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Regular Solids

© Copyright 1998, Jim Loy

Regular solids (regular polyhedra, or Platonic solids which were described by Plato) are solid geometric figures, with identical regular polygons (such as squares) as their faces, and with the same number of faces meeting at every corner (vertex).

If each corner consisted of two squares, then we end up with a "solid" made up of two squares, back to back. That's not an interesting solid. So it is not legal, as a regular solid. This eliminates the infinite number of regular polygons (triangle, square, pentagon, ...) back to back. These cannot form regular solids. So, we deduce that each vertex must have at least three faces.

If each corner consisted of four squares, we cannot enclose a space. The object that we do get is a plane, covered with squares. We say that squares can tile the plane. So we cannot produce a regular solid with four squares at each vertex. If the angles at our vertex add up to 360°, then we tile the plane, and cannot have a regular solid.

If we try to add a fifth square, we find that we have no room for our fifth square. We would have to deform our squares. And then they would cease to be squares. So, if the angles of the vertex add up to more than 360°, then we cannot fit these polygons into one corner, and we get no regular solid.

So, 360° is the upper limit on the sum of the angles at a vertex. And, we cannot have fewer than three faces at a vertex. So, there are a limited number of regular solids:

	The equilateral triangle is the simplest regular polygon. Let's start with three equilateral triangles at a vertex (total angle 180°). And we get a tetrahedron (4 faces, 4 vertices).
	We next try four equilateral triangles at each vertex (total angle 240°). And we get an octahedron (8 faces, 6 vertices).
	Now we try five equilateral triangles at each vertex (300°) We end up with an icosahedron (20 faces and 12 vertices). A sixth equilateral triangle (at a vertex) will tile the plane (360°).
	So, we try the second simplest regular polygon, the square. Three squares at each corner (270°) forms a cube, or hexahedron (6 faces and 8 vertices). And a fourth square tiles the plane (360°), as we saw with six triangles.
	The next simplest regular polygon is the regular pentagon. Three pentagons at each vertex (324°) produces a dodecahedron (12 faces and 20 vertices). Three regular hexagons, at each vertex, tile the plane (360°). And there is no room, at a vertex, for more complicated regular polygons.

So, there are only five regular solids:

As you can see, there is a kind of symmetry between pairs of regular solids. And they follow Euler's formula for general convex (not necessarily regular) polyhedra (solids): faces + vertices=edges + 2. It works for some non-convex polyhedra, and does not work for others (those with holes in them or those with points coming out of faces).

Addendum #1:

Science Fair Project: Making the five regular solids (out of toothpicks or paper or something else) may make a good science fair project. Addendum #5 (below) shows models for the five regular polyhedra. If your project involves the five regular solids, be prepared to explain (as part of the project) why there are only five of them (as I do above). I think that is what makes this subject so interesting. If you don't cover that part of the subject, some skeptics will certainly ask you about it.

If you make models out of toothpicks, you will notice that some of these solids are rigid (keep their shape) while others are not. Can you guess what feature of these solids makes them rigid? It turns out that the solids made of triangles are rigid (usually); the others (cube and dodecahedron) are not. The cube twists all over. And the dodecahedron squashes fairly flat. You may have to make cross-braces to make these rigid. The cube is made up of squares, which are equilateral quadrilaterals. There are other equilateral quadrilaterals (rhombuses). So, the square can be distorted into other rhombuses (the angles can change, in certain ways). And the cube (being made up of squares) can be distorted as well. An equilateral triangle cannot be distorted. Although it can be moved from place to place, and rotated; its angles cannot be changed from their fixed 60 degrees. All equilateral triangles have the same shape.

Models of regular solids make nice Christmas tree ornaments.

Addendum #2:

Let's prove that there are an infinite number of polyhedra (but not regular) made up entirely of identical regular polygons. These would have regular polygons as faces. But the vertices may vary.

