The Topology and Combinatoricsof Soccer Balls

When mathematicians think about soccer balls,the number of possible designs quickly multiplies

Dieter Kotschick

With the arrival of the quadrennialsoccer World Cup this summer,

more than a billion people around theworld are finding their television andcomputer screens filled with depictionsof soccer halls. In Germany, where theWorld Cup matches are being played,soccer balls are tuming up on all kindsof merchandise, much of it having noth-ing to do with soccer.

Although a soccer ball can be put to-gether in many different ways, there isone design so ubiquitous that it has be-come iconic. This standard soccer ball isstitched or glued together from 32 poly-gons, 12 of them five-sided and 20 six-sided, arranged in such a way that everypentagon is surrounded hy hexagons.Postmodern paint jobs notwithstandiiig,the traditional way to color such a ballis to paint the pentagons black and thehexagons white. This color scheme wasreportedly introduced for the WorldCup in 1970 to enhance the visibilityof the ball on television, although thedesign itself is older.

Most people associate the soccer-ballimage with hours spent on the field orthe sidelines, or perhaps just with ad-vertisements for sport merchandise. Butto a mathematician, a soccer bail is anintriguing puzzle. Why does it look theway it does? Are there other ways ofputting it together? Could the penta-

Dider Kotschick earned his doetorate in ninthcinaticsfrom the University of Oxford in T989. After holding

postdoctoral positions at Prhtceton and Cambridge

(England), lie became a professor ofniatliaiiatics

at the Unhvrsity of Basel in Szvitzerland l'>efore

moving to the Ludung-MaximiHans-Universitdt

Miinchen, where lie holds the ckiir ofgeometiy. He

is grateftd to Volker Braungardt and Allyn Jackson

for help with tlw preparation of this article. Address:

MathematisclitTS Institut, LMU MUnclieji, Tliere-

sienstrasse 39,80333 Miinchen, Germany. Internet:

dieter@niember.ams.org

gons and hexagons be arranged differ-ently? Could other polygons be usedinstead of pentagons and hexagons?These questions can be tackled using thelanguage of mathematics—in particulargeometry, group theory, topology andgraph theory. Each of these subjects pro-vides concepts and a natural context forphrasing questions such as those aboutthe design of soccer balls, and some-times for answering them as well.

An important aspect of the applicationof mathematics is that different ways ofmaking mathematical sense of everydayquestions lead to different answers. Thismay come as a bit of a surprise to read-ers who are used to schoolbook prob-lems that have only one right answer.Properly framing questions is just as im-portant a part of the art of mathematicsas answering them. Moreover, a genuinemathematical exploration of an open-ended question does not stop with find-ing "the answer" (if there is one), butinvolves understanding why the answeris what it is, and how it changes whenthe underlying assumptions are modi-fied. The questions posed by the designof soccer balls provide a wonderful il-lustration of this process.

Soccer Balls and FullerenesMathematicians like to begin by definingtheir terms. What, then, is a soccer ball?An official soccer ball, to be approved bythe Federation Internationale de Foot-ball Association (FIFA), must be a spherewith a circumference between 68 and 70centimeters, with at most a 1.5 percentdeviation from sphericit\' when inflatedto a pressure of 0.8 atmospheres.

Alas, such a definition says nothingabout how the ball is put together, andis therefore not suitable for a mathemat-ical exploration of the design. A betterdefinition is that a soccer ball is approxi-

mately a sphere made of polygons, orwhat mathematicians call a sphericalpolyhedron. The places where the poly-gons come together—the \'ertices andedges of the polyhedron—trace out amap on the sphere, which is called agraph. (Such a graph has nothing to dowith graphs of functions. The word hastwo completely different mathematicalmeanings.) Examined from the perspec-tive of graph theory, the standard soccerball has three important properties:

(1) it is a polyhedron that consistsonly of pentagons and hexagons;

(2) the sides of each pentagon meetonly hexagons; and

(3) the sides of each hexagon alter-nately meet pentagons and hexagons.

As a starting point, then, we can definea soccer ball to be any spherical poly-hedron with properties (1), (2) and (3).If the pentagons are painted black andthe hexagons are painted white, thenthe definition does capture the iconicimage, though it does not determineif uniquely.

This definition places the problemof soccer ball design into the context ofgraph theory and topology. Topology,often described as "rubber-sheet ge-ometry," is the branch of mathematicsthat studies properties of objects thatare unchanged by continuous defor-mations, such as the inflation of a soc-cer ball. For the purposes of topology,it doesn't matter how long the edges ofa polyhedron are, or whether we aredealing with a round polyhedron orone with flat sides.

