TOPOLOGY AND COMBINATORICS OF SOCCER BALLS

D. KOTSCHICK

With the quadrennial soccer World Cup taking place again in June andJuly of 2006, more than a billion people around the world are finding theirTV and computer screens covered with depictions of soccer balls. Althoughthere is now a large number of different designs for soccer balls, these dif-ferent designs usually do not reflect the way the ball is actually put together,but are just painted on. In the vast majority of cases, the underlying ballis stitched or glued together in the classical way from 12 pentagons and20 hexagons, arranged so that every pentagon is surrounded by hexagons.Postmodern paint jobs notwithstanding, the traditional way is to paint thepentagons black and the hexagons white. The resulting image is ubiquitous,particularly in Europe, not only during the World Cup, as it is used to pro-mote all kinds of merchandise, not all of it soccer-related. This traditionalsoccer ball pattern also arises in chemistry as the spatial structure of thefullerene C60, and it was used by the architect Buckminster Fuller for theconstruction of cupolas and domes, now referred to as buckyball domes.

So why does the standard soccer ball look the way it does? Are thereother ways of putting it together? Perhaps pentagons and hexagons couldbe arranged differently. Perhaps other polygons could be used instead ofpentagons and hexagons. These questions can be tackled using the lan-guage of mathematics, particularly geometry, group theory, topology andgraph theory. Each of these subjects provides concepts and a natural con-text for phrasing questions such as those about the design of soccer balls,and sometimes for answering them as well.

SOCCER BALLS AND FULLERENES

A convex polyhedron in three-dimensional space, like the cube or thetetrahedron, can always be inflated to a sphere. The vertices and edges ofthe polyhedron then trace out a graph on the sphere, and the combinatoricsof this graph reflects some of the structure of the polyhedron. What is lost isthe geometric information about angles and side lengths. In this way poly-hedra can be thought of as objects of graph theory or topology, which is of-ten described as “rubber-sheet geometry”, precisely because it concentrates

Date: December 13, 2006; c© D. Kotschick 2006.1

2 D. KOTSCHICK

on properties of objects that are unchanged by continuous deformations,like the inflation of a soccer ball.

In this article we discuss soccer balls in the context of graph theory andtopology. For the purposes of this discussion a soccer ball is defined to be aspherical polyhedron consisting of pentagons and hexagons, satisfying thefollowing conditions:

(1) the sides of each pentagon meet only hexagons, and(2) the sides of each hexagon alternately meet pentagons and hexagons.

It is important that there are no geometric constraints, for example the an-gles and side lengths of the pentagons and hexagons are not specified, be-cause in terms of graph theory and topology these concepts cannot even beformulated, as they are not preserved by continuous deformation.

If one thinks of the pentagons painted black and the hexagons paintedwhite, these conditions do capture the familiar image—but they do not de-termine it uniquely. It turns out that there are infinitely many polyhedrasatisfying these conditions. A recent paper [1] exhibits a complete descrip-tion of the infinite variety of soccer balls.

The carbon molecules referred to as fullerenes are also spherical poly-hedra consisting of pentagons and hexagons, with the vertices occupied bycarbon atoms and the edges corresponding to chemical bonds. Fullerenesmust satisfy a different constraint instead of (1) and (2) above, in that theyare required to have precisely three edges meeting at every vertex. Some-times condition (1) is imposed as an additional constraint, to define a re-stricted class of fullerenes. Having disjoint pentagons is expected to berelated to the chemical stability of fullerenes.

The standard soccer ball design with 12 pentagons and 20 hexagons isalso a fullerene. Placing carbon atoms at its vertices, one obtains the C60

molecule, whose discovery was honored by the 1996 Nobel Prize for chem-istry. It is quite remarkable that although there is an infinite number of soc-cer balls and an infinite number of fullerenes, the two families of polyhedrahave only the standard soccer ball in common.

To see that this is so, one has to delve into the analysis of polyhedra usingEuler’s formula. This formula, found by Swiss mathematician LeonhardEuler in the mid-18th century, is a basic tool in graph theory and topology.For a spherical polyhedron with v vertices, e edges and f faces, Euler’sformula reads

v − e + f = 2 .

