TOPOLOGY AND COMBINATORICS OF SOCCER BALLS
D. KOTSCHICK
With the quadrennial soccer World Cup taking place again in June and
July of 2006, more than a billion people around the world are finding their
TV and computer screens covered with depictions of soccer balls. Although
there is now a large number of different designs for soccer balls, these different 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 ball
is stitched or glued together in the classical way from 12 pentagons and
20 hexagons, arranged so that every pentagon is surrounded by hexagons.
Postmodern paint jobs notwithstanding, the traditional way is to paint the
pentagons black and the hexagons white. The resulting image is ubiquitous,
particularly in Europe, not only during the World Cup, as it is used to promote all kinds of merchandise, not all of it soccer-related. This traditional
soccer ball pattern also arises in chemistry as the spatial structure of the
fullerene C60 , and it was used by the architect Buckminster Fuller for the
construction of cupolas and domes, now referred to as buckyball domes.
So why does the standard soccer ball look the way it does? Are there
other ways of putting it together? Perhaps pentagons and hexagons could
be arranged differently. Perhaps other polygons could be used instead of
pentagons and hexagons. These questions can be tackled using the language of mathematics, particularly geometry, group theory, topology and
graph theory. Each of these subjects provides concepts and a natural context for phrasing questions such as those about the design of soccer balls,
and sometimes for answering them as well.
S OCCER BALLS AND
FULLERENES
A convex polyhedron in three-dimensional space, like the cube or the
tetrahedron, can always be inflated to a sphere. The vertices and edges of
the polyhedron then trace out a graph on the sphere, and the combinatorics
of this graph reflects some of the structure of the polyhedron. What is lost is
the geometric information about angles and side lengths. In this way polyhedra can be thought of as objects of graph theory or topology, which is often described as “rubber-sheet geometry”, precisely because it concentrates
c D. Kotschick 2006.
Date: December 13, 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 and
topology. For the purposes of this discussion a soccer ball is defined to be a
spherical polyhedron consisting of pentagons and hexagons, satisfying the
following 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 angles and side lengths of the pentagons and hexagons are not specified, because in terms of graph theory and topology these concepts cannot even be
formulated, as they are not preserved by continuous deformation.
If one thinks of the pentagons painted black and the hexagons painted
white, these conditions do capture the familiar image—but they do not determine it uniquely. It turns out that there are infinitely many polyhedra
satisfying these conditions. A recent paper [1] exhibits a complete description of the infinite variety of soccer balls.
The carbon molecules referred to as fullerenes are also spherical polyhedra consisting of pentagons and hexagons, with the vertices occupied by
carbon atoms and the edges corresponding to chemical bonds. Fullerenes
must satisfy a different constraint instead of (1) and (2) above, in that they
are required to have precisely three edges meeting at every vertex. Sometimes condition (1) is imposed as an additional constraint, to define a restricted class of fullerenes. Having disjoint pentagons is expected to be
related to the chemical stability of fullerenes.
The standard soccer ball design with 12 pentagons and 20 hexagons is
also a fullerene. Placing carbon atoms at its vertices, one obtains the C60
molecule, whose discovery was honored by the 1996 Nobel Prize for chemistry. It is quite remarkable that although there is an infinite number of soccer balls and an infinite number of fullerenes, the two families of polyhedra
have only the standard soccer ball in common.
To see that this is so, one has to delve into the analysis of polyhedra using
Euler’s formula. This formula, found by Swiss mathematician Leonhard
Euler 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’s
formula reads
v−e+f = 2.
We want to apply Euler’s formula to a polyhedron consisting of b black
pentagons 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 is
e = 12 (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, and
as the pentagons meet vertices 5b times and the hexagons meet vertices
6w times, the number of vertices is v = 13 (5b + 6w). If we substitute
these values for f , e and v into Euler’s formula, then the terms involving w
cancel out, and the formula reduces to b = 12. Therefore, every fullerene
has exactly 12 pentagons, but we know nothing about the number w of
hexagons. 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 the
molecule. 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 each
of its 20 vertices. This is the dodecahedron, which, like the cube and the
tetrahedron, is a Platonic solid, about which we will have more to say later
on. For positive values of w one obtains the fullerenes that have more than
20 atoms and that consist of both pentagons and hexagons.
If one imposes the additional requirement that in a fullerene no two pentagons share an edge, then one can show that w has to be at least 20. The
standard soccer ball realizes this minimal value, for which there are 60 vertices, corresponding to the 60 atoms in the C60 molecule.
For arbitrary soccer balls the number of faces meeting at a vertex is not
determined by their definition, other than that this number is at least 3.
Therefore, one has to replace the equation v = 13 (5b + 6w) by the inequality
v ≤ 31 (5b + 6w). Substituting into Euler’s formula, the terms involving w
again cancel out, and one obtains the inequality b ≥ 12. Thus every soccer
ball contains at least 12 pentagons, but, unlike a fullerene, may well contain
more. This happens exactly if there is at least one vertex at which more than
three faces meet.
Also unlike fullerenes, soccer balls have a precise relationship between
the number of pentagons and the number of hexagons. Counting the number
of edges along which pentagons and hexagons meet, condition (1) says that
this number equals 5b, and condition (2) says that this number equals 21 ·
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 is
the only(!) spherical polyhedron made up of 12 pentagons and 20 hexagons
satisfying (1) and (2).
4
D. KOTSCHICK
We have now verified that the standard soccer ball with 12 pentagons and
20 hexagons is the only soccer ball that is also a fullerene. This shows that
conditions (1) and (2) defining soccer balls are remarkably restrictive, because there are 1812 distinct fullerenes with 12 pentagons and 20 hexagons,
see [2], for example. All but one have adjacent pentagons, and therefore fail
condition (1).
N EW
SOCCER BALLS FROM OLD
The standard soccer ball with 12 pentagons and 20 hexagons has three
faces 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 precisely
three faces meeting at every vertex. What other soccer balls are there, with
more than three faces meeting at some vertex, and how can we understand
them?
It turns out that one can generate infinite sequences of other examples by
a topological construction called a branched covering. To explain what this
means, we place the standard soccer ball pattern on the surface of the earth
in such a way that two vertices are at the north and south poles. We select a
sequence of edges connecting the two poles, and we distort the soccer ball
pattern on the surface of the earth so that our path along edges from pole to
pole is a path of constant geographical longitude, longitude zero, say.—It’s
alright 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, and
we slice open the earth along this semicircle. Next we shrink the slicedopen coat of the earth in the east-west direction holding the north and south
poles fixed, until the coat covers exactly half the sphere, say the western
hemisphere. We can take a second copy of this shrunken coat of the sphere
and place it over the eastern hemisphere, so that it is the image of the western 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 pentagons and hexagons as we started with! The reason is that at each of the
two seams running between the north and south poles the two sides of the
seam 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 back
together again, so that the adjacency conditions (1) and (2) defining soccer
balls are fulfilled.
One says that the new soccer ball constructed in this way is obtained
from the old one by passage to a two-fold branched covering whose branch
points 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 the
standard soccer ball with three faces meeting at every vertex, then the new
ball has two distinguished vertices, one at the north and one at the south
poles, with six faces meeting at each of them, and it has 2(60-2)=116 other
vertices, with three faces meeting at each of them.
It is clear that there are a number of variations on this construction. First
we need not start with the standard soccer ball, but we can start with any
soccer ball. In particular, we can iterate the passage to branched coverings.
Second we can choose the vertices at which to place the branch points arbitrarily, by distorting a given soccer ball in such a way that two chosen
vertices 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 that
after slicing open the coat of the earth, we do not shrink it to fit over a hemisphere, but we shrink it further, so that it fits precisely over one segment of
a segmentation of the sphere into D orange segments. Then we spin this
around the north-south axis to cover the other D-1 orange segments, and
fit everything together along the D seams. For all this it is important that
one 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 of
soccer balls, obtained by applying some other modification to the standard
example, or that just appear sporadically and have some large number of
faces, without any apparent connection to the standard example. That this
does not happen is the main Theorem proved in [1]. It is shown there that
every 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 this
vertex, 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 where
only hexagons meet. Thus at every vertex there is a pentagon. Its sides meet
hexagons, and the sides of the hexagons alternately meet pentagons and
hexagons. Therefore the number of faces meeting at a vertex is a multiple of
3, 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 means
that locally, around a vertex, the structure looks like that of a branched
covering around a branch point. This control over the local configurations
around arbitrary vertices is essential for the analysis of soccer balls.
6
D. KOTSCHICK
T OROIDAL SOCCER BALLS
While a convex polyhedron always inflates to form a sphere, there are
other, non-convex, polyhedra whose surfaces are more complicated. The
surfaces that arise as the boundaries separating the interior and exterior
of 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 double torus, the triple torus, which is the boundary of a pretzel, the quadruple
torus, 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, the
double torus has genus two, and so on.
Starting from a sphere, one can construct a torus by removing two small
disks from the sphere, and gluing the two boundary circles to each other.
This procedure can be applied several times to construct surfaces of arbitrarily large genera.
Mathematicians are not satisfied with only spherical soccer balls. They
are just as interested in studying soccer balls of higher genera. They then
consider polyhedra, not necessarily convex, that consist of pentagons and
hexagons satisfying the same conditions (1) and (2) satisfied by spherical
soccer balls. There are soccer balls of all genera, for example, because every
surface is a branched cover of the sphere (in a slightly more general way
than we discussed before). One can arrange the branch points to be vertices
of some soccer ball graph on the sphere, and look at the preimage of this
graph on the higher genus surface. This gives the surface the structure of a
soccer ball.
Here is an easier construction for the torus. Let us take a spherical soccer
ball, and cut it open along two disjoint edges. We open up the sphere along
each cut, and what we obtain looks rather like a sphere from which two
disks have been removed, but now we have a soccer ball pattern on it, and
the two boundary circles at which we have opened the sphere each have two
vertices on them, which are the endpoints of the cut edges. If the cut edges
are of the same type, meaning that along both of them two hexagons met
in the original spherical soccer ball, or that along both of them a pentagon
met a hexagon, then we can glue the two boundary circles together so as to
match vertices with vertices, and so that the resulting pattern on the torus
satisfies conditions (1) and (2).
This toroidal soccer ball has the same number of faces and edges as the
original spherical one, but the number of vertices is smaller by two. Therefore the alternating sum v − e + f is zero, rather than 2. This is an instance
of the general Euler formula, which says that for a non-empty connected
graph 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 classical case of the Euler formula. For the torus we have g = 1 and therefore
2 − 2g = 0.
Let us perform the above construction starting from the standard spherical soccer ball. Then we obtain a toroidal soccer ball with 12 pentagons
and 20 hexagons. The number of vertices is 58 rather than 60, although
the number of edges is the same as in the spherical case. But now there
are two special vertices at each of which 6 faces meet, whereas at all the
others three faces meet. Could this be a branched covering of the standard
spherical soccer ball? The answer is no!
There are many other examples of toroidal soccer balls, and of soccer
balls of higher genera, which are not branched coverings of the standard
spherical one.
B EYOND PENTAGONS AND HEXAGONS
Clearly there is very little in the analysis of soccer balls that depends on
them being made from pentagons and hexagons. One can consider convex
polyhedra made of two kinds of polygons: black k-gons having k vertices
and edges each, and white l-gons having l vertices and edges each. Then
one can require that the sides of a k-gon meet only sides of l-gons, and the
sides of l-gons alternately meet k-gons and l-gons. Of course the alternating condition only makes sense if l happens to be an even number. More
generally, if l equals a product n · m, then one can require that every nth
side of every l-gon meet a k-gon, and all its other sides meet l-gons. In the
special case when n = 1, this means that all sides of the l-gons meet sides
of 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, not
necessarily 5, 6 and 2. But precisely which values are possible? Perhaps
surprisingly, only a few. The determination of all the possibilities, carried
out in [1], is closely related to the regular polyhedra known as Platonic
solids.
T HE P LATONIC SOLIDS
Geometrically, a regular polyhedron is made up of equilateral polygons
in the most symmetric way possible, so that the same number of faces meet
at all vertices. It was known to Euclid and Plato in antiquity, that there are
only five such completely regular polyhedra: the tetrahedron, the cube, the
octahedron, the dodecahedron and the icosahedron.
8
D. KOTSCHICK
For the purposes of topology or graph theory, one dispenses with geometric properties like having all edges of the same length. Thus one considers
polyhedra consisting of polygons with some fixed number K of edges, and
requires that M of these polyhedra meet at every vertex, without imposing
any geometric conditions. The basic question then is, which values of K
and M are possible for spherical polyhedra? The answer is that only the
values realised by the Platonic solids are possible. Thus, there is in fact
no additional freedom gained by replacing the rigid geometric definition of
regular polyhedra by the more flexible topological definition.
The key to the topological determination of the Platonic solids is Euler’s
formula v − e + f = 2. Suppose that one has f polygonal faces, all of them
K-gons. Then the number of edges is e = 12 K · f . If M faces meet at every
vertex, then the number of vertices is v = M1 K · f . Substituting these values
in Euler’s formula, elementary algebra leads to the equation
1
1
1
1
+ =
+
.
K ·f
4
2K 2M
If both K and M are at least 4, then the right hand side of this equation is at
most 14 , which is not possible, as the left hand side is strictly larger than 14 .
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 all
of these values correspond to actual regular polyhedra: the tetrahedron, the
octahedron and the icosahedron.
If M = 3, then similarly K has to be 3, 4 or 5, corresponding to the
tetrahedron, the cube and the dodecahedron.
There is a duality here given by interchanging the rôles of K and M: the
cube and the octahedron are dual to each other, and so are the dodecahedron
and the icosahedron. The tetrahedron is self-dual. Dual polyhedra have the
same numbers of edges, and the duality interchanges the numbers of faces
and vertices.
The above equation does have other solutions in positive integers, which
correspond to so-called degenerate Platonic solids, that are not bona fide
polyhedra. 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 of
a 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 = v
is unconstrained, and f = M = 2. These values are realized by the division
of a sphere into two K-gons meeting along their edges.
G ENERALIZED SOCCER BALLS
The standard soccer ball with 12 pentagons and 20 hexagons is derived
from the icosahedron by a procedure known as truncation. The icosahedron
is 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 of
the 12 vertices one obtains a new face. These faces are pentagons, because
there were 5 faces meeting at each of the vertices of the icosahedron. The
triangular faces of the icosahedron are having their corners sliced off, so
they become hexagons. The sides of such a hexagon are of two kinds, which
occur alternately: the remnants of the sides of the original triangular faces
of 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. For
example, performing it on the tetrahedron produces a polyhedron consisting of triangles and hexagons, such that the sides of each triangle meet only
sides of hexagons, and the sides of the hexagons alternately meet triangles
and hexagons. Concerning the question which values of k and l are possible
for a generalized soccer ball consisting of k-gons and l-gons, the truncated
icosahedron of course gives the values (k, l) = (5, 6). The truncations of
the 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 dodecahedron.
The pairs (k, l) do not look very symmetric, and the duality between
pairs of Platonic solids seems to have disappeared. The reason is that we
neglected 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: it
is 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 spherical
polyhedron made of k-gons and l-gons with l = n · m, in such a way that
every side of a k-gon meets a side of an l-gon, and every nth side of every
l-gon meets a k-gon, and all its other sides meet l-gons? All we have to do
is 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 the
number of l-gons by w, thinking of them as white. Given k and m, the two
numbers 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, we
find b · k = w · m.
The total number of faces is f = b + w, and the number of edges is
e = 12 (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 have
a total number of w · l. As for the soccer balls made from pentagons and
hexagons, we do not know how many faces meet at a vertex, except that
this number has to be at least 3. This tells us that the number of vertices
is v ≤ 13 (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
n+1
1
1
+
≤
+
,
k·b
12
2k 2m
which is obviously very similar to the equation we had to solve to find the
Platonic solids1. Of course the situation now is more complex, because
there is the additional variable n, and because we have only an inequality,
not an equation. Nevertheless, one can analyze this inequality, and compile
a list of all the possible solutions in positive integers (k, m, n) with k ≥ 3
and l = n · m ≥ 3.
Alas, the story does not end there. Satisfying the above inequality is
only a necessary condition a triple (k, m, n) must meet; it is not a sufficient
condition. 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 do
not arise from generalized soccer balls.
Thus determining a list of candidate triples is only the first step. The
necessary second step then is to find out which of these candidates can be
realized by actual polyhedra. The search for examples is guided by the
following observation. Once we fix a triple (k, m, n), the above inequality
gives 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 minimal
numbers 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 five
different 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) with
arbitrary 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 truncate
it at its two poles, we get exactly what is required: a polyhedron with two
k-gons and k quadrilaterals arranged in the correct adjacency pattern.
1
This 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 by
the truncated Platonic solids, if we allow the degenerate American football
among 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 realizations
with three faces meeting at every vertex. All these minimal realizations are
cooked up from Platonic solids in one way or another. When n = 1, the
minimal number of faces meeting at a vertex is 4, and the minimal realizations are given by an octahedron, whose faces are painted black and white in
a suitable way, and by the cuboctahedron and the icosidodecahedron. These
last two polyhedra, like the truncated Platonic solids, are examples of socalled Archimedean solids, which generalize the Platonic solids, cf. [3].
The complete list of triples (k, m, n) that do occur, together with a minimal realization for each of them, is given in [1], and we reproduce it here
as Figure 1.