I first encountered the above defini-tion in 1983, in a problem posed in theBundeswettbewerb Mathematik, a Ger-man mathematics competition for highschool students. The problem was: Givenproperties (1H3), determine how many

350 American Scientist, Volume 94

Figure 1. Twelve pentagons and 20 hexagons form a figure known to mathematicians as a truncated icosahedron, to chemists as the buckmin-sterfullerene molecule—and to nearly everybody else as the standard soccer ball. As this summer's World Cup competition approached, a soc-cer bail-shaped information pavilion toured the German cities preparing to host World Cup matches. Here the "football globe" makes a stopin Leipzig. The iconic black and white soccer ball is also an intriguing puzzle amenable to mathematical analysis. Other soccer balls that maynever be seen on the playing field also offer interesting solutions to the mathematical questions posed by the standard design.

pentagons and hexagons a soccer ball ismade of. Thinking about this problemat the time, I assumed that the ball is aconvex polyhedron in space made upof regular polygons. Tlnis geometric as-sumption, together with rules (1), (2) and(3), implies that there are 12 pentagonsand 20 hexagons. Moreover, there is aunique way of putting them together,giving rise to the iconic standard soccerball. Without the geometric assumption,the graph-theory problem has infiiiitelymany other solutions, which have largernumbers of pentagons and hexagons.

I began thinking about this problemagain after I was invited to give a lec-ture at a prize ceremony for tbe samecompetition in 20D1. Eventually, one ofmy postdoctoral fellows, Volker Braun-gardt, and I found a way to characterizea!! tlie solutions, a characterization tliat Iwill describe below.

interestingly, a related problem arosein chemistry in the 1980s after the 60-atom carbon molecule, called the buck-minsterfullerene or "buckyball," was

discovered. The spatial shape of this C ,molecule is identical to the standard soc-cer-ball polyhedron consisting of 12 pen-tagons and 20 hexagons, with the 60 car-bon atoms placed at the vertices and theedges corresponding to chemical bonds.The discovery of the buckyball, whichwas honored by tlie 1996 Nobel Prize forchemistry, created enormous interest ina class of carbon molecules called fuller-enes, which satisfy assumption (1) abovetogether with a further condition:

(3') precisely three edges meet atevery vertex.

This property is forced by the chemi-cal bonding properties of carbon. Inaddition, assumption (2) is sometimesimposed to define a restricted class offullerenes. Having disjoint pentagonsis expected to be related to the chemi-cal stability of fullerenes. There are infi-nitely many fullerene polyhedra—C^,,was merely the first one discovered as anactual molecule—and it is quite remark-able that the two infinite families of poly-

hedra, tbe soccer balls and the fullerenes,have only the standard soccer ball incommon. Thus (1H3) together witli (3')give a unique description of the standardscKcer ball without imposing geometricassumptions. (Assumptions like regular-ity in fact imply condition {3')-)

To see that this is so requires a briefexcursion into properties of polybedra,starting with a beautiful formula dis-covered by the Swiss mathematician Le-onhard Euler in the 18th century. Euler'sformula (see "Euler's formula," next page),a basic tool in graph theory and topol-ogy, says that in any spherical polyhe-dron, the number of vertices, v, minusthe number of edges, e, plus the numberof faces,/, equals 2:

v-c+f=2

Let's apply Euler's formula to a poly-hedron consisting of b black pentagonsand 10 white hexagons. The total num-ber/of faces isb + w. In all, the penta-gons have 5b edges, because there are 5edges per pentagon and b pentagons in

www.americanscicntist.org 2006 July-August 351

Figure 2. Soccer-ball design over the years has been driven by the demand for a round ball that holds its shape and by the technology available. Eightpanels of vulcanized rubber were glued together to create the hall at left, used in the earliest soccer championship in the United States in 1863. Theleather ball at center, used in the 1950 World Cup, has a design typical of its era. The small number of large, irregularly shaped flat pieces adverselyaffected its roundness. Thanks to improvements in materials and manufacturing, curved pieces in more complicated shapes can now be used. Thisyear's World Cup ball (right) is made from 14 synthetic-leather panels cut in intricately curved shapes. In graph-theory terms, this ball is a truncatedoctahedron. (Historic photographs courtesy of Jack Huckel, National Soccer Hall of Fame; right photograph courtesy of firosportfoto.de.)