We want to apply Euler’s formula to a polyhedron consisting of b blackpentagons and w white hexagons. The total number of faces is f = b + w.Two faces meet along every edge, the pentagons have a total of 5b edges,

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 3

and the hexagons a total of 6w edges. Therefore, the number of edges ise = 1

2(5b + 6w). What about the number v of vertices?

Well, if one knows how many faces meet at a vertex, then v is determined;otherwise it is not. For a fullerene, three faces meet at every vertex, andas the pentagons meet vertices 5b times and the hexagons meet vertices6w times, the number of vertices is v = 1

3(5b + 6w). If we substitute

these values for f , e and v into Euler’s formula, then the terms involving wcancel out, and the formula reduces to b = 12. Therefore, every fullerenehas exactly 12 pentagons, but we know nothing about the number w ofhexagons. In any case, every fullerene satisfies the equations

f = 12 + w ,

e = 30 + 3w ,

v = 20 + 2w ,

with the number v of vertices indicating the number of carbon atoms in themolecule. If one sets w equal to zero, so that there are no hexagons at all,then the polyhedron is made up of 12 pentagons with 3 meeting at eachof its 20 vertices. This is the dodecahedron, which, like the cube and thetetrahedron, is a Platonic solid, about which we will have more to say lateron. For positive values of w one obtains the fullerenes that have more than20 atoms and that consist of both pentagons and hexagons.

If one imposes the additional requirement that in a fullerene no two pen-tagons share an edge, then one can show that w has to be at least 20. Thestandard soccer ball realizes this minimal value, for which there are 60 ver-tices, corresponding to the 60 atoms in the C60 molecule.

For arbitrary soccer balls the number of faces meeting at a vertex is notdetermined by their definition, other than that this number is at least 3.Therefore, one has to replace the equation v = 1

3(5b+6w) by the inequality

v ≤ 13(5b + 6w). Substituting into Euler’s formula, the terms involving w

again cancel out, and one obtains the inequality b ≥ 12. Thus every soccerball contains at least 12 pentagons, but, unlike a fullerene, may well containmore. This happens exactly if there is at least one vertex at which more thanthree faces meet.

Also unlike fullerenes, soccer balls have a precise relationship betweenthe number of pentagons and the number of hexagons. Counting the numberof edges along which pentagons and hexagons meet, condition (1) says thatthis number equals 5b, and condition (2) says that this number equals 1

2·

6w = 3w. Thus, 5b = 3w, and because b is at least 12, w is at least 20.These minimal numbers are realized by the standard soccer ball, which isthe only(!) spherical polyhedron made up of 12 pentagons and 20 hexagonssatisfying (1) and (2).

4 D. KOTSCHICK

We have now verified that the standard soccer ball with 12 pentagons and20 hexagons is the only soccer ball that is also a fullerene. This shows thatconditions (1) and (2) defining soccer balls are remarkably restrictive, be-cause there are 1812 distinct fullerenes with 12 pentagons and 20 hexagons,see [2], for example. All but one have adjacent pentagons, and therefore failcondition (1).

NEW SOCCER BALLS FROM OLD

The standard soccer ball with 12 pentagons and 20 hexagons has threefaces meeting at every vertex: one black pentagon, and two white hexagons.As we have seen, this is the only soccer ball for which there are preciselythree faces meeting at every vertex. What other soccer balls are there, withmore than three faces meeting at some vertex, and how can we understandthem?

It turns out that one can generate infinite sequences of other examples bya topological construction called a branched covering. To explain what thismeans, we place the standard soccer ball pattern on the surface of the earthin such a way that two vertices are at the north and south poles. We select asequence of edges connecting the two poles, and we distort the soccer ballpattern on the surface of the earth so that our path along edges from pole topole is a path of constant geographical longitude, longitude zero, say.—It’salright to distort, because remember, we are not doing geometry here, but“rubber-sheet geometry”!

