Wednesday, December 24, 2008

Wednesday Math, Vol. 53: The Platonic Solids

A few weeks back, I did a post about tiling the plane with regular polygons. A regular polygon is defined as all sides the same length and all angles the same measure, like a square or a triangle where all the angles are 60 degrees. Given any number of sides n, a regular polygon can be constructed, and the formula for the equal angle measure is (180 - n/360) degrees.

Can this idea be generalized to three dimensional objects? Yes, hypothetical question asker, it can.

In three dimensions, we need faces to be the same size and shape and edges to be the same length. The shapes that meet these criteria are called the Platonic Solids. These are solids that really like you, but... not in that way.

Okay, no they aren't. Just a little math joke.

Our buddy Plato was all about the idea of ideal objects, and the simplicity of how these three dimensional shapes are defined makes them ideal. Unlike regular polygons, of which there are infinitely many types, there are only five types of regular polyhedra, which is the plural of polyhedron. Besides the idea of same shaped faces and same length edges, we also have the rule that the same number of faces have to meet at each corner.

Given all those restrictions, the number of options is five. Here's the roll call.

Face shape: Triangle.
Number of faces:4
Number of corners: 4

Face shape: Square.
Number of faces: 6
Number of corners: 8

Face shape: Triangle.
Number of faces: 8
Number of corners: 6

Face shape: Pentagon.
Number of faces: 12
Number of corners: 20

Face shape: Triangle.
Number of faces: 20
Number of corners: 12

In the picture and in my color coded fact lists, the solids have naturally linked relationships. The number of faces on a cube is the number of corners on an octahedron, and vice versa. If you put a dot in the exact middle of every cube face and linked those dots together with straight lines, you would get an octahedron, and vice versa. The 12-sided and 20-sided solids have the same relationship, and the tetrahedron stands alone. Connect the dots in the middle of each face of a tetrahedron and you get a smaller tetrahedron.

The name for this relationship in math is duality. The cube is the octahedron's dual, the icosahedron and dodecahedron are duals of each other, and the tetrahedron stands alone, because it is its own dual.

There is another "simple" relationship between the Platonic Solids, and that is between the tetrahedron and the octahedron. If you wanted to make a tetrahedron that was twice as tall as another tetrahedron, you would need four copies of the original tetrahedron stacked up as shown, but you would be missing a shape in the middle. The missing middle shape is an octahedron. Since the twice as tall tetrahedron is three dimensional, it has eight times the volume of the original tetrahedron, because eight is two cubed. We used four tetrahedra to mark the corners, the octahedron must have the volume of four tetrahedra of equal side length, since four plus four equals eight.

If you have the right materials, it's fun constructing platonic solids. Using toothpicks and clay, with small dots of clay as the connective tissue at the corners, the only shape that is rigid is the simplest, the all-triangle tetrahedon. The cube and other shapes will bend under the weight of the materials. This idea was covered in an earlier post about making things rigid.

In the comments, Lockwood mentioned there is also a simple relationship between the cube and the tetrahedron. If you pick any corner of a cube and connect it to the corners diagonal to it across the faces of the cube, those four corners define a regular tetrahedron. I nicked this diagram from Antonio Gutierrez which shows both shapes separately and then combines them. The volume of the inscribed tetrahedron is one third the volume of the cube that contains it.

Thanks to Lockwood for pointing this out.


Lockwood said...

There is a realtionship between the cube, octahedron and tetrahedron as well: if you connect alternating corners of the cube (i.e. choose a corner, move along an edge, skip the next corner you come to, then move to another corner), you get a tetrahedron. Each of these solids has four three-fold axes of rotational symmetry. In mineralogy, the cubic system is often described as having three axes of four-fold rotational symmetry of equal length, but the better, clearer, and more accurate definition is having four three-fold axes of rotational symmetry. These are also the three of the five platonic solids that occur in nature.

Matty Boy said...

Hi, Lockwood. Thanks for stopping by.

In math, the symmetries of the tetrahedron have twelve elements, while the symmetries of the cube and octahedron have twenty four elements, which would be your four three-fold axes. If you spin the tetrahedon 90 degrees, it still fits in the cube you mentioned, but the cuts are on the opposite diagonals.

Technically, the buckyball C-60 has the same symmetries as the two larger platonic solids, and it shows up in nature, though it usually doesn't last very long.

Lockwood said...

Does the c-60 solid have a name?

Matty Boy said...

C-60 is a molecule with 60 carbon atoms. It's called the fullerene or the buckyball in honor of Buckminster Fuller. You can look up truncated icosahedron on Google to get a picture.

Lockwood said...

OK, truncated icosahedron is the term I was wondering about.

CDP said...

Hey, I actually know something...when I read the comment referencing the C-60, I wondered "hey, is that the same as a fullerene"? Anyway, Merry Christmas, Matty Boy!

DCup said...

I'm going off topic to wish you a wonderful holiday, Matty Boy!

Please forgive me.

Matty Boy said...

Thanks, DCup. Best wishes to you nad your pretty fambly as well.

Distributorcap said...

i am running out at taking the SATs right now

Matty Boy said...

Really, D-Cap? I took the SATs for fun a few years back, before the new essay part was added. I did much better in English than I did when it counted back in the day, but I hang my head in shame that I did a little worse in Math.

(790-710 when it counted. 760-800 when it didn't.)