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Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron.

Is there some insight into why the alternation of pentagons and hexagons yields an approximated sphere? Is this special, or are there an arbitrary number of assorted n-gons sets that may be joined together to create regular sphere-like surfaces?

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related (not duplicate): math.stackexchange.com/questions/15438/… –  Isaac Jan 20 '11 at 19:01
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A related question on MathOverflow: mathoverflow.net/questions/19747/the-symmetry-of-a-soccer-ball –  Jonas Meyer Jan 20 '11 at 21:05

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up vote 32 down vote accepted

The possible ways to put polygons together to form a sphere-like object are constrained by Euler's formula $V - E + F = 2$ (where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces). Equivalently you can think of this as a statement about planar graphs.

Suppose we use $f_3$ triangles, $f_4$ squares, $f_5$ pentagons, etc. Every edge meets exactly two faces, and an edge of type $f_n$ meets $n$ faces, so let's double-count the number of pairs of an edge and a face next to it: on the one hand, this is $2E$, and on the other hand, this is

$$3f_3 + 4f_4 + 5f_5 + ...$$

Plugging this into Euler's formula gives $V - \frac{f_3 + 2f_4 + 3f_5 + ...}{2} = 2$. If in addition the polyhedron is convex and the polygons are regular, there are constraints on the faces that can meet at each vertex coming from the fact that the angles must sum up to less than $360^{\circ}$. (This is one way to prove the classification of Platonic solids.) For example, at most $5$ faces can meet at each vertex if we allow arbitrary faces; this means $3f_3 + 4f_4 + ... \le 5V$. (If you really want to, you can allow six triangles to touch at one point, but I would just count this as a hexagon.) If we don't allow triangles, exactly $3$ faces meet at each vertex; this means $4f_4 + 5f_5 + ... = 3V$.

Here is an application in chemistry: a fullerene is a certain type of molecule made from carbon atoms. (One of these, the buckyball, looks just like a soccer ball.) It gives a convex polyhedron in which each face is either a regular pentagon or hexagon. This gives $V - \frac{3f_5 + 4f_6}{2} = 2$ on the one hand, and $3V = 5f_5 + 6f_6$ on the other. Together these equations give $f_5 = 12$ and $V - 2f_6 = 20$; in other words, any fullerene must have exactly twelve pentagons.

(Hexagons are special. One way to interpret this result is that an infinite plane can be tiled with hexagons, so hexagons correspond to zero curvature, whereas since pentagons have a smaller angle at each vertex they correspond to positive curvature. What the above statement says, roughly, is that the total amount of curvature is a constant. This is a simple form of the Gauss-Bonnet theorem, which is closely related to Euler's formula.)

Here are some other things you can prove, again under the assumptions of convexity and regularity:

  • If you only use triangles (as opposed to triangles and hexagons), then $f_3 \le 20$ and $4 | f_3$.
  • If you only use squares, then $f_4 = 6$.
  • You cannot use only hexagons or higher. (In the Gauss-Bonnet picture, attempting to use heptagons corresponds to negative curvature, and negative curvature does not interact well with Euclidean or spherical geometry; the natural setting for putting heptagons together is instead hyperbolic geometry.)

This is already most of the way to the classification of Platonic solids. If you're interested in learning more about Euler's formula, I highly recommend David Richeson's Euler's Gem. Extremely well-written and informative. You might also enjoy David Eppstein's Nineteen ways to prove Euler's formula.

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+1. If I'm reading the fullerene article correctly, the assertion about exactly 12 pentagons should be limited to fullerenes that form closed surfaces, right? –  Isaac Jan 20 '11 at 19:12
    
Yes, it needs to be homeomorphic to a closed surface or else Euler's formula doesn't apply. (I wonder if there are fullerenes homeomorphic to a torus...) –  Qiaochu Yuan Jan 20 '11 at 19:13
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slightly related: nature.com/nature/journal/v468/n7326/full/nature09620.html @Qiaochu: carbon nanotori are studied theoretically and claimed to have been observed ; though I haven't been able to find more recent experimental work on it. –  Willie Wong Jan 20 '11 at 20:06
    
@Qiaochu: This seems similar: math.stackexchange.com/questions/10648/… –  PEV Jan 20 '11 at 20:26
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Love your answer. You clearly know what you're talking about and it's appreciated. –  ErikE Jan 20 '11 at 23:40

The soccer ball shape can be formed by truncating an icosahedron—if each of the 12 vertices is cut off, it will leave a pentagonal face and cause each of the 20 triangular faces to become hexagonal.

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You'd probably be interested in the Platonic solids, Archimedean solids, Catalan solids, and some of the Johnson solids.

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Wonderful links, thank you. It seems obvious in retrospect that any regular solid could be truncated to create a new solid, and wherever edges join in equal angles further truncation is possible. –  Phrogz Jan 21 '11 at 4:04
    
@Phrogz obvious in theory, but the results are sometimes surprising, and factors like the precise nature of the truncation matter as well. I do a lot of geometric artwork (modular origami, magnet sphere constructions, rope structures, etc) so being familiar with all of those solids and their properties is important for me. –  Sparr Jan 25 '11 at 4:31

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