Now our path from pole to pole is the semicircle of zero longitude, andwe slice open the earth along this semicircle. Next we shrink the sliced-open coat of the earth in the east-west direction holding the north and southpoles fixed, until the coat covers exactly half the sphere, say the westernhemisphere. We can take a second copy of this shrunken coat of the sphereand place it over the eastern hemisphere, so that it is the image of the west-ern hemisphere under half of a full rotation around the north-south axis.Now something remarkable happens: the two pieces can be sewn together,to give the sphere a new structure of a soccer ball with twice as many pen-tagons and hexagons as we started with! The reason is that at each of thetwo seams running between the north and south poles the two sides of theseam look like the two sides of the cut we made in our original soccer ball.Because we were cutting an actual soccer pattern, the two sides now fit backtogether again, so that the adjacency conditions (1) and (2) defining soccerballs are fulfilled.

One says that the new soccer ball constructed in this way is obtainedfrom the old one by passage to a two-fold branched covering whose branchpoints are the poles. The new ball looks the same as the old one everywhere

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 5

except near the branch points. In the above example, if we start out with thestandard soccer ball with three faces meeting at every vertex, then the newball has two distinguished vertices, one at the north and one at the southpoles, with six faces meeting at each of them, and it has 2(60-2)=116 othervertices, with three faces meeting at each of them.

It is clear that there are a number of variations on this construction. Firstwe need not start with the standard soccer ball, but we can start with anysoccer ball. In particular, we can iterate the passage to branched coverings.Second we can choose the vertices at which to place the branch points ar-bitrarily, by distorting a given soccer ball in such a way that two chosenvertices are placed at the poles. Third, instead of taking two-fold coverings,we can take D-fold coverings for every positive integer D. This means thatafter slicing open the coat of the earth, we do not shrink it to fit over a hemi-sphere, but we shrink it further, so that it fits precisely over one segment ofa segmentation of the sphere into D orange segments. Then we spin thisaround the north-south axis to cover the other D-1 orange segments, andfit everything together along the D seams. For all this it is important thatone thinks of soccer balls as combinatorial or topological—not geometric—objects, so that the polygons can be distorted arbitrarily.

At this point one might think that there could be many more examples ofsoccer balls, obtained by applying some other modification to the standardexample, or that just appear sporadically and have some large number offaces, without any apparent connection to the standard example. That thisdoes not happen is the main Theorem proved in [1]. It is shown there thatevery soccer ball is in fact a suitable branched covering of the standard one.The proof, using covering space theory, a fundamental part of topology,depends on the following crucial observation.

If one looks at a vertex of a soccer ball, then for every face meeting thisvertex, there are two consecutive edges of the face that meet at this vertex.As every other edge of a hexagon meets a pentagon, there is no vertex whereonly hexagons meet. Thus at every vertex there is a pentagon. Its sides meethexagons, and the sides of the hexagons alternately meet pentagons andhexagons. Therefore the number of faces meeting at a vertex is a multiple of3, and the faces are ordered around the vertex in the sequence black, white,white, black, white, white, etc. as one travels around the vertex. This meansthat locally, around a vertex, the structure looks like that of a branchedcovering around a branch point. This control over the local configurationsaround arbitrary vertices is essential for the analysis of soccer balls.

6 D. KOTSCHICK

TOROIDAL SOCCER BALLS

While a convex polyhedron always inflates to form a sphere, there areother, non-convex, polyhedra whose surfaces are more complicated. Thesurfaces that arise as the boundaries separating the interior and exteriorof polyhedra in three-dimensional space are well understood topologically.The most basic examples are the sphere, which is the boundary of a ball,and the torus, which is the boundary of a doughnut. Then there are the dou-ble torus, the triple torus, which is the boundary of a pretzel, the quadrupletorus, etc. These surfaces are distinguished from each other by their genus,the number of holes: the sphere has genus zero, the torus has genus one, thedouble torus has genus two, and so on.

Starting from a sphere, one can construct a torus by removing two smalldisks from the sphere, and gluing the two boundary circles to each other.This procedure can be applied several times to construct surfaces of arbi-trarily large genera.

Mathematicians are not satisfied with only spherical soccer balls. Theyare just as interested in studying soccer balls of higher genera. They thenconsider polyhedra, not necessarily convex, that consist of pentagons andhexagons satisfying the same conditions (1) and (2) satisfied by sphericalsoccer balls. There are soccer balls of all genera, for example, because everysurface is a branched cover of the sphere (in a slightly more general waythan we discussed before). One can arrange the branch points to be verticesof some soccer ball graph on the sphere, and look at the preimage of thisgraph on the higher genus surface. This gives the surface the structure of asoccer ball.

