. Otherwise 2(−3*m*/2)=−3*m*≡0(mod 3), hence in that case, *v*_{1} is the centre of an edge.

Although he used different words, Goldberg proposed to glue a copy of the interior of this triangle *v*_{0},*v*_{1},*v*_{2}, which we will call a *Goldberg right triangle*, into each chamber of a polyhedron—that is, to decorate the polyhedron—in such a way that the vertices *v*_{0},*v*_{1},*v*_{2} are identified with the corresponding vertices of the chambers. This construction glues each side of a triangle against the same side in the mirror image of a copy of the triangle. If the sides in the hexagonal lattice are part of a mirror axis, the edges nicely match to form a tiling of the sphere because they match in the hexagonal tiling. If they are not, edges do not necessarily match. The side *v*_{2},*v*_{0} has direction (*l*,*m*), which is part of a mirror axis through the origin for *lm* if and only if *m*=0 or *l*=*m*. If *m*=0, the side *v*_{2},*v*_{1} has direction (*l*/2,*l*/2) and starts at a face centre, and the side *v*_{1},*v*_{0} has direction (*l*/2,−*l*/2) starting at a face centre (*l* even) or the centre of an edge parallel to this direction. If *l*=*m*, the side *v*_{2},*v*_{1} has direction (0,3*l*/2) starting at a face centre, and side *v*_{0},*v*_{1} has direction (−*l*,*l*/2) also starting at a face centre. In each case, all three sides of the triangle are mirror axes.

When applied to a polyhedron or tiling, new faces not containing a corner point of the subdivided chamber are hexagons as in the lattice, and vertices not forming a corner point remain 3-valent. At *v*_{1} four copies are glued together in the same way as in the hexagonal lattice, so the local situation is exactly like that in the hexagonal lattice. As the dodecahedron is 3-valent, at *v*_{0} six copies are glued together, just as they are situated in the hexagonal lattice, so this gluing process will give a 3-valent vertex if *v*_{0} is a vertex and a hexagon if *v*_{0} is the centre of a face. Only at *v*_{2} can face sizes differ from 6: In the resulting polyhedron, the vertex *v*_{2} will be the centre of a face of the same size as before, thus a pentagon in the case of decorating the dodecahedron.

If *l*≠*m* and *m*>0, the result of the gluing process can give disconnected graphs or double edges, because the right triangles are glued to parts that are different from the parts they are glued to in the hexagonal lattice. For *l*≠*m* and *m*>0, the sides are not mirror axes, so the contents of the congruent triangles on the two sides of each edge of each right triangle are different (figure 3). To this end, in addition to *v*_{0},*v*_{1} and *v*_{2}, another vertex *v*′_{0} with coordinates (−*b*,*a*+*b*) must be used. We mark the triangle *v*′_{0},*v*_{2},*v*_{1} black, whereas we mark the triangle *v*_{0},*v*_{1},*v*_{2} white. Gluing copies of the black triangles to all black triangles of the chamber system of the dodecahedron and copies of the white triangles to all white triangles of the chamber system of the dodecahedron produces the desired icosahedral fullerene. The mirror symmetries of the dodecahedron are lost, as the content of black and white triangles cannot be mapped onto each other by a reflection, but all orientation-preserving symmetries remain.

The fact that *v*_{1} is a centre of a twofold rotation, that *v*_{2} is a centre of a sixfold rotation, and that *v*_{0} and *v*′_{0} are centres of threefold rotations ensures that the triangles match up. As each *n*-gon contains *n* black and *n* white triangles that can be paired, it is not necessary to distinguish between the white and the black triangles, but the larger quadrangle *v*_{2},*v*_{0},*v*_{1},*v*′_{0} (that has in fact a equilateral triangular shape due to the angle of 180^{°} at *v*_{1}), containing a black right triangle and a white right triangle, can be used to decorate the paired chambers in the polyhedron.