all. Similarly, the hexagons have a totalof 6w edges. Adding these two numbersshould give the total number of edges—except that 1 have counted each edgetwice because each edge lies in two dif-ferent faces. To compensate I divide by2, and hence the number of edges is:

Finally, to count the number of verti-ces, I note that the pentagons have 5/'vertices in all and the hexagons have6 1 vertices. In the case of a fullerene,assumption (X) says that each vertexbelongs to three different faces. Thus ifI compute 5/? + 6w, I have counted eachvertex exactly three times, and hence Imust divide by 3 to compensate:

v = {\/3){5b + 6w)

Substituting these values for/, eand V into Euler's formula, I find thatthe terms involving w cancel out, andthe formula reduces to b = 12. Everyfullerene, therefore, contains exactly12 pentagons! However, there is noa priori limit to the number of hexa-gons, w, and therefore no limit on thenumber of vertices. (This is implicitin the title of a 1997 article on fuller-enes in American Scientist "FullereneNanotubes: C j^,^, |, ^ and Beyond.")If I impose the additional condition(2), then I can show that the numberof hexagons has to be at least 20. Thestandard soccer ball or buckvball real-

izes this minimum value, for whichthe number v of vertices equals 60, cor-responding to the 60 atoms in the C .,molecule. However, it can be shownthat there are indeed infinitely manyother mathematical possibilities forfullerene-shaped polyhedra. Which ofthese correspond to actual molecules isa subject of research in chemistry.

For soccer balls, we are allowed touse only assumptions (l)-(3), but not(3'), the carbon chemist's requirementthat three edges meet at every vertex.In this case the number of faces meet-ing at a vertex is not fixed, but thisnumber is at least 3. Therefore, theequation v = (l/3)(5f' + 6w) becomesan inequnlity: u < (1 /3){5b + 6w). Substi-

Euler's formula

Any non-empty connected finite graph on the sphere satis-fies Euler's formula v ~ e + / - 2. Here v and e are the num-bers of vertices and edges, and/is the number of regionsinto u'hich the sphere is divided. A proof of Euler 's formulaprtKeeds by repeatedly simplifying the graph by the fol-lowing two operations:

The first operation consists of deleting any vertex thatmeet5 only one edge, and in addition deleting the edgethat meets it (a). This operation does not change thenumber of regions, while it decreases both i' and f by 1.The second operation consists of collapsing a region toa single vertex, together with all the edges and verticeson its boundary (b). If the collapsed region had k verticeson its boundary, then this collapsing reduces v by fc-1,reduces c by k and reduces/by 1. Thus v - e + fis notchanged by either of the two operations.

A finite iteration of these two simplifications reducesany graph to a graph with only one vertex and no edges.Then there is one region, and v - e +f- 1-0 + 1 - 2 .

352 American Scientist, Volume 94

tuting into Euler's formula, the termsinvolving w again cancel out, leavingthe inequality b > 12. Thus every soc-cer ball contains at least 12 pentagons,but, unlike a fullerene, may well con-tain more.

Also unlike fullerenes, soccer ballshave a precise relation between thenumber of pentagons and the number ofhexagons. Counting the number of edg-es along which pentagons and hexagonsmeet, condition (2) says that all edges ofpentagons are also edges of hexagons,and condition (3) says that exactly halfof the edges of hexagons are also edgesof pentagons. Hence (l/2)(6n') = 5b, or3iv = 5b. Because b > 12, w is at least 20.These minimal values are realized bythe standard soccer ball, and the realiza-tion is combinatorially unique becauseof conditions (2) and (3). But there arealso infinitely many other numerical so-lutions, and the problem arises whetherthese non-minimal numerical solutionscorrespond to soccer-ball polyhedra.It turns out that they do, as we'll seeshortly, so that there is indeed an infinitecollection of soccer balls.

Thus we see that there are infinitelymany fuilerenes (satisfying assumptions(1), (2) and (3')) and infinitely many soc-cer balls {satisfying (1), (2) and (3)).However, if we combine the two defini-tions, there is only one possibility! Fora fullerene, b -12, and for a soccer ball,5b = 3w. Consequently, for a soccer ballto also be a fullerene, we must concludethat 5 X 12 = 3zv, or w = 20. Any soccerball that is also a fullerene must thereforehave 12 pentagons and 20 hexagons. It isknown that there are 1,812 distinct fuller-enes witli 12 pentagons and 20 hexagons,but 1,811 of them have adjacent penta-gons somewhere and are therefore not

soccer balls, because they violate condi-tion (2). The standard soccer ball is theonly one with no adjacent pentagons.

