Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to dwh2@cornell.edu.

Chapter 19

Polyhedra

The right way is next in order after the second dimension to take the third. This, I suppose, is the dimension of cubes and of everything that has depth.

— Plato, Republic, VII.528b [A: Plato]

Definitions and Terminology

A tetrahedron, DABCD, in 3-space [in a 3-sphere or a hyperbolic 3-space]¹ is determined by any four points, A,B,C,D, called its vertices, such that all four points do not lie on the same plane [great 2-sphere, great hemisphere] and no three of the points lie on the same line [geodesic]. The faces of the tetrahedron are the four [small] triangles DABC, DBCD, DCDA, DDAB. The edges of the tetrahedron are the six line [geodesic] segments AB, AC, AD, BC, BD, CD. The interior of the tetrahedron is the [smallest] 3-dimensional region that it bounds.

Tetrahedra are to 3-dimensions as triangles are to 2-dimensions. Every polyhedron can be dissected into tetrahedra, but the proofs are considerably more difficult than the ones from Problem 7.5b, and in the discussion to Problem 7.5 there is a polyhedron that is impossible to dissect into tetrahedra without adding extra vertices. There are numerous congruence theorems for tetrahedra, analogous to the congruence theorems for triangles. All of the problems below apply to tetrahedra in Euclidean 3-space or a 3-sphere or a hyperbolic 3-space.

The dihedral angle, ÐAB, at the edge AB is the angle formed at AB by DABC and DABD. The dihedral angle is measured by intersecting it with a plane which is perpendicular to AB at a point between A and B. The solid angle at A, ÐA, is that portion of the interior of the tetrahedron "at" the vertex A. See Figure 19.1.

Figure 19.1. Dihedral and solid angles.

Problem 19.1. Measure of a Solid Angle

The measure of the solid angle is defined as the ratio,

m(ÐA) