However, this is not true for other values of n. The minimal realization of type 8 is combinatorially the same as the World Cup ball shown in Figure 2, whereas type 10 is the standard soccer ball.
Are these the only possibilities for generalized soccer ball patterns, or are there others? Just as we did for the Platonic solids, we can express the number of faces, edges and vertices in terms of our basic data.
Here this is the number b of black faces, the number w of white faces, and the parameters k , m and n. Now, because the number of faces meeting at a vertex is not fixed, we do not obtain an equation, but an inequality expressing the fact that the number of faces meeting at each vertex is at least 3. The result is a constraint on k, m and n that can be put in the following form :.
This may look complicated, but it can easily be analyzed, just like the equation leading to the Platonic solids. It is not hard to show that n can be at most equal to 6, because otherwise the left-hand side would be greater than the right-hand side.
With a little more effort, it is possible to compile a complete list of all the possible solutions in integers k, m and n. Alas, the story does not end there. However, Braungardt and I were able to determine the values of k, m, n that do have realizations as soccer balls; these are shown in the table in Figure 9, where we also illustrate the smallest realizations for a few types. The numbers of hexagons in these examples are 30, 60 and 2, respectively.
Note that in the latter case the color scheme is reversed, so the hexagons are black rather than white. The numbers of carbon atoms are 80, and 24, respectively. The last of these is the only fullerene with 24 atoms. In the case of 80 atoms, there are 7 different fullerenes with disjoint pentagons, but only one occurs in our table of generalized soccer balls. For atoms, there are , fullerenes with disjoint pentagons. Figure Braungardt and I discovered something very intriguing when we tried to see whether every generalized soccer ball comes from a branched covering of one of the entries in our table.
However, it is not true for other values of n! The minimal example is just a tetrahedron with one face painted black Figure 10a. Another realization is an octahedron with two opposite faces painted black Figure 10b. This is not a branched covering of the painted tetrahedron!
A branched covering of the tetrahedron would have 3, 6, 9, … faces meeting at every vertex—but the octahedron has 4. In the tetrahedron example, there are two different kinds of vertices: a vertex at which only white faces meet, and three vertices where one black and two white faces meet. Moreover, the painted octahedron has yet another kind of vertex. Every vertex has the same sequence of colors, which goes black, white, white, black, white, white, …, with only the length of the sequence left open.
Because the definition of soccer balls through conditions 1 , 2 and 3 does not specify that soccer-ball polyhedra should be spherical, there is a possibility that they might also exist in other shapes.
Besides the sphere, there are infinitely many other surfaces that might occur: the torus which is the surface of a doughnut , the double torus, the triple torus which is the surface of a pretzel , the quadruple torus, etc. These surfaces are distinguished from one another by their genus , informally known as the number of holes: The sphere has genus zero, the torus has genus one, the double torus has genus two, and so on.
Toroidal soccer balls are of two kinds: those that are branched coverings of spherical ones, and those that are not. A branched double cover of the standard spherical soccer ball produces a toroidal ball with 24 black and 40 white faces a.
Opening up the standard soccer ball along two edges, deforming it to a tube and then matching the ends of the tube produces a toroidal soccer ball with 12 black and 20 white faces b. This pattern cannot be obtained as a branched covering. There are soccer balls of all genera, because every surface is a branched covering of the sphere in a slightly more general way than we discussed before. By arranging the branch points to be vertices of some soccer ball graph on the sphere, we can generate soccer ball graphs on any surface.
Figure 11a shows a toroidal soccer ball obtained from a two-fold branched covering of the standard spherical ball. In this case there are four branch points. Note that a two-fold branched covering always doubles the number of pentagons and hexagons. Here is an easier construction of a toroidal soccer ball. Take the standard spherical soccer ball and cut it open along two disjoint edges. Opening up the sphere along each cut produces something that looks rather like a sphere from which two disks have been removed.
This surface has a soccer-ball pattern on it, and the two boundary circles at which we have opened the sphere each have two vertices on them.
If the cut edges are of the same type, meaning that along both of them two white faces met in the original spherical soccer ball, or that along both of them a black face met a white face, then we can glue the two boundary circles together so as to match vertices with vertices. See Figure 11b for step-by-step illustrations of this construction.
The surface built in this way is again a torus. It has the structure of a polyhedron that satisfies conditions 1 , 2 and 3 , and is therefore a soccer ball. This second toroidal soccer ball is not a branched covering of the standard spherical ball, because it has the same numbers of pentagons and hexagons 12 and 20 respectively as the standard spherical ball. For a branched covering these numbers would be multiplied by the degree of the covering. In this case, the failure is not caused by loss of control over the local structure of the pattern as in the previous section , but by a global property of the torus the hole.
Thus the basic result that all spherical soccer balls are branched coverings of the standard one is not true for soccer balls with holes. Soccer balls provide ample illustrations of the intimate connection that exists between graphs on surfaces and branched coverings. This circle of ideas is also connected to subtle questions in algebraic geometry, where the combinatorics of maps on surfaces encapsulates data from number theory in mysterious ways.
Skip to main content. Login Register. Page DOI: Examined from the perspective of graph theory, the standard soccer ball has three important properties: 1 it is a polyhedron that consists only of pentagons and hexagons; 2 the sides of each pentagon meet only hexagons; and 3 the sides of each hexagon alternately meet pentagons and hexagons.
The discovery of the buckyball, which was honored by the Nobel Prize for chemistry, created enormous interest in a class of carbon molecules called fullerenes, which satisfy assumption 1 above together with a further condition: 3' precisely three edges meet at every vertex. The complete list of possible values for the pairs K, M is: 3, 3 for the tetrahedron 4, 3 and 3, 4 for the cube and the octahedron 5, 3 and 3, 5 for the dodecahedron and the icosahedron.
Michael Trott and Barbara Aulicino. Bibliography Braungardt, V. The classification of football patterns. A constructive enumeration of fullerenes. Journal of Algorithms Bundeswettbewerb Mathematik. Stuttgart, Germany: Ernst Klett Verlag. The 20 hexagons have 6 edges apiece, giving us edges in total, while the 12 pentagons have 5 edges each, or 60 in total. That gives us edges that need to meet up with each other.
Each one takes another one away from the pile, so there are 90 pairs. Her goal is to make math as playful for kids as it was for her when she was a child. Her mom had Laura baking before she could walk, and her dad had her using power tools at a very unsafe age, measuring lengths, widths and angles in the process. The buckyball, or "buckminster" ball, is the name given to the most common type of soccer ball today. It's official shape name is a spherical polyhedron but it is affectionately known as the buckyball after Richard Buckminster Fuller, an architect who was trying to find a way to construct buildings with minimum materials, according to Soccer Ball World Website.
The modern soccer ball has 32 panels of leather or synthetic plastic tightly stitched together. Twenty of these panels are hexagons and 12 of them are pentagons. The hexagons and pentagons are equally important as they fit together like a puzzle to form a perfectly spherical shape.
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What is the numbers hexagons and pentagons on a official soccer ball? Study Guides. Trending Questions.
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