New Soccer Balls from OldLeaving behind chemistry and fullerenegraphs, let us now consider the crucialquestion: What other, nonstandard, soc-cer balls are there, with more than threefaces meeting at some vertex, and howcan we understand them? It turns outthat we can generate infinite sequencesof different soccer balls by a topologicalconstruction called a branched covering.You can visualize this by imagining thestandard soccer-ball pattern superim-posed on the surface of the Earth andaligned so that there is one vertex at theNorth Pole and one vertex at the SouthPole. Now distort the pattern so that oneof the zigzag paths along edges frompole to pole straightens out and lies on ameridian, say the prime meridian of zerogeographical longitude (see Figure 4b). Itis all right to distort the graph, becausewe are doing "rubber-sheet geometry."

Next, imagine slicing the Earth openalong the prime meridian. Shrink thesliced-open coat of the Earth in the east-west direction, holding the poles fixed,until the coat covers exactly half thesphere, say the Western Hemisphere. Ei-nally, take a copy of this shrunken coatand rotate it around the north-southaxis until it covers the Eastern Hemi-sphere. Remarkably, the two pieces canbe sewn together, giving the sphere anew structure of a stKcer ball with tu'iceas many pentagons and hexagons asbefore. The reason is that at each of thetwo seams running between the Northand South Poles, the two sides of theseam are indistinguishable from the twosides of the cut we made in our original

Figure 4. New soccer balls can be made from existing ones by a mathematical con-struction called a branched covering. First one chooses a seam of the old soccer ballalong the edges of polygons (a). This seam is straightened out and sliced open (b,c). The whole surface of the soccer ball is shrunk to cover only a hemisphere (d}. Asecond copy of this hemisphere is rotated around and stitched to the first (e, f). Thisbuilds a new soccer ball, which can be deformed as ing. Conforming to the definitionof a soccer ball, black faces in the new ball are adjacent to only white faces (faces thatmeet only at vertices are not considered adjacent), and white faces have an alternatingsequence of white and black faces around their edges. (Soccer-ball images in Figures4, 5,6,9 and 11 were calculated and created by Michael Trott using Mathematica.)

Figure 3. Fullerenes are large carbon mol-ecules whose shapes are made up of penta-gons and hexagons that meet three at a time,in such a way that no two pentagons areadjacent. Every fullerene contains exactly12 pentagons, but there is no limit to thenumber of hexagons. The simplest fuller-ene molecule, C^ has the iconic soccer-ballshape. Other fullerenes, such as C^^^^, havebeen made in the laboratory. Mathematically,the combinatorics of fullerenes is an applica-tion of Euler's formula.

soccer ball. Therefore, the two pieces fittogether perfectly, in such a way thatthe adjacency conditions (2) and (3) arepreserved. (See Eigure 4 for step-by-stepillustrations of this construction.)

The new soccer ball constructed inthis way is called a two-fold branched cov-ering of the original one, and the polesare called branch points. The new balllooks the same as the old one (from thetopological or rubber-sheet geometrypoint of view), except at the branchpoints. There are now six faces (insteadof three) meeting at those two vertices,and there are 116 other vertices (the 58vertices that weren't pinned at the poles,plus their duplicates), with three facesmeeting at each of them.

There is a straightforward modifica-tion we can make to this construction.Instead of taking two-fold coverings.

www.americanscientist.org 2006 July-August 353

Figure 5, Infinitely many soccer balls can beconstructed by the method used in Figure 4.For example, an eight-fold branched cover-ing of the standard soccer ball can be built byusing eight copies of the sliced-open coat ofthe standard ball to create a soccer ball with96 pentagons and 160 hexagons. The eightpieces fit together like sections of an orange.The author and his collaborator Volker Braun-gardt have proved that every soccer ball is abranched covering of the standard one.

