Polytopes and Polyhedra

Table of Contents

Introduction

five pictures of polyhedra

The pictures above are pictures of the five regular polyhedra in three-space. There are no others. (Click on any of them to be able to play with it.) All of the regular polyhedra (singular polyhedron) are constructed from regular polygons. A regular polygon is constructed from equal-length segments joined by equal angles. In the same way, a solid regular polyhedron is constructed using equal-sized regular polygons joined at their edges by equal angles. In most of the discussions here, the polygons must all be the same polygon (all squares or all triangles or all pentagons, no mixing allowed).

We humans find it easy to think of things in three perpendicular dimensions; for example, we look at the corner of a room where the ceiling meets two walls, we understand easily that we are seeing three meeting planes. We are not so used to thinking in four dimensions. In four-space, there must be a fourth line, perpendicular to all three of the lines you see in the corner, that meets that corner. Constructing objects in this kind of space is much harder to imagine. We can use our powers of analogy and our knowledge of geometry to make that construction less difficult.

It is possible to construct an answer to this question: polygon : polyhedron :: polyhedron : ??? The answer is a regular polytope. A regular polytope is bounded by regular polyhedra joined at the faces by regular angles in four-space. If this doesn't seem to make sense, fear not, we will try to explain it more clearly.

Even if our brilliant explication leaves you in the dark, you can still travel through our site to look at the pretty pictures.

For a good exploration of three-space polyhedra, including a walk-through of Euclid's proof of the existence of only five, try George Hart's Virtual Polyhedra Page at Hofstra University.

Slicing Polyhedra

Most people know that we live in a three-dimensional world. But have you ever wondered what it would be like to live in a world that wasn't three-dimensional? In 1885, Edwin A. Abbott wrote Flatland, a novel describing the life of creatures in a flat space, limited to only two dimensions. At one point in the story, A. Square, the protagonist, encounters a sphere of our three-dimensional world. As it passes through Flatland, A. Square sees the sphere as a single circle, growing gradually until A. Sphere's equator intersects the plane, then diminishing once again to a point.

Through the circular slices, A. Square gains insight into the nature of the higher dimensional object. In some respects, polyhedra are more complicated than spheres, but we too can gain insight into the nature of solid objects, both three dimensional and higher dimensional, through examining their slices.

If we start with a cube, a polyhedron that is very familiar to us, we notice that we can look at it from three different perspectives: from a face, an edge, or a vertex. Friedrich Froebel, the inventor of kindergarten, noticed the importance of these different perspectives back in the early 1800's when he was building gifts for his children to play with. So what do we get if we "slice" the cubes from these different perspectives? First, what do we mean by "slicing"? In three dimensions slicing is analogous to what happened with A. Sphere in Flatland: the intersection of a two dimensional plane with a three dimensional object. Think again about A. Square now. Did it matter which way the Sphere was facing when it entered Flatland? No, it didn't. But it does matter which way we slice the cube.

When we slice a cube starting with a face and moving parallel through the solid, we simply get a series of squares. What about when we start parallel to an edge? This is a bit more involved, but we realize that the first slice will be a line (the edge), and the line will grow into rectangles, and then shrink back to a line (the opposite edge). Now, what about slicing from a vertex? This is more complicated, but also very interesting.

To slice a cube from its vertex, click here.

When we move to four-dimensional shapes --polytopes-- our method of slicing changes a bit. Instead of slicing with a two-dimensional plane, we can gain the same kind of information if we slice with a three-dimensional hyper-plane. Thus our slices will be three-dimensional rather than two-dimensional.

So what happens when we slice the hypercube? For instance, what would a visiting hypercube look like if it were to pass through our three-dimensional space? Does the orientation of our slicing hyper-plane matter? Yes, it does. In fact, the slices of the hypercube very much resemble those of the cube. For example, what do you think a hypercube looks like when it passes through our space head-on (starting with one of its eight three-dimensional cubic faces)? You guessed it, it looks like a series of cubes, all equal in size, just like slicing the cube face first gave a series of squares. See the parallels?

Slicing the hypercube from a two-dimensional side first gives analogous results to the cubic slices from an edge. Cubic slices from a vertex, as seen in the interactive model, has its analogy when a hypercube is sliced from a one-dimensional edge. But we still have one more "perspective" from which to slice the hypercube that returns a new phenomenon that we did not experience with the cube. To think about the shape of the hypercube by slicing it from a vertex, click here.

To see a discussion (and pictures) on slicing rotating polyhedra that incorporates slicing, click here.

Polyhedral Fold-Outs

Slicing is a good way to understand shapes better because it breaks them down into a series of lower-dimensional objects. Another good way to understand a shape is to try to build it from a lower-dimensional "fold-out," like a cardboard box unfolded and placed flat on the floor. To define the original shape, all one needs is the fold-out and a set of instructions for putting it together again. This is how Euclid, ca. 300 B.C., proved the existence of exactly five regular polyhedra. To see a discussion of the five Platonic solids that uses the same proof method that Euclid did, click here.

Consider this picture of an unfolded cube:

Does it look like a cube to you? What if we were to give you instructions that said "connect the blue edges" to build a cube? The best Flatlanders could do would be if they stretched the squares and aligned the blue edges together in the plane.

But this was not what we were looking for. We wanted them to fold up the fold-out without stretching the squares, something that requires three dimensions, something they would not be able to do. The fold-out and instructions for putting it together could nonetheless help the Flatlanders gain a better appreciation for what a real cube is.

What about other three dimensional polyhedra? What are the shapes of these fold-outs? What are the instructions for putting them together? What could a Flatlander learn from looking at them? For answers and images to some of these questions click here.