For a more formal way to describe the combinatorial structure of the resulting fullerene or for implementing the operations in a computer program, it is better to switch to chamber systems completely. The symmetry group operates on the chambers of the hexagonal tiling, and the barycentric subdivision can be chosen in a way that complete chambers are contained in the Goldberg right triangle. This way the gluing of the Goldberg right triangle comes down to subdividing a chamber of the dodecahedron into smaller chambers. In the chiral case, the set of chambers of the hexagonal tiling can be chosen in various ways as fundamental domains of the group generated by the rotations required for the sides of the triangles. For a given barycentric subdivision, it is possible that when drawing the triangle with straight lines chambers are intersected and only partially contained in the triangle. The set of chambers must be chosen in a way that the boundary is a simple cycle through *v*_{0},*v*_{1},*v*′_{0},*v*_{2}, so that the path from *v*_{1} to *v*_{0} is the image of the path from *v*_{1} to *v*′_{0} under a rotation of 180^{°} around *v*_{1} and that the path from *v*_{2} to *v*′_{0} is the image of the path from *v*_{2} to *v*_{0} under a rotation of 60^{°} around *v*_{2}. Figure 4 gives an example for parameters (5,3).

Details on operations not necessarily preserving orientation reversing symmetries will be given in a later article—a first sketch is given as electronic supplementary material.

## 3. The approach of Buckminster fuller

The first geodesic dome is believed to be the provisional dome of the Zeiss-Planetarium on the roof of the Zeiss factory in Jena around 1924. The structure was designed by Walter Bauersfeld, an optical engineer who worked for Zeiss, and was patented and constructed by the firm of Dykerhoff and Wydmann. It was already based on a subdivided icosahedron in order to get close to a round surface suitable for the projection of the stars.

Fuller reinvented this idea about 20 years later and popularized it in the USA. Nowadays, it is mainly his name that is associated with geodesic domes. Refinements of the icosahedron and the cuboctahedron, the latter that he called *vector equilibrium* or *Dymaxion* (figure 5), were also used for the early version of his *Dymaxion map*—a map of the world, projected onto the surface of an inscribed polyhedron.

As his work preceded that of Caspar and Klug, it is of course interesting to know whether, for his refinements of the icosahedral and Dymaxion structures, he had already used the methods of Caspar and Klug based on the corresponding Euclidean lattices. In [8], p. 50 Fuller and Marks write as follows:

*The geometric principles underlying the Dymaxion map are the same as those used to develop the basic pattern of Fuller’s domes.*

and

*To produce Fuller’s Dymaxion map, we reverse this process. We start with a sphere, on whose surface a spherical icosahedron has been drawn. Next we subtriangulate the icosahedron’s 20 triangular faces with symmetric, three way, great circle grids of a chosen frequency. Then we transfer this figure’s configuration of points to the faces of an ordinary (non-spherical) icosahedron which has been symmetrically subtriangulated in frequency of modular subdivision corresponding to the frequency of the spherical icosahedron’s subdivisions.*

So Fuller’s construction was less general. He *subtriangulated* the triangles without referring to the triangular lattice and without the possibility to produce chiral structures where some smaller triangles would be cut by the edges of the original icosahedron in a way producing quadrangles.

## 4. The approach of Caspar and Klug

Caspar and Klug use the Euclidean triangular lattice equipped with a coordinate system, so that—with the origin in a vertex—the points (0,1) and (1,0) are on vertices neighbouring the origin, and (0,1) can be obtained from (1,0) by a 60^{°} rotation in counterclockwise direction around the origin. In this lattice the vertices are exactly the points with both coordinates integers (figure 6). Note that when we use the coordinate system of Goldberg that is defined in the dual hexagonal lattice (figures 2 and 3), the length of the unit vectors used by Goldberg is shorter by a factor of

Caspar and Klug propose to look at the triangle formed by the vertices *v*_{1}=(0,0), *v*_{2}=(*a*,*b*) and *v*_{3}=(−*b*,*a*+*b*) for integer values *a*≥*b*≥0. This triangle has all corners in vertices, and vertices of the triangular lattice are centres of rotational symmetries of the tiling by 60^{°}. The product of two rotations by 60^{°} in clockwise order—first around *v*_{1}, then around *v*_{2}—is a rotation by 120^{°} around the centre of the chosen triangle, proving that this rotation is a symmetry of the triangle, its interior, and the tiling.

These observations show that however we join identical copies of this Caspar-Klug triangle by identifying the copies pairwise along a side to form a closed structure, all faces will be triangles and all vertices that are not a corner of a Caspar-Klug triangle will have degree 6. If the triangles are glued into a triangulation of some surface, the degrees of the vertices of the original triangulation stay the same.

Automorphisms of a triangulation of the sphere map triangles to triangles. As the Caspar-Klug triangles always have the full set of orientation-preserving symmetries of the triangle, all orientation-preserving symmetries can be extended to the interior of the triangles. In other words, each orientation-preserving automorphism of the triangulation is also an automorphism of the decorated triangulation.

If (and only if) the parameters (*a*,*b*) fulfil *b*=0 or *a*=*b*, the Caspar-Klug triangles also have mirror axes perpendicular to the midpoints of the edges. In this case, the Caspar-Klug triangles have the full symmetry group of the triangle, and all automorphisms of a triangulation of the sphere that is decorated can be extended to the decorated triangulation, so that the whole automorphism group is preserved.

## 5. A general approach to local symmetry-preserving operations on polyhedra

The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Platonic solids and the Archimedean solids that can be constructed from the Platonic solids by simple operations that preserve the symmetry of the original object. When he rediscovered the Archimedean solids in his book *Harmonices Mundi* [15], Johannes Kepler coined the names now used for the Archimedean solids. The name *truncated octahedron* shows clearly that he considered this polyhedron to be constructed by truncation—a symmetry-preserving operation—from the octahedron.

Although these operations are often used and well-studied in mathematics, there currently exists no systematic way to describe them. There exists an extensive naming scheme using terms such as *ambo*, *kis*, *truncate*, *cantellate*, *runcinate* etc., and for several subclasses there exist different techniques to describe them, e.g. the Conway polyhedron notation [16], Ch. 21, Schläfli symbols [17] or Wythoff symbols [18]. Nevertheless, there is no definition of *local symmetry-preserving operations* and therefore also no technique to systematically describe all possible local symmetry-preserving operations or theorems about what local symmetry-preserving operations can do or cannot do. A list of popular operations can also be found on the Wikipedia site for the Conway polyhedron notation.

In addition to these mathematically motivated operations that produce polyhedra from polyhedra, there is a long tradition in art and design of decorating polyhedra and other objects with parts of periodic tilings. Most famous are probably the *Sphere with Fish* and the *Sphere with Angels and Devils* by M.C. Escher. For more information and examples see [19–21].

While operations on polyhedra were mainly interpreted as manipulating the solid object (e.g. truncation as cutting off vertices and thereby producing new faces), the results of the decorations were interpreted as decorated polyhedra—and not just new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected.

The term *symmetry-preserving* can easily be transformed into an exact requirement: the symmetry group of the polyhedron to which the operation is applied must be a subgroup of the symmetry group of the polyhedron that is the result of the operation. In most cases it will be the whole symmetry group of the result. This goal could also be obtained by taking global properties into account—e.g. the symmetry group itself—so the terms *local* and *operation* are still to be made precise. The classical operations like *truncation* are obviously what must be included in the definition, so checking that all classic operations are covered by the definition is a first test of the usefulness of the following definition.

### Definition 5.1

Let *T* be a periodic 3-connected tiling of the Euclidean plane with chamber system *C*_{T} that is given by a barycentric subdivision that is invariant under the symmetries of *T*. Let *v*_{0},*v*_{1},*v*_{2} be points in the Euclidean plane so that, for 0≤*i*<*j*≤2, the line *L*_{i,j} through *v*_{i} and *v*_{j} is a mirror axis of the tiling.

If the angle between *L*_{0,1} and *L*_{2,1} is 90^{°}, the angle between *L*_{2,1} and *L*_{2,0} is 30^{°} and, consequently, the angle between *L*_{0,1} and *L*_{0,2} is 60^{°}, then the triangle *v*_{0},*v*_{1},*v*_{2} subdivided into chambers as given by *C*_{T} and the corners *v*_{0},*v*_{1},*v*_{2} labelled with their names *v*_{0},*v*_{1},*v*_{2} is called a local symmetry-preserving operation, *lsp operation* for short.