Figure 6. The proof that branched coveringsproduce all soccer balls depends on an analy-sis of the sequence of colors around any ver-tex. Because at least one of the edges meetingat each vertex bounds a pentagon (black),there is no vertex where only hexagons(white) meet. The sequence of faces arounda vertex is always black-white-white, black-white-white, and closes up after a number offaces that is a multiple of three.

we can take rf-fold branched co\'eringsfor any positive integer d. Instead ofshrinking the sphere halfway, we imag-ine an orange, made up of d orange sec-tions, and for each section we shrink acopy of the coat of the sphere so thatit fits precisely over the section. Onceagain the different pieces fit togetheralong the seams (see Figure 5). For all ofthis it is important that we think of soc-cer halls as comhinatorial or topologi-cal—not geometric—objects, so that thepolygons can be distorted arbitrarily.

At this point you might think thatthere could be many more examples ofsoccer balls, perhaps generated from thestandard one by other modifications, or

perhaps sporadic examples having noapparent connection to the standard soc-cer ball. But this is not the case! Braun-gardt and I proved that every soccer ballis in fact a suitable branched co\'ering ofthe standard one (possibly with slightlymore complicated branching than wasdiscussed above).

The proof involved an interesting in-terplay between the local structure ofsoccer balls around each vertex and theglobal structure of braiiched coverings.Consider any vertex of any soccer ball(see Figure 6). For every face meeting thisvertex, there are tw.'o consecutive edgesthat meet tliere. Because at least one ofthose two edges bounds a pentagon, bycondition (3), there is no vertex whereonly hexagons meet. Thus at every ver-tex there is a pentagon. Its sides meethexagons, and the sicies of the hexagonsalternately meet pentagons and hexa-gons. This condition can be met only ifthe faces are ordered around the ver-tex in the sequence black, vvliite, white,black, white, white, etc. (Remember thatthe pentagons are black.) In order for thepattern to close up around the vertex, thenumber of faces that meet at tliis vertexmust be a multiple of 3. This means thatlocally, around aiiy vertex, the structurelooks just like that of a branched cover-ing of the standard stKcer ball around abranch point. Covering space theory—the part of topology that investigates re-lations between spaces that look locallyalike—then enabled us to prove that anystxcer ball is in fact a branched coveringof tlie standard one.

Beyond Pentagons and HexagonsTo mathematicians, generalization issecond nature. Even after somethinghas been proved, it may not be appar-ent exactly why it is true. Testing theargument in slightly different situationswhile probing generalizations is an im-portant part of really understanding it,and seeing which of the assumptionsused are essential, and which can bedisper\sed with.

A quick look at the arguments abovereveals that there is very little in theanalysis of soccer balls that depends ontheir being made from pentagons andhexagons. So it is natural to define "gen-eralized soccer balls" allowing otherkinds of polygons. Imagining that weagain color the faces black and white,we assume that the black faces have kedges, and the white faces have / edgeseach. For conventional soccer balls, kequals 5, and / equals 6. As before, the

edges of black faces are required to meetonly edges of white faces, and the edgesof the white faces alternately meet edgesof black and white faces. The alternationof colors forces / to be an even number.

Going one step further in this processof generalization, we can require thatevery nth edge of a white face meets ablack face, and all its other edges meetwhite faces. This forces / to be a mul-tiple of //; that is, I - m x u for someinteger m. Of course we still require thatthe edges of black faces meet only whitefaces. Let us call such a polyhedron ageneralized soccer ball. Thus the patternof a generalized soccer ball is describedby the three integers (k, m, n), where k isthe number of sides in a black face, / -m X n is the number of sides in a whiteface, and every nth side of a white facemeets a black face. The first questionwe must ask is: Which combinationsof k, HI and n are actually possible for ageneralized soccer ball? It turns out thatthe answer to this question is closelyrelated to the regular polyhedra.

Regular PolyhedraAncient Creek mathematicians and phi-losophers were fascinated by the regularpolyhedra, also known as Platonic sol-ids, attributing to them many mysticalproperties. The Platonic solids are poly-hedra with the greatest possible degreeof symmetry: All their faces are equilat-eral polygons with the same numberof sides, and the same number of facesmeet at every vertex. Euclid proved inhis Elements that there are only fi\'e suchpolyhedra: the tetrahedron, the cube,the octahedron, the dodecahedron andthe icosahedron (sec Figure 7).

Although Euclid used the geometricdefinition of Platonic solids, assumingall the polygons to be regular, modemmathematicians know that the argu-ment does not depend on the geometry.In fact, a topological argument usingonly Euler's formula shows that thereare no possibilities other than the fiveshown in Eigure 7.