Proof: We can simply make a chain of octahedra, end to end, of any length we want. So we can make infinitely many different polyhedra this way, each of which has equilateral triangles as faces.

We can make more complicated polyhedra, with more than one kind of polygon for faces. If we try to make a chain of tetrahedra (in an effort to simplify the above proof), we may often come up with rhombuses (each with a cross brace) for faces. So it won't have regular polygons for faces. This is not a problem with octahedra.

The Pyramids of Egypt are, of course, not regular solids, having square bases. They also do not have equilateral triangles for sides. I wonder if maybe the pyramids are not pyramids after all, but octahedra, each with a buried vertex. There is some evidence that the Egyptians preferred much taller, skinnier pyramids, but had to settle for short squat ones to keep them from crumbling.

Also see Yahoo's Polyhedra Page, George W. Hart - Virtual Polyhedra (rotate them), and Gijs K. Altes - Paper Models of Polyhedra (print them and cut them out). These are not just regular polyhedra.

Addendum #3:

An important concept, concerning polyhedra, is the dual of a polyhedron. For any polyhedron, we would like to inscribe another polyhedron inside it, in the following way: Place a vertex of the new polyhedron be the center of each face of the old polyhedron. This new polyhedron is the dual of the old one. If you make a dual of the new polyhedron, you get a replica of the first polyhedron, but smaller. It is easy to show that the edges are parallel to the original, and proportional in length. So, duals come in pairs, hence the name. Concerning the above regular polyhedra, the dual of an octahedron is a cube (hexahedron), and vice versa. The dual of an icosahedron is a dodecahedron, and vice versa. And the dual of a tetrahedron is a tetrahedron.

For some purposes, a polyhedron is equivalent to a two dimensional graph. See The Bridges of Königsberg for an introduction to graphs. Both a graph and a polyhedron have edges and vertices. And if faces must be considered, then the graph has faces defined by the edges. The graph is distorted, to fit in two dimensions, so features like area and volume are not represented by the graph.

A geodesic dome (discovered and named by R. Buckminster Fuller) is a polyhedron with many many (usually) faces. There are many varieties of these domes. But the edges are of varying lengths in order to maintain a nearly constant curvature. In other words, the vertices are all nearly of equal distance from the center of the dome. And the vertices are not all the same.

Addendum #4:

A polyhedron can be deformed so that all of its vertices, edges, and faces lie on a plane (or a sphere for that matter). Just stretch one of the polygons until it is very large, with all of the other polygons within it. See the diagram of a dodecahedron on the left. The area around the outside of the largest polygon (around the biggest pentagon here) is the interior of that polygon. That face is now infinite. Since this polyhedron was connected, this plane object is what is called a "connected graph." And what we prove about connected graphs will be true for connected polyhedra.

Addendum #5:

Here are models for the five regular polyhedra:

If you are using Netscape or Internet Explorer, you can probably right click on the image and choose to save it on your computer. Expand them with a paint program. If you want to print them, view these models separately, so you don't print this whole page.

Addendum #6:

Besides finding the five regular solids above, we also discovered the three regular tilings (tessellations) of the plane (equilateral triangles, squares, and hexagons):

These are the only ones possible with identical regular polygons. See Tesselations.

Addendum #7:

Here are animations of a rotating tetrahedron, cube, octahedron, icosahedron, and dodecahedron.

I am told that some dictionaries say that a polygon has five or more sides. My dictionaries say three or more sides. And of course, the mathematical definition says three or more sides.

I hear that Euclid's definition of the above solids is incomplete (as are a few other of his definitions throughout his books). He calls the above polyhedra, not regular polyhedra, and just says that each face is a regular polygon. He obviously means that they are regular, but does not define that. His definition of the icosahedron is a polyhedron with 20 faces. Well there are several such solids, some very irregular (combinations of tetrahedra and octahedra stuck together at odd angles, perhaps). But from the theorem (that there are only five solids), he obviously means that they are regular.

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