The result of applying an lsp operation *O* to a tiling or polyhedron *P* is given by subdividing each chamber *C* of the chamber system *C*_{P} of *P* with *O* by identifying, for 0≤*i*≤2, the vertices of *O* labelled *v*_{i} with the vertices labelled *i* in *C*. Note that this operation is purely combinatorial and that the angles of *O* are not preserved.

The choice of 30^{°} and 60^{°} is a reference to Goldberg’s paper. For people familiar with tilings and Delaney-Dress symbols as in [13], it is immediately clear that one could also interchange the values, require 45^{°} for both or even choose other values and go to hyperbolic or spherical tilings and still get the same set of operations. For example, if one would use 45^{°} for both angles, the dual would come from the regular square tiling but would be the same combinatorial operation. As another example, figure 7 shows the tiling from which truncation can be obtained with two angles of 45^{°}.

An alternative way to understand that different choices of angles give the same set of operations is as follows: Let *T*_{0} be a tiling of the sphere or the Euclidean or hyperbolic plane where the symmetry group acts transitively on the chambers. Then any lsp operation as defined in definition 5.1 can be applied to *T*_{0} to give another tiling *T*_{1} from which the operation can be recovered by requiring the set of angles as in the chambers of *T*_{0}.

The symmetry group of the tiling *P* is an automorphism of the chamber system *C*_{P}. This implies immediately that it is also an automorphism of the result *O*(*P*), justifying the term *symmetry-preserving operation*. The fact that operations are defined on the level of chambers justifies the term *local*. Nevertheless, one must ask whether all operations one wants to call *local symmetry-preserving operation* are covered. We motivate our definition by showing that all well-known and often used operations on polyhedra are covered by our definition. In figure 8, we give the chamber operations for the operations *identity*, *dual*, *ambo*, *truncate*, *chamfer* and *quinto*. Other operations (such as *join*, *kis*, *expand*, *ortho*, *bevel*, *meta*, etc.) can be written as products of these operations. These operations are basic to the field. For example, the *dual* of the dodecahedron is the icosahedron. The *ambo* of the cube is the cuboctahedron. *Truncation* of the icosahedron produces the truncated icosahedron. And, *chamfering* of a dodecahedron produces the icosahedral fullerene with 80 vertices. People who are not familiar with these operations (or at least their names) can take their representations as lsp operations as the definition.

Though initially described as purely geometrical operations, lsp operations have also been studied combinatorially (see e.g. [22]) and have been described as subdivisions of chambers of the tiling or polyhedron. Figure 7 in [22] describes how the chamfering operation can be implemented as a chamber operation and can be applied to a map given as an abstract chamber system. Representing operations on polyhedra as chamber operations means that the operation is given and coded or described as a chamber operation, so the operation defines the decomposition of the chamber. Definition 5.1 establishes the reverse direction: it says which decompositions of chambers into smaller chambers define an operation.

This characterization is necessary as not every subdivision of a chamber defines an operation that transforms a chamber system into another chamber system of a tiling or polyhedron—see figure 9 for an easy counterexample. One of the properties of chamber systems of tilings or polyhedra is that each 1-vertex is contained in exactly four chambers. Applying the subdivision in figure 9 to a tiling or polyhedron, the result would not have this property.

While in [3,5] it is simply assumed that the result of applying the construction to a dodecahedron or an icosahedron is again a polyhedron, this assumption should be proved in general. To justify our approach to lsp operations, we have to prove that after applying such an operation to a polyhedron, the result is again a polyhedron. The theorem of Steinitz allows translation of this requirement to a purely combinatorial theorem:

### Theorem 5.2

*If P is a polyhedron and O an lsp operation, then O(P) is a polyhedron.*

### Proof.

The fact that the resulting graph is plane is obvious. The fact that it is 3-connected needs some more work. Let *G* be a plane graph or tiling and *B*_{G} be the barycentric subdivision of *G*.

It is well known and easy to prove that *G* is 3-connected if and only if *B*_{G} has the following properties:

(i)

*B*_{G}is a simple graph—that is: there are no cycles of length 2.(ii) The only cycles in

*B*_{G}of length 3 are boundaries of chambers.(iii) The only cycles in

*B*_{G}of length 4 have on one side two triangles sharing an edge or four triangles sharing a 1-vertex.