Eacli Platonic solid can be described bytwo numbers: tlie number K of verticesin each face and the number M of facesmeeting at each vertex. If /"is the numberof faces, then the total number of edges ise = (1/1)K x/, and the number of \'ertices isV = (1/M)K xf. Substituting these values inEuler's formLila/- ii + e = 2, we find thatelementary algebra leads to the equation:

KfX + J_2K 2M

354 American Scientist, Volume 94

The possible solutions can be deter-mined quite easily. The complete list ofpossible values for the pairs (K, M) is:

(3,3) for the tetrahedron(4,3) and (3,4) for the cube and theoctahedron(5, 3) and (3, 5) for the dodecahe-dron and the icosahedron.

Strictly speaking, this is only the list ofgenuine polyhedra satisfying the aboveequation. Tlie equation does have otherstilutions in positive integers. These so-lutions correspond to so-called degener-ate Platonic solids, which are not bonafide polyhedra. One family of thesedegenerate polyhedra has K=2 and Marbitrary, and the other has M-2 and Karbitrary. The first case can be thoughtof as a beach ball that is a sphere di-\'ided into M sections in the marmer ofa citrus fniit.

Finding Generalized Soccer BallsThe Platonic solids give rise to general-ized soccer balls by a procedure knownas truncation. Suppose we take a sharpknife and slice off each of the comers ofan icosahedron. At each of the 12 verti-ces of the icosahedron, five faces cometogether at a point. When we slice offeach vertex, we get a small pentagon,with one side bordering each of the facesthat used to meet at that vertex. At thesame time, we change the shape of the20 triangles that make up the faces of theicosahedron. By cutting off the comersof the triangles, we turn them into hexa-gons. The sides of the hexagons are oftwo kinds, which occur alternately: theremnants of the sides of the original tri-angular faces of the icosahedron, and thenew sides produced by lopping off thecorners. The first kind of side bordersanother hexagon, and the second kindtouches a pentagon. Tn fact, the polyhe-dron we have obtained is nothing butthe standard soccer ball. Mathematicianscall it the truncated icosahedron.

The same truncation procedurecan be applied to the other Platonicsolids. For example, the truncatedtetrahedron consists of triangles andhexagons, such that the sides of thetriangles meet only hexagons, whilethe sides of the hexagons alternatelymeet triangles and hexagons. This is ageneralized soccer ball with /r-3, / H - 3 ,n-1 (and 1 - mx n - 6). The truncatedicosahedron gives values for k, m andn of 5, 3 and 2. The remaining trunca-tions give (k, in, n) - (4, 3, 2) for the oc-tahedron, (3,4, 2) for the cube, and (3,

cubetetrahedron

octahedron

icosahedron dodecahedron

Figure 7. The five basic Platonic solids shown here have been known since antiquity. Exam-ples of all generalized soccer-ball patterns can be generated by altering Platonic solids.

5, 2) for the dodecahedron. In addition,we can truncate beach balls to obtaingeneralized soccer balls with (k, m, n)= (k, 2, 2), where k can be any integergreater than 2.

Are these the only possibilities forgeneralized soccer ball patterns, or arethere others? Again, we can answertliis question by using Fuler 's formula,

f-c + v = 2. Just as we did for the Pla-tonic solids, we can express the numberof faces, edges and v'ertices in terms ofour basic data. Here tliis is the numberb of black faces, the number zv of whitefaces, and the parameters k, m and n.Now, because the number of faces meet-ing at a vertex is not fixed, we do notobtain an equation, but an inequalityexpressing the fact that the number offaces meeting at each \'ertex is at least 3.The result is a constraint on k, m and nthat can be put in the following form:

u) = {4, 4, 1), that satisfy the inequalityfor suitable values of b but do not arisefrom generalized soccer balls. However,Braungardt and 1 were able to determinethe values of (k, m, n) that do have real-izations as soccer balls; these are shownin the table in Figure 9, where we alsoillustrate the smallest realizations fora few types. Notice that all of the oneswith n-2 come from truncations of Pla-tonic solids.

The polyhedra listed here have vari-ous interesting properties, of which I'llmention just one. Besides entry 10 inthis table, which is of course the stan-dard soccer ball, the table contains three

kb 12 " 2/c 2 m

This may look complicated, but it caneasily be analyzed, just like the equationleading to the Platonic solids. It is nothard to show that n can be at most equalto 6, because otherwise the left-hand sidewould be greater than the right-handside. With a little more effort, if is pos-sible to compile a complete list of all thepossible solutions in integers k, in and n.

Alas, the story does not end there.There are some triples, such as (k, m.