Here is an easier construction for the torus. Let us take a spherical soccerball, and cut it open along two disjoint edges. We open up the sphere alongeach cut, and what we obtain looks rather like a sphere from which twodisks have been removed, but now we have a soccer ball pattern on it, andthe two boundary circles at which we have opened the sphere each have twovertices on them, which are the endpoints of the cut edges. If the cut edgesare of the same type, meaning that along both of them two hexagons metin the original spherical soccer ball, or that along both of them a pentagonmet a hexagon, then we can glue the two boundary circles together so as tomatch vertices with vertices, and so that the resulting pattern on the torussatisfies conditions (1) and (2).

This toroidal soccer ball has the same number of faces and edges as theoriginal spherical one, but the number of vertices is smaller by two. There-fore the alternating sum v − e + f is zero, rather than 2. This is an instanceof the general Euler formula, which says that for a non-empty connectedgraph on a surface of genus g, the number v − e + f always equals 2− 2g.

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 7

For the sphere the genus g is zero, so that 2 − 2g = 2, which is the classi-cal case of the Euler formula. For the torus we have g = 1 and therefore2− 2g = 0.

Let us perform the above construction starting from the standard spher-ical soccer ball. Then we obtain a toroidal soccer ball with 12 pentagonsand 20 hexagons. The number of vertices is 58 rather than 60, althoughthe number of edges is the same as in the spherical case. But now thereare two special vertices at each of which 6 faces meet, whereas at all theothers three faces meet. Could this be a branched covering of the standardspherical soccer ball? The answer is no!

There are many other examples of toroidal soccer balls, and of soccerballs of higher genera, which are not branched coverings of the standardspherical one.

BEYOND PENTAGONS AND HEXAGONS

Clearly there is very little in the analysis of soccer balls that depends onthem being made from pentagons and hexagons. One can consider convexpolyhedra made of two kinds of polygons: black k-gons having k verticesand edges each, and white l-gons having l vertices and edges each. Thenone can require that the sides of a k-gon meet only sides of l-gons, and thesides of l-gons alternately meet k-gons and l-gons. Of course the alternat-ing condition only makes sense if l happens to be an even number. Moregenerally, if l equals a product n · m, then one can require that every nthside of every l-gon meet a k-gon, and all its other sides meet l-gons. In thespecial case when n = 1, this means that all sides of the l-gons meet sidesof k-gons, so that the situation is symmetric in k and l.

Such polyhedra should exist for many different values of k, l and n, notnecessarily 5, 6 and 2. But precisely which values are possible? Perhapssurprisingly, only a few. The determination of all the possibilities, carriedout in [1], is closely related to the regular polyhedra known as Platonicsolids.

THE PLATONIC SOLIDS

Geometrically, a regular polyhedron is made up of equilateral polygonsin the most symmetric way possible, so that the same number of faces meetat all vertices. It was known to Euclid and Plato in antiquity, that there areonly five such completely regular polyhedra: the tetrahedron, the cube, theoctahedron, the dodecahedron and the icosahedron.

8 D. KOTSCHICK

For the purposes of topology or graph theory, one dispenses with geomet-ric properties like having all edges of the same length. Thus one considerspolyhedra consisting of polygons with some fixed number K of edges, andrequires that M of these polyhedra meet at every vertex, without imposingany geometric conditions. The basic question then is, which values of Kand M are possible for spherical polyhedra? The answer is that only thevalues realised by the Platonic solids are possible. Thus, there is in factno additional freedom gained by replacing the rigid geometric definition ofregular polyhedra by the more flexible topological definition.

The key to the topological determination of the Platonic solids is Euler’sformula v− e+ f = 2. Suppose that one has f polygonal faces, all of themK-gons. Then the number of edges is e = 1

2K · f . If M faces meet at every

vertex, then the number of vertices is v = 1M

K ·f . Substituting these valuesin Euler’s formula, elementary algebra leads to the equation

1

K · f+

1

4=

1

2K+

1

2M.

If both K and M are at least 4, then the right hand side of this equation is atmost 1

4, which is not possible, as the left hand side is strictly larger than 1

4.