The polyhedra listed as minimal realizations of the generalized soccer
ball patterns have various interesting properties. We mention just one,
which is relevant to our earlier discussion of fullerenes. Entry 10 in the table
is of course the classical soccer ball C60 , but there are four other fullerenes
in the table: numbers 14 and 20, shown here in the margin, and the case
k = 6 of number 17. The numbers of hexagons in these examples are
30, 60 and 2 respectively, so that the numbers of vertices, i. e. the numbers
of carbon atoms in the molecules, are 80, 140, and 24 respectively. In the
examples 14 and 20 the pentagons are disjoint, whereas in the k = 6 case
of number 17, the pentagons meet along edges. In fact, this is the only
fullerene with 24 atoms. In the case of 80 atoms there are 7 fullerenes with
disjoint pentagons, but only one occurs in our table of generalized soccer
balls. For 140 atoms the number of fullerenes with disjoint pentagons is
121354, see [2].
W HERE BRANCHED COVERINGS DO NOT SUFFICE
We have seen that there are infinitely many combinatorially distinct spherical 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 can
understand these infinitely many examples as branched coverings of the
minimal 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 course
take branched coverings to produce infinitely many distinct realizations.
The only difficulty is in proving that there are no other realizations, which
12
D. KOTSCHICK
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
3
4
3
5
3
3
4
3
5
3
2
2
2
2
1
1
1
1
1
1
n
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
4
5
6
6
6
minimal realization
octahedron
cuboctahedron
cuboctahedron
icosidodecahedron
icosidodecahedron
truncated tetrahedron
truncated cube
truncated octahedron
truncated dodecahedron
truncated icosahedron = standard soccer ball
prism or truncated American football
variation on the tetrahedron
variation on the cube
variation on the dodecahedron
pyramid or partially truncated American football
double tin can
zigzag tin can
subdivision of the tetrahedron
subdivision of the cube
subdivision of the dodecahedron
b
4
8
6
20
12
4
8
6
20
12
2
4
6
12
1
2
2
4
6
12
w
4
6
8
12
20
4
6
8
12
20
k
6
12
30
k
2k
2k
12
24
60
F IGURE 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., by
switching the roles of k and m, which for n = 1 equals l.
Case 1. is self-dual. Similarly, cases 7. and 8., respectively
9. and 10., are dual to each other with the duality induced by
the duality of Platonic solids, and case 6. is self-dual.
are not branched coverings of the minimal one. It is shown in [1] that for
all triples with n = 2 it is true that all possible spherical realizations are
branched 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 a
way that the sides of each black triangle meet only sides of white ones, and
that each white triangle has exactly one side that meets a black triangle. The
minimal realization listed under number 15 in the table is a pyramid over a
triangular base. We can also think of this as a painted tetrahedron, in which
one 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 all
others 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 the
cases n > 2. In the example we have n = 3, and there is no way to control
what happens at a vertex. The painted tetrahedron has two very different
types of vertices: a vertex at which only white faces meet, and three vertices
where there are one black and two white faces. The painted octahedron has
all vertices the same, but they are different from the vertices of the painted
tetrahedron, 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 the
adjacency conditions of a soccer ball imply that around each vertex one has
a particular sequence of faces. This local structure is crucial for the proof
that all spherical realizations arise as branched coverings of the minimal
one, cf. [1].
Acknowledgements: I am grateful to V. Braungardt and to A. Jackson for help
with the preparation of this article.
This article is a slight revision of a text I wrote in March of 2006 to popularize
the results of my joint paper [1] with V. Braungardt. After a lot of editing, an article
loosely based on my text was published in the American Scientist [4], and a partial
German translation of the American Scientist article also appeared in Spektrum der
Wissenschaft [5]. I have decided to make this version of my original text available
because I believe it has certain qualities that were lost in the editing process of
American Scientist and Spektrum der Wissenschaft. I am grateful to M. Trott of
Wolfram Research for permission to use some of his Mathematica pictures that he
originally produced to illustrate the American Scientist article [4].
R EFERENCES
1. V. Braungardt and D. Kotschick, The classification of football patterns, Preprint
arXiv:math.GT/0606193; to appear in German translation in Math. Semesterberichte.
2. G. Brinkmann and A. W. M. Dress, A constructive enumeration of fullerenes, Journal
of Algorithms 23 (1997), 345–358
3. H. S. M.Coxeter, Regular Polytopes, Methuen & Co. Ltd. London 1948.
4. D. Kotschick, The Topology and Combinatorics of Soccer Balls, American Scientist
94 (2006), 350–357.
5. D. Kotschick, Topologie und Kombinatorik des Fußballs, Spektrum der Wissenschaft,
Juli 2006, 108–115.