Figure 8. Chopping off comers, or truncation,converts any Platonic solid into a general-ized soccer ball. In particular, the standardsoccer ball is a truncated icosahedron. Aftertruncation, the 20 triangular faces of the ico-sahedron become hexagons; the 12 vertices,as shown here, turn into pentagons.

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type

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

k

3

3

4

3

5

3

3

4

3

5

>3

3

4

5

>3

>3

>3

3

4

5

m n

3

4

3

5

3

3

4

3

5

3

2

2

2

2

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

4

5

6

6

6

minimal realization b w ^ H

octahedron

cuboctahedron

cuboctahedron

icosi dodecahedron

icosidodecahedron

truncated tetrahedron

truncated cube

truncated octahedron

truncated dodecahedron

truncated icosahedron = standard soccer ball

truncated beach ball

variation on the tetrahedron

variation on the cube

variation on the dodecahedron

partially truncated beach ball

double tin can

zigzag tin can

subdivision of the tetrahedron

subdivision of the cube

subdivision of the dodecahedron

4 4

8 6

6 8

20 12

12 20

4 4

8 6

6 8

20 12

12 20

2 k

4 6

6 12

12 30

1 k

2 2k

2 2k

4 12

6 24

12 6020

Figure 9. Generalized soccer balls fall into 20 types. In this table, k represents the number of sides in any black face; the product in x ti is thenumber of sides in any white face. Every side of a black face meets a white face. Every »th side of a white face meets a hlack face. The columnsb and w represent the number of black and white faces in the simplest representative of each type. For the types with n=2, every generalizedsoccer ball of that type is a branched covering of the simplest one. However, this is not true for other values of ii. The minimal realization oftype 8 is combinatorially the same as the 2006 World Cup ball shown in Figure 2, whereas type 10 is the standard soccer ball.

other fullerenes: numbers 14 and 20,and the case k=6 of entry 17. The num-bers of hexagons in these examples are30, 60 and 2, respectively. (Note that inthe latter case the color scheme is re-versed, so the hexagons are black rather

Figure 10. The tetrahedron with just one blackface (a} is the minimal realization of soccer-ball type 15 in Figure 9, where (k, m, n) = (3,1, 3). Another realization, an octahedron withtwo opposite black faces (b), is not a branchedcovering of a, showing that it is not possibleto produce all generalized soccer balls with«>2 using branched coverings.

than white.) The numbers of carbon at-oms are 80,140 and 24, respectively. Thelast of these is the only fullerene with 24atoms. In the case of 80 atoms, thereare 7 different fullerenes with disjointpentagons, but only one occurs in ourtable of generalized soccer balls. For 140atoms, there are 121,354 fullerenes withdisjoint pentagons.

Braungardt and I discovered some-thing very intriguing when we tried tosee whether every generalized soccerball comes from a branched coveringof one of the entries in our table. This istrue, we found, for all the triples withn=2, that is, for generalized soccer ballsfor which black and white faces alter-nate around the sides of each whiteface. However, it is not true for othervalues of n\ The easiest example dem-onstrating this failure arises for the tri-ple (k, in, n) = (3,1, 3), meaning that wehave black and white triangles arraiiged

in such a way that the sides of eachblack triangle meet only white ones,and each white triangle has exactly oneside that meets a black one. The mini-mal example is just a tetrahedron withone face painted black (Figure lOa). An-other realization is an octahedron withtwo opposite faces painted black (FigurelOb). This is not a branched covering ofthe painted tetrahedron! A branchedcovering of the tetrahedron would have3, 6, 9, ... faces meeting at every ver-tex—but the octahedron has 4.

The reason for this strange behavior isa subtle difference between the case H=2and the cases n>2. In the tetrahedronexample, there are two different kindsof vertices: a vertex at which only whitefaces meet, and three vertices where oneblack and two white faces meet. More-over, the painted octahedron has yet an-other kind of vertex. But in the case n=2,all the vertices look essentially the same.

356 American Scientist, Volume 94

Figure 11. Toroidal soccer balls are of twokinds: those that are branched coveringsof spherical ones, and those that are not. Abranched double cover of the standard spheri-cal soccer ball produces a toroidal ball with 24black and 40 white faces (a). Opening up thestandard soccer ball along two edges, deform-ing it to a tube and then matching the ends ofthe tube produces a toroidal soccer ball with12 black and 20 white faces (b>. This pattemcannot be obtained as a branched covering.