Therefore, either K or M (or both) equals 3.Suppose first that K = 3. Substituting this into the above equation, one

sees that M is at most 5. Therefore M can only be 3, 4 or 5, and indeed allof these values correspond to actual regular polyhedra: the tetrahedron, theoctahedron and the icosahedron.

If M = 3, then similarly K has to be 3, 4 or 5, corresponding to thetetrahedron, the cube and the dodecahedron.

There is a duality here given by interchanging the roles of K and M: thecube and the octahedron are dual to each other, and so are the dodecahedronand the icosahedron. The tetrahedron is self-dual. Dual polyhedra have thesame numbers of edges, and the duality interchanges the numbers of facesand vertices.

The above equation does have other solutions in positive integers, whichcorrespond to so-called degenerate Platonic solids, that are not bona fidepolyhedra. If K = 2, then M = f = e is unconstrained, and v = K =2. These values are realized by a sphere divided up as the boundary ofa segmented citrus fruit with M segments, which also happens to be the

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 9

standard design of an American football. Dually, if M = 2, then K = e = vis unconstrained, and f = M = 2. These values are realized by the divisionof a sphere into two K-gons meeting along their edges.

GENERALIZED SOCCER BALLS

The standard soccer ball with 12 pentagons and 20 hexagons is derivedfrom the icosahedron by a procedure known as truncation. The icosahedronis made up of 20 triangles, with 5 of them meeting at each of its 12 vertices.One slices off the vertices of the icosahedron, so that in place of each ofthe 12 vertices one obtains a new face. These faces are pentagons, becausethere were 5 faces meeting at each of the vertices of the icosahedron. Thetriangular faces of the icosahedron are having their corners sliced off, sothey become hexagons. The sides of such a hexagon are of two kinds, whichoccur alternately: the remnants of the sides of the original triangular facesof the icosahedron, and the new sides produced by lopping off the corners.The new pentagonal faces are surrounded by hexagons.

This truncation can of course be applied to the other Platonic solids. Forexample, performing it on the tetrahedron produces a polyhedron consist-ing of triangles and hexagons, such that the sides of each triangle meet onlysides of hexagons, and the sides of the hexagons alternately meet trianglesand hexagons. Concerning the question which values of k and l are possiblefor a generalized soccer ball consisting of k-gons and l-gons, the truncatedicosahedron of course gives the values (k, l) = (5, 6). The truncations ofthe other Platonic solids lead to the values (k, l) = (3, 6) for the tetrahedron,(4, 6) for the octahedron, (3, 8) for the cube, and (3, 10) for the dodecahe-dron.

The pairs (k, l) do not look very symmetric, and the duality betweenpairs of Platonic solids seems to have disappeared. The reason is that weneglected the factorization of l as a product n ·m. If instead of pairs (k, l)we look at triples (k,m, n), then the truncated icosahedron gives (5, 3, 2),and the truncations of the other Platonic solids, in the same order as above,give (3, 3, 2), (4, 3, 2), (3, 4, 2) and (3, 5, 2). Now the duality is restored: itis given by interchanging k and m, not k and l.

So how do we find out which triples (k,m, n) are possible for a sphericalpolyhedron made of k-gons and l-gons with l = n ·m, in such a way thatevery side of a k-gon meets a side of an l-gon, and every nth side of everyl-gon meets a k-gon, and all its other sides meet l-gons? All we have to dois mimic the argument that led to the classification of Platonic solids.

We denote the number of k-gons by b, thinking of them as black, and thenumber of l-gons by w, thinking of them as white. Given k and m, the twonumbers b and w determine each other by the adjacency conditions. Indeed,

10 D. KOTSCHICK

if we count the number of edges separating black and white polygons, wefind b · k = w ·m.