As the barycentric subdivision *B*_{O(P)} is a subdivision of *B*_{P}, for each edge *e* of *B*_{O(P)} there is a chamber *C*(*e*) of *B*_{P} containing it. If the edge is included in the boundary of a chamber, *C*(*e*) is not unique. To prove that *O*(*P*) is 3-connected, we have to prove that *B*_{O(P)} has properties (i)–(iii).

Figures 10 and 11 give all possibilities for a 4-cycle in *B*_{P} together with all chambers containing edges of the cycles. If vertices displayed in these figures as distinct vertices would represent the same vertex in *B*_{P}, this would contradict at least one of the properties (i)–(iii), so all vertices shown are pairwise distinct. Figures 10 and 11 also show corresponding areas in the tiling for an lsp operation. From the definition of an operation (that is: the requirement for the location of mirror planes), it follows that all vertices in the displayed area are pairwise distinct. After the decoration, there is an isomorphism from the vertices of the barycentric subdivision *B*_{O(P)} inside the drawn area onto the vertices of the barycentric subdivision of the tiling in the drawn area.

For a cycle *e*_{0},…,*e*_{k} in *B*_{O(P)}, let *C*(*e*_{0}),…,*C*(*e*_{k})=*C*_{0},…,*C*_{k} be the cyclic sequence of chambers of *B*_{P}, so that, for 0≤*i*≤*k*, edge *e*_{i} is contained in *C*(*e*_{i}). As long as the sequence of chambers has more than one chamber and we have *C*_{i}=*C*_{i+1} for some *i* with the indices taken modulo *k*+1, we can remove *C*_{i} (and renumber the chambers). This way we get the reduced sequence *C*_{0},…,*C*_{m}. Let now *e*_{0},…,*e*_{k} be a cycle in *B*_{O(P)} with length at most 4. The reduced sequence *C*_{0},…,*C*_{m} also has length *m*+1≤4. If *m*+1≤2, the cycle is completely contained in one chamber or in two chambers sharing an edge. As the image of *e*_{0},…,*e*_{k} under *ϕ* is a cycle of the same length, it must satisfy the requirements described in (i)–(iii), hence also *e*_{0},…,*e*_{k} must satisfy these requirements.

Assume now that *m*+1≥3. Depending on whether *e*_{0},…,*e*_{k} crosses the border between two chambers in a vertex or an edge, for 0≤*i*≤*m* the chambers *C*_{i} and *C*_{i+1} (with the indices modulo *m*+1) share one or two vertices. Furthermore, each *C*_{i} shares in total at least two vertices with the others. Checking the different configurations of three or four chambers in a chamber system that have these properties, one can prove that there is a cycle *Z* of length *m*+1≥3 in *B*_{P} with an edge in each of *C*_{0},…,*C*_{m}.

If *m*=2, *Z* is the boundary of a chamber in *B*_{P}. This implies that the cycle *e*_{0},…,*e*_{k} lies completely in the region described by a chamber *C* of *B*_{P} and the chambers sharing an edge with *C*, so that again the cycle together with its interior can be mapped into the tiling by an isomorphism.

If *m*=3, *Z* must either be the boundary of the union of two chambers sharing an edge or the boundary of the union of four chambers sharing a 1-vertex. In each case, we can map the interior of *Z* and the decoration of chambers neighbouring *Z* isomorphically into the tiling, which implies that *e*_{0},…,*e*_{4} satisfies property (iii).

Together these observations imply that *B*_{O(P)} fulflls (i)–(iii) and that *O*(*P*) is 3-connected. □

### (a) A first step towards general local operations preserving orientation-preserving symmetries but not necessarily orientation-reversing automorphisms

The emphasis of this paper is on the work of Goldberg, Fuller, Caspar & Klug, and on operations preserving all symmetries. General operations that preserve orientation-preserving automorphisms but not necessarily orientation-reversing automorphisms can be described in a similar way. In the case of Goldberg, these operations could be described by two subdivisions—one of a black triangle and one of a white. However, such a description is not always possible, and it is necessary to describe how a quadrangle containing a white and a black triangle that share an edge must be subdivided. Details will be presented in a later paper, but a first sketch is provided as electronic supplementary material.