E\'er\' vertex lias the same sequence ofcolors, which goes black, white, white,black, white, white, ..., with only thelength of the sequence left open. Thusthe adjacency conditions provide a de-gree of control over the local structureof any generalized soccer ball with ii=2.This control is lacking in the n>2 case.At present, therefore, it is possible to de-scribe all generalized soccer balls withn=2: They are branched coverings oftruncated Platonic solids. But there is nosimple way to produce all the general-ized soccer balls with n>2.

Toroidal Soccer BallsFrom a topologist's point of view,spherical scKcer balls are just one par-ticular example of maps drawn on sur-faces. Because the definition of soccerballs through conditions (1), (2) and (3)does not specify that soccer-ball poly-hedra should be spherical, there is apossibility that they might also exist inother shapes. Besides the sphere, thereare infinitely many other surfaces thatmight occur: the torus (which is the sur-face of a doughnut), the double torus,the triple torus (which is the surface of apretzel), the quadn.iple torus, etc. Thesesurfaces are distinguished from one an-other by their gemis, informally knownas the number of holes: The sphere hasgenus zero, the tonts has genus one, thedouble torus has genus two, and so on.

There are scxcer balls tif all genera,because every surface is a branched cov-ering of the sphere (in a slightly moregeneral way than we discussed before).By arranging the branch points to bevertices of some soccer ball graph onthe sphere, we can generate soccer ballgraphs on any surface. Figure 11a showsa toroidal soccer ball obtained from a

tv\'i>-fc>ld branched covering of the stan-dard spherical ball. In this case there arefour branch points. Note that a two-foldbranched co\'ering always doubles thenumber of pentagons and hexagons.

Flere is an easier construction of atoroidal soccer ball. Take the standardspherical soccer ball and cut it openalong two disjoint edges. Openingup the sphere along each cut produc-es something that looks rather like asphere from which two disks have beenremoved. This surface has a soccer-ballpattern on it, and the two boundarycircles at which we have opened thesphere each have two vertices on them.If the cut edges are of the same type,meaning that along both of them twowhite faces met in the original sphericalsoccer ball, or that along both of them ablack face met a white face, then we canglue the two boundary circles togetherso as to match vertices with vertices.(See Figure l ib for step-by-step illustra-tions of this construction.) The surfacebuilt in this way is again a torus. It hasthe structure of a polyhedron that satis-fies conditions (1), (2) and (3), and istherefore a soccer ball.

This second toroidal soccer ball is not abranched covering of the standard spher-ical ball, because it has the same numbersof pentagons and hexagons (12 and 20respectively) as the standard sphericalball. For a branched covering these num-bers would be multiplied by the degreeof the covering. In tfiis case, the failureis not caused by loss of control over thelocal structure of the pattem (as in theprevious section), but by a global prop-erty of the torus (the hole). Thus the basicresult that all spherical soccer balls arebranched coverings of the standard oneis not true for soccer bails with holes.

CodaSoccer balls provide ample illustra-tions of the intimate connection thatexists between graphs on surfacesand branched coverings. This circle ofideas is also connected to subtle ques-tions in algebraic geometry, where thecombinatorics of maps on surfaces en-capsulates data from number theoryin mysterious ways. Following theterminology introduced by Alexan-der Grothendieck, one of the leadingmathematicians of the 20th century,the relevant graphs on the sphere arenowadays called dcssins d'eiifants.

BibliographyBraungardt, V., and D. Kotschick. 2006. The

classification of fcKitball patterns. Preprint.http://129.187.ni.185/~dieter/ftX3tball.pdf

Brinkmann,G., and A. V '. M. Dresa. 1997. A con-structive eniuneration of fullerenes. loiiriial of

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und UKuni^en 1983-1487. Stuttgart, Germany:Ernst KlettVerlag.

Chimg, R, and S. Stemberg. 1943. Mathematicsand the buckyball./ij/ifWaiiJ Sn'tvifist 81:56-71.

Coxeter, H. S. M. 1948. Regular Poli/topes. Lon-don: Methuen & Co. Ltd.

Schneps, L. (ed.). 1994. Tlie Grothendieck Theory ofDcisiiis d'Eiifmits. London Mathematical Soci-ety Lecture Note Series vol. 20(1. Cambridge,U.K.: Cambridge University Press.

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For relevant Web links, consult thisissue of American Scientist Online:

http://www.americanscientist.org/lssueTOC/issue/861

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