For us, who live in Spaceland, we can gain a better understanding of four-dimensional polytopes by examining their three-dimensional fold-outs and the instructions that go along with them.

The minimum material required to build a cube, as Euclid outlined, is three squares around a point (the red-and-blue squares above). When we talk about polytopes, the situation is analogous: the minimum material required to build a hypercube is three cubes arranged around an edge with instructions to connect the blue faces without stretching the cubes.

As spacelanders we cannot complete this task, but a four-dimensional creature could fold the fold-out into a hypercube with no problem if they filled in the spaces. Here is the complete hypercube fold-out:

Notice the similarities of this fold-out to the cube fold-out: we have a central object (square or cube) surrounded on each face with other identical objects, and then a final one stuck on the bottom. It is easy to think of the center object as being the "base" of the folded-up polytope, and the extra object on the end of the fold-out as being the "top" or "cap" of the polytope.

Duality

We've discussed lots of these regular shapes in both three- and four-space, and seen several interesting ways of examining the polyhedra and polytopes (folding and slicing).

But folding and slicing are useful ways to examine the poly-objects because they increase (folding) or reduce (slicing) the dimension of examination. How can we examine the relationships between these polyobjects and leave them in their original dimension? Another way of phrasing the question is to ask "what relationships hold among the polyhedra?"

One fascinating relationship among the polyhedra is duality. Duality is a complex concept, with applications across the board. The Encyclopedia Brittanica gives its definition as:

Duality,
a pervasive property of algebraic structures, holds that two operations or concepts
are interchangeable, all results holding in one formulation also holding in the
other, the dual formulation.

However, in geometry, it is in some respects more complex. The EB has more expanded definitions. If you want to read them, click here, and have a field day.

More useful than a definition is a practical application or two. The cube and the octahedron are mutually dual, that is, the cube is the octahedron's dual and the octahedron is the cube's dual. How do we know?

Here's a method for finding a polyhedron's dual:

connect the center of each face to the center of each face that shares an edge with 
that face.
By connecting the centers of each of the cube's faces, we get an octahedron:

Applet:cube w/inscribed octahedron

And the converse is true too: by connecting the centers of an octahedron's faces, a cube appears:

Applet: octahedron w/inscribed cube

A way to make the relationship more intuitive is to think about the faces which come together around a vertex. In the cube, three square faces come together around a vertex. By connecting their centers, you get a triangle--one of the building blocks of an octahedron. Similarly (dually), by connecting the four triangular faces of an octahedron that come together at a vertex, we get a square, a piece of the construction of a cube. Play with the octahedron and cube until you're convinced that the structure of one implicitly defines the structure of the other.

In fact, all five of the three-space polyhedra are in dual relationships. The dodecahedron and icosahedron are mutually dual, just as the cube and octahedron we just saw are.

Applet: dodecahedron w/inscribed icosa

Applet: icosahedron w/inscribed dodeca

Can we get these on each others' right and left rather than up and down?

Here, the pentagon formed by connecting the five triangular faces of the icosahedron that come together at a vertex is one face of a dodecahedron, and conversely, by connecting the centers of the three pentagons at a dodecahedral vertex, we get one of the triangles of an icosahedron.

But with five polyhedra, it seems difficult to give each one a dual partner: since there are an odd number, one has to be a wallflower. But no--the tetrahedron dances with itself: it is self-dual.

Applet: self-dual tetrahedron

In our discussion so far, we've talked about two kinds of regular shapes: three-space (polyhedra) and four-space (polytopes). But we've only talked about one kind of duality so far: the duality of the polyhedra. On analogy with the algorithm to generate polyhedral duals above, we can write an algorithm to generate polytope duals:

connect the center of each cell to the center of each cell that shares a face with
that cell.

As you can see, this formulation just raises the dimension of each term in the previous formulation by one: we've replaced "face" with "cell," and "edge" with "face." This is appropriate, because, just as regular polyhedra are bounded by regular polygons, the regular polytope is bounded by regular polyhedra ("cells"). We are connecting the centers of three-space polyhedra that have been folded up into four-space to form the boundary of a regular polytope.

Four-space objects are hard to visualize. Even without the ability to see the whole object, we can still learn about the polytopes from the dual relationships into which they enter.

By performing this algorithm on the hypercube (the "8-cell," because it uses eight cubes to bound itself), we can generate another polytope: the 16-cell. Because in the hypercube four cubes come together in a point, we learn that the 16-cell must be made up of tetrahedra (each of which has four vertices--one from the center of each cube). How many tetrahedra around a vertex in the 16-cell? Eight--because there are eight vertices around a cell (cube) in the hypercube. (For a more in-depth discussion of this kind of pattern, click here.)

Semi-regularity

All of the polyhedral principles that we have discussed thus far, specifically slicing, duality, and fold-outs, come into play within the discussion of solids known as semi-regular polyhedra. Semi-regular polyhedra deviate from regular polyhedra in that they are composed of more than one type of regular cell.

For a discussion of semi-regularity and its relationship to duality (with images) click here.

Other people have explored the semi-regular polyhedra:

Tom Getty, in the department of Mathematics at California State University in Chico, has lots of material about polyhedra. Click here to see his material on Archimedean solids.

Maeder in Zurich has another interesting collection of pictures of semi-regular polyhedra.

V. Bulatov, in the U.K., has a collection of beautiful pictures of regular, semi-regular, stellated, and other polyhedra. Most of what he does is far outside the scope of what we address in this page, but click here to see it anyway.

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Credits:

David Akers

Jeremy Kahn

Michael Matthews

Alison Tarbox