The total number of faces is f = b + w, and the number of edges ise = 1

2(b · k + w · l), because there are two faces meeting along every edge,

and the k-gons have a total number of b · k edges, whereas the l-gons havea total number of w · l. As for the soccer balls made from pentagons andhexagons, we do not know how many faces meet at a vertex, except thatthis number has to be at least 3. This tells us that the number of verticesis v ≤ 1

3(b · k + w · l). Substituting these values into Euler’s formula

f − e + v = 2, and using the equations l = n · m and b · k = w · m,elementary algebra leads to the inequality

1

k · b+

n + 1

12≤ 1

2k+

1

2m,

which is obviously very similar to the equation we had to solve to find thePlatonic solids1. Of course the situation now is more complex, becausethere is the additional variable n, and because we have only an inequality,not an equation. Nevertheless, one can analyze this inequality, and compilea list of all the possible solutions in positive integers (k,m, n) with k ≥ 3and l = n ·m ≥ 3.

Alas, the story does not end there. Satisfying the above inequality isonly a necessary condition a triple (k,m, n) must meet; it is not a sufficientcondition. In other words, there are triples, such as (k, m, n) = (4, 4, 1),which solve this inequality for large enough values of b, but which still donot arise from generalized soccer balls.

Thus determining a list of candidate triples is only the first step. Thenecessary second step then is to find out which of these candidates can berealized by actual polyhedra. The search for examples is guided by thefollowing observation. Once we fix a triple (k,m, n), the above inequalitygives a lower bound for the number b of k-gons. By the equation b·k = w·n,this also gives a lower bound for the number w of l-gons. These minimalnumbers are realized if and only if the inequality happens to be an equality,so that three faces meet at every vertex.

Let us look at some examples of admissible triples. If n = 2, then fivedifferent such triples are realized by the truncations of the Platonic solids.The only other admissible triples with n = 2 are of the form (k, 2, 2) witharbitrary k ≥ 3. The minimal values for b and w are 2 and k respectively.Can we find a polyhedron realizing these numbers? The answer is yes,because if we take an American football made of k segments and truncateit at its two poles, we get exactly what is required: a polyhedron with twok-gons and k quadrilaterals arranged in the correct adjacency pattern.

1This similarity is even more obvious if we restrict to n = 2.

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 11

Thus all the admissible triples (k, m, n) with n = 2 are realized bythe truncated Platonic solids, if we allow the degenerate American footballamong them.

What about other values of n? It turns out that n can be at most equal to 6,and as long as n is not equal to one, all admissible triples have realizationswith three faces meeting at every vertex. All these minimal realizations arecooked up from Platonic solids in one way or another. When n = 1, theminimal number of faces meeting at a vertex is 4, and the minimal realiza-tions are given by an octahedron, whose faces are painted black and white ina suitable way, and by the cuboctahedron and the icosidodecahedron. Theselast two polyhedra, like the truncated Platonic solids, are examples of so-called Archimedean solids, which generalize the Platonic solids, cf. [3].

The complete list of triples (k,m, n) that do occur, together with a min-imal realization for each of them, is given in [1], and we reproduce it hereas Figure 1.

The polyhedra listed as minimal realizations of the generalized soccerball patterns have various interesting properties. We mention just one,which is relevant to our earlier discussion of fullerenes. Entry 10 in the tableis of course the classical soccer ball C60, but there are four other fullerenesin the table: numbers 14 and 20, shown here in the margin, and the casek = 6 of number 17. The numbers of hexagons in these examples are30, 60 and 2 respectively, so that the numbers of vertices, i. e. the numbersof carbon atoms in the molecules, are 80, 140, and 24 respectively. In theexamples 14 and 20 the pentagons are disjoint, whereas in the k = 6 caseof number 17, the pentagons meet along edges. In fact, this is the onlyfullerene with 24 atoms. In the case of 80 atoms there are 7 fullerenes withdisjoint pentagons, but only one occurs in our table of generalized soccerballs. For 140 atoms the number of fullerenes with disjoint pentagons is121354, see [2].

WHERE BRANCHED COVERINGS DO NOT SUFFICE

We have seen that there are infinitely many combinatorially distinct spher-ical soccer balls made up from pentagons and hexagons in the usual way,so that conditions (1) and (2) are satisfied. We have also seen that one canunderstand these infinitely many examples as branched coverings of theminimal example, with the smallest number of pentagons and hexagons.This naturally begs the question whether the same is true for arbitrary triples(k,m, n) arising from generalized spherical soccer balls in place of (5, 3, 2).

