Why are there exactly five Platonic solids?

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You know what a cube is: a solid three-dimensional object withsix identical sides, each of which is a square. The cube is one of thefive Platonic solids: highly symmetrical shapes known at leastas far back as the ancient Greeks. In addition to the six-sided cube,we have the four-sided tetrahedron, theeight-sided octahedron, the twelve-sided dodecahedron,and the twenty-sided icosahedron. Their common feature is thatall their sides are identical. For example, all four sides of thetetrahedron are equilateral triangles, and all twelve sides of thedodecahedron are regular pentagons. Since all of their faces areidentical, they are all well suited to use as dice. Many games usecubes (common six-sided dice) since they are easy to manufacturer andship, but the popular (at least when I was growing up, before computergames appeared) fantasy role-playing gameDungeons and Dragons uses all five kinds, as shown below.

It has long been known that these five are the only shapes withthis sort of complete symmetry. Today I want to discuss the question"Why are there exactly five Platonic solids?" Why not more, or fewer?Why these particular five?

First, let's be clear about the definition of a Platonic solid. It isa geometric shape like the ones pictured above, all of whose faces areidentical regular polygons. What's a "polygon", and what is "regular"?

Well, start by drawing some dots on a piece of paper. We will refer tothem as vertices. Now connect the dots with straight lines(called edges) that do not cross, to form a closedshape. That's a polygon. For example, atriangle has 3 vertices and 3 edges.

A regular polygon is one for which all the edges are the samelength, and all the angles between the edges are the same. Forinstance, a triangle with all three edges of equal length is regular;we call it an equilateral triangle. If a four sided polygon isregular, we call it a square. If a five sided one is regular,we call it a regular pentagon.

So, to make a Platonic solid, you pick a regular polygon: either anequilateral triangle, a square, or a regular pentagon. You make abunch of copies of the polygon; we call these the faces.Finally, you glue them together along adjacent edges to form a solidthree-dimensional shape.

However, if you try this with regular hexagons (6-sided polygons), youwill find that you cannot glue them into a solid "dice-like" shape. Itjust does not work: the angles are not right. This provides a clue tofiguring out why there are only five possibilities.

Imagine we draw three squares together as shown in (A) below, then cutout the L-shape and fold on the edges between colors. The three facesmeet at a common vertex, forming half of a cube. But if we hadincluded the fourth square, as in (B), we could not achieve this: anyattempt to fold the edges down would not fit together.

Let's see if we can generalize this observation, so we can characterizewhich shapes will and will not fit together.

If we draw several identical regular polygons around a common vertex,like the three squares above, we have to leave a gap, like in (A), ifwe want to be able to fold it into a three-dimensional shape. Sincesquares have 90 degree angles between adjacent edges, four squarestake up all 360 degrees: no gap. We are limited to using threesquares, and we wind up creating the cube.

Two squares are insufficient, since we need three to form a threedimensional corner. So the cube is the only possible Platonicsolid built from squares.

Equilateral triangles have an interior angle of 60 degrees, so six ofthem completely fill the space around a vertex, just like four squaresdo. So, we are limited to using 3, 4, or 5 triangles around a vertex,if we want to build a Platonic solid. These give the tetrahedron,octahedron, and icosahedron, respectively.

With just three triangles around each vertex, the tetrahedron is very"spiky": the points are sharp, because we have to fold the papertightly to glue the edges together. Try it yourself with a piece ofpaper: print the picture above, cut out the three-triangle strip, foldwhere the colors change, and try to bend it to make a threedimensional corner. In contrast, when we make the icosahedron, with five trianglesaround each vertex we do not need to fold the paper as sharply, so thecorners are not as sharp. In fact, the icosahedron is much more like asphere (smooth all over), in comparison to the tetrahedron.

We can also use regular pentagons. The angle between adjacentedges is 3*180/5 = 108 degrees, so we can fit three together, leavinga gap, much like the three squares in (A), but four will not fit. Sothere is only one possible Platonic solid made from pentagons, namelythe dodecahedron.

Finally, if we try something with 6 or more sides, like ahexagon, the interior angles will be 120 degrees (for hexagons) ormore (for 7 or more sided shapes), so even three will fill or overfillthe 360 degrees available, with no room for a gap. So there are noPlatonic solids with faces of six or more sides. That means the five wefound are all of them: there are no others.

It turns out that there is an entirely different approach wecould have used. The analysis of angles we discussed above was knownto Euclid around 300 B.C., but this second approach is much morerecent. Various mathematicians, including Descartes in 1639, Euler in1751, and Cauchy in 1811, contributed to the discovery that for any"sphere-like polyhedron" (i.e. any geometric solid built frompolygons, without holes, such as the Platonic solids), there is arelationship between the number of vertices V, edges E, and faces F,often called Euler's formula:

V - E + F = 2

Start with a cube, for example. It has V=8 corners, E=12edges, and F=6 faces; 8-12+6 = 2 as claimed. Now imagine we take oneof the square faces of the cube and cut it into 2 triangles, asillustrated below:

What does this do? The number of vertices does not change, but thenumber of faces goes up by one, as does the number of edges. So theoverall value of V-E+F does not change, since the extra face cancelsthe extra edge. A similar argument shows that no matter how youadd or remove vertices, faces and edges, the value of V-E+F does notchange, provided you do not cut holes through the solid (i.e. turn itinto a doughnut or inner-tube shape).

Using Euler's formula, we can classify the Platonic solidswithout having to calculate with angles. Suppose the Platonic solid ismade from n-sided polygons, and suppose k of them meet at everyvertex. For instance, the cube has k=3 polygons around each vertex,and each of the polygons is a square, with n=4 edges.

If there are F faces, then before we glue them together to build thesolid, we have nF edges and nF vertices. We glue edges in pairs, sothere will be E = nF/2 edges in the final solid. We glue k verticestogether at each corner, so there will be V = nF/k overall in thefinal solid. By Euler's formula, we have

nF/k - nF/2 + F = 2.

Dividing by twice the number of edges (2E) gives

1/k + 1/n = 1/E + 1/2.

Since E>0, we know 1/k + 1/n must exceed 1/2. We also know that n isat least 3 (triangles have 3 sides, other polygons have more) and thatk is at least 3 (we need at least three faces per vertex to get athree dimensional solid).

What whole numbers n and k, each at least 3, satisfy the constraintthat 1/k + 1/n must exceed 1/2? We can list them out.

When k=3, 1/n must exceed 1/6, so n can only be 3, 4, or 5.

When k=4, 1/n must exceed 1/4, so n can only be 3.

When k = 5, 1/n must exceed 0.3, so n can only be 3.

And if k is 6 or more, 1/n must exceed 1/3, so there are no solutionsfor n (since n must be at least 3).

So once again, we have exactly five solutions, no more and noless, and once again they correspond exactly to the known Platonicsolids. The tetrahedron has k=3 triangles (n=3) meeting at eachvertex. The cube has k=3 squares (n=4) and the dodecahedron has k=3pentagons (n=5) meeting at each vertex. The octahedron has k=4triangles (n=3) at each vertex, and the icosahedron has k=5 triangles(n=3) at each vertex.

There are other interesting questions you can ask aboutgeometric shapes. For instance, someday I intend to do an article on"tessellation", which is the process of covering a sheet of paper witha repeating grid of shapes (not just squares, but triangles andhexagons work too). One can also ask more exotic questions, such as:how many four-dimensional Platonic "hypersolids" are there? But thatrequires being able to visualize geometry in 4 dimensions, which istoo tricky to delve into today.

If you enjoyed this article, you might also like Algorithmic Art. Or perhaps Counting and Number Systems.

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