Figure 6. The proof that branched coverings produce all soccer balls depends on an analysis of the sequence of colors around any vertex. Because at least one of the edges meeting at each vertex bounds a pentagon black , there is no vertex where only hexagons white meet. The sequence of faces around a vertex is always black-white-white, black-white-white, and closes up after a number of faces that is a multiple of three.
There is a straightforward modification we can make to this construction. Instead of taking two-fold coverings, we can take d -fold branched coverings for any positive integer d. Instead of shrinking the sphere halfway, we imagine an orange, made up of d orange sections, and for each section we shrink a copy of the coat of the sphere so that it fits precisely over the section.
Once again the different pieces fit together along the seams see Figure 5. For all of this it is important that we think of soccer balls as combinatorial or topological—not geometric—objects, so that the polygons can be distorted arbitrarily. At this point you might think that there could be many more examples of soccer balls, perhaps generated from the standard one by other modifications, or perhaps sporadic examples having no apparent connection to the standard soccer ball.
But this is not the case! Braungardt and I proved that every soccer ball is in fact a suitable branched covering of the standard one possibly with slightly more complicated branching than was discussed above. The proof involved an interesting interplay between the local structure of soccer balls around each vertex and the global structure of branched coverings. Consider any vertex of any soccer ball see Figure 6. For every face meeting this vertex, there are two consecutive edges that meet there.
Because at least one of those two edges bounds a pentagon, by condition 3 , 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. This condition can be met only if the faces are ordered around the vertex in the sequence black, white, white, black, white, white, etc.
Remember that the pentagons are black. In order for the pattern to close up around the vertex, the number of faces that meet at this vertex must be a multiple of 3.
This means that locally, around any vertex, the structure looks just like that of a branched covering of the standard soccer ball around a branch point. Covering space theory—the part of topology that investigates relations between spaces that look locally alike—then enabled us to prove that any soccer ball is in fact a branched covering of the standard one.
To mathematicians, generalization is second nature. Even after something has been proved, it may not be apparent exactly why it is true. Testing the argument in slightly different situations while probing generalizations is an important part of really understanding it, and seeing which of the assumptions used are essential, and which can be dispensed with.
A quick look at the arguments above reveals that there is very little in the analysis of soccer balls that depends on their being made from pentagons and hexagons. Imagining that we again color the faces black and white, we assume that the black faces have k edges, and the white faces have l edges each. For conventional soccer balls, k equals 5, and l equals 6. As before, the edges of black faces are required to meet only edges of white faces, and the edges of the white faces alternately meet edges of black and white faces.
The alternation of colors forces l to be an even number. Going one step further in this process of generalization, we can require that every n th edge of a white face meets a black face, and all its other edges meet white faces. Of course we still require that the edges of black faces meet only white faces.
Let us call such a polyhedron a generalized soccer ball. The first question we must ask is: Which combinations of k, m and n are actually possible for a generalized soccer ball? It turns out that the answer to this question is closely related to the regular polyhedra. Ancient Greek mathematicians and philosophers were fascinated by the regular polyhedra, also known as Platonic solids , attributing to them many mystical properties.
The Platonic solids are polyhedra with the greatest possible degree of symmetry: All their faces are equilateral polygons with the same number of sides, and the same number of faces meet at every vertex.
Euclid proved in his Elements that there are only five such polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron see Figure 7. Figure 7. The five basic Platonic solids shown here have been known since antiquity. Examples of all generalized soccer-ball patterns can be generated by altering Platonic solids.
Although Euclid used the geometric definition of Platonic solids, assuming all the polygons to be regular, modern mathematicians know that the argument does not depend on the geometry. Each Platonic solid can be described by two numbers: the number K of vertices in each face and the number M of faces meeting at each vertex. The possible solutions can be determined quite easily.
The complete list of possible values for the pairs K, M is:. Strictly speaking, this is only the list of genuine polyhedra satisfying the above equation. The equation does have other solutions in positive integers. These solutions correspond to so-called degenerate Platonic solid s, which are not bona fide polyhedra. The first case can be thought of as a beach ball that is a sphere divided into M sections in the manner of a citrus fruit.
The Platonic solids give rise to generalized soccer balls by a procedure known as truncation. Suppose we take a sharp knife and slice off each of the corners of an icosahedron. At each of the 12 vertices of the icosahedron, five faces come together at a point.
When we slice off each vertex, we get a small pentagon, with one side bordering each of the faces that used to meet at that vertex. At the same time, we change the shape of the 20 triangles that make up the faces of the icosahedron. By cutting off the corners of the triangles, we turn them into hexagons. The sides of the hexagons 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 first kind of side borders another hexagon, and the second kind touches a pentagon. In fact, the polyhedron we have obtained is nothing but the standard soccer ball. Mathematicians call it the truncated icosahedron. Figure 8. Chopping off corners, or truncation, converts any Platonic solid into a generalized soccer ball.
In particular, the standard soccer ball is a truncated icosahedron. After truncation, the 20 triangular faces of the icosahedron become hexagons; the 12 vertices, as shown here, turn into pentagons. The same truncation procedure can be applied to the other Platonic solids. For example, the truncated tetrahedron consists of triangles and hexagons, such that the sides of the triangles meet only hexagons, while the sides of the hexagons alternately meet triangles and hexagons.
The truncated icosahedron gives values for k, m and n of 5, 3 and 2. Moreover, there is a unique way of putting them together, giving rise to the iconic standard soccer ball. The side length of one of the pentagons measures 2 inches and the apothem measures about 1. What is the area of one of the pentagons? A model of a soccer ball is made up of regular pentagons and hexagons.
According to conservative FIFA estimates, over million people play soccer worldwide. We all know the FIFA regulation size soccer ball is a size 5. How many tiles are on a soccer ball? Category: sports soccer. How many pentagons make a sphere?
Why do soccer balls have pentagons? What is the correct pressure for a size 5 soccer ball? Is a soccer ball a dodecahedron? What is a striker in soccer? What is the football shape called? What country has lost the most World Cups? What is the weight of a soccer ball? How big are the hexagons on a soccer ball? What is the name of a 20 sided polygon?
How do you make an icosahedron? Build a folded paper icosahedron. Search for:. Hint if needed: Every shape edge is shared with 1 other shape… Answers: Wee ones: 5 sides. Little kids: 6 shapes. Bonus: 11 pentagons. Big kids: 32 faces.
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