## 6. Conclusion and future work

We have described the differences between the approaches of Goldberg, Fuller and Caspar and Klug. We have also made clear that what is generally referred to as the Goldberg-Coxeter operation is due to Caspar and Klug. Furthermore, we have explained the error in Goldberg’s paper and how it can be corrected.

By exactly defining—based on Goldberg’s approach—local symmetry-preserving operations (lsp), it becomes possible not only to study specific operations but also to study the classes of all such operations. This definition makes it possible to design general proofs instead of examining each operation individually (theorem 5.2 is an example for this).

The results presented in this paper also open new fields for investigation. Here are some examples:

### Definition 6.1

Two operations *O*, *O*′ are called equivalent if, for all polyhedra *P*, the results of applying *O* and *O*′ to *P* are isomorphic.

While it is immediately evident that there are non-equivalent operations *O*, *O*′ that produce isomorphic results for some polyhedra and non-isomorphic results for others (see, for example, the operations *identity* and *dual* applied to self-dual versus not self-dual polyhedra), it is not clear how exceptional these cases are.

Thus we have the following question: Are self-dual polyhedra the only polyhedra for which the application of non-equivalent operations can give isomorphic results?

This last question seems to be related to another one: lsp operations preserve the symmetry group of the polyhedron to which they are applied. In some cases they can also add new symmetries. An example is the operation *ambo* applied to self-dual polyhedra. By studying lsp operations applied to tilings of the torus, it is easy to see that there exist operations that can increase symmetry but cannot be written as a product of other operations involving *ambo*. Consider, for example, a 4-regular toroidal tiling with 4-gons with automorphism group *Z*_{n}×*Z*_{n}. If we apply the lsp operation that subdivides each 4-gon into nine smaller 4-gons in a 3×3 pattern, we get the automorphism group *Z*_{3n}×*Z*_{3n}. The number of chambers grows by a factor of 9, while for all products with *ambo* the numbers of chambers grow by an even factor—so the operation cannot be written as a product of ambo and other operations. Nevertheless, such examples are not known for polyhedra.

Thus we have the following question: Can all lsp operations that can increase the number of symmetries of a *polyhedron* be written as a product of operations involving *ambo*?

To measure the impact of an operation *O* on the size or complexity of a polyhedron, the ratios of the numbers of vertices or faces before and after applying the operation are not good measures, as they do not depend on the operation alone. This can already be seen at the example of the operation *dual*. Focusing on the edges of the polyhedra, we get such an invariant: the inflation rate *g*(*O*), defined as the number of edges after applying the operation to a polyhedron divided by the number of edges before the operation, is an invariant of the operation. It is equal to the number we would get by counting chambers before and after the operation or by counting the number of chambers in an lsp operation *O*.

Thus we have the following task: Develop a computer program that can generate all non-equivalent lsp operations for a given inflation rate.

Of course one could also ask for operations with a given inflation rate that produce only polyhedra with a given degree of the vertices or given face sizes.

Symmetric polyhedra and periodic tilings can be seen as containing redundant information, as all the information is given by the structure of the fundamental domain and the symmetry group distributing this information. This interpretation would not describe polyhedra with a trivial symmetry group as redundant, but the situation is very similar to that of subdivisions of chambers in the role of the fundamental domain and polyhedra in the role of the group: when we apply operations with a large inflation factor to polyhedra with a trivial symmetry group, in most cases the product will have a trivial symmetry group too and the information of the operation—the subdivided chamber—is distributed via the chamber system of the polyhedron.

Thus we have the following task: Develop an efficient algorithm that can determine whether a given polyhedron can be described as an operation with inflation factor *g*>1 on a smaller polyhedron.

## Data accessibility

This article has no additional data.

## Authors' contributions

All authors contributed critically in ideas and writing.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

The authors want to thank Nico Van Cleemput (Ghent University) for helpful discussions.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3900511.

- Received April 12, 2017.
- Accepted September 20, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.