Once a triple (k, m, n) is realized by some soccer ball, we can of coursetake branched coverings to produce infinitely many distinct realizations.The only difficulty is in proving that there are no other realizations, which

12 D. KOTSCHICK

k m n minimal realization b w1. 3 3 1 octahedron 4 42. 3 4 1 cuboctahedron 8 63. 4 3 1 cuboctahedron 6 84. 3 5 1 icosidodecahedron 20 125. 5 3 1 icosidodecahedron 12 206. 3 3 2 truncated tetrahedron 4 47. 3 4 2 truncated cube 8 68. 4 3 2 truncated octahedron 6 89. 3 5 2 truncated dodecahedron 20 1210. 5 3 2 truncated icosahedron = standard soccer ball 12 2011. ≥ 3 2 2 prism or truncated American football 2 k12. 3 2 3 variation on the tetrahedron 4 613. 4 2 3 variation on the cube 6 1214. 5 2 3 variation on the dodecahedron 12 3015. ≥ 3 1 3 pyramid or partially truncated American football 1 k16. ≥ 3 1 4 double tin can 2 2k17. ≥ 3 1 5 zigzag tin can 2 2k18. 3 1 6 subdivision of the tetrahedron 4 1219. 4 1 6 subdivision of the cube 6 2420. 5 1 6 subdivision of the dodecahedron 12 60

FIGURE 1. The generalized soccer ball patterns. For n = 1,there is a complete symmetry between k and l. Therefore,the cases 2. and 3. are dual to each other, as are 4. and 5., byswitching the roles of k and m, which for n = 1 equals l.Case 1. is self-dual. Similarly, cases 7. and 8., respectively9. and 10., are dual to each other with the duality induced bythe duality of Platonic solids, and case 6. is self-dual.

are not branched coverings of the minimal one. It is shown in [1] that forall triples with n = 2 it is true that all possible spherical realizations arebranched coverings of the minimal ones listed in the table reproduced here.However, for other values of n, this fails!

The easiest example demonstrating this failure arises for the triple (k,m, n) =(3, 1, 3), meaning that we have black and white triangles arranged in such away that the sides of each black triangle meet only sides of white ones, andthat each white triangle has exactly one side that meets a black triangle. Theminimal realization listed under number 15 in the table is a pyramid over atriangular base. We can also think of this as a painted tetrahedron, in whichone face has been painted black, and all the others white. For a different

TOPOLOGY AND COMBINATORICS OF SOCCER BALLS 13

realization take an octahedron, and paint two opposite faces black, and allothers white. This is not a branched covering of the painted tetrahedron!

What we see here is a subtle difference between the case n = 2 and thecases n > 2. In the example we have n = 3, and there is no way to controlwhat happens at a vertex. The painted tetrahedron has two very differenttypes of vertices: a vertex at which only white faces meet, and three verticeswhere there are one black and two white faces. The painted octahedron hasall vertices the same, but they are different from the vertices of the paintedtetrahedron, because at each of them one black and three white faces meet.As we discussed earlier for the usual triple (5, 3, 2), in the case n = 2 theadjacency conditions of a soccer ball imply that around each vertex one hasa particular sequence of faces. This local structure is crucial for the proofthat all spherical realizations arise as branched coverings of the minimalone, cf. [1].

Acknowledgements: I am grateful to V. Braungardt and to A. Jackson for helpwith the preparation of this article.

This article is a slight revision of a text I wrote in March of 2006 to popularizethe results of my joint paper [1] with V. Braungardt. After a lot of editing, an articleloosely based on my text was published in the American Scientist [4], and a partialGerman translation of the American Scientist article also appeared in Spektrum derWissenschaft [5]. I have decided to make this version of my original text availablebecause I believe it has certain qualities that were lost in the editing process ofAmerican Scientist and Spektrum der Wissenschaft. I am grateful to M. Trott ofWolfram Research for permission to use some of his Mathematica pictures that heoriginally produced to illustrate the American Scientist article [4].

REFERENCES

1. V. Braungardt and D. Kotschick, The classification of football patterns, PreprintarXiv:math.GT/0606193; to appear in German translation in Math. Semesterberichte.

2. G. Brinkmann and A. W. M. Dress, A constructive enumeration of fullerenes, Journalof Algorithms 23 (1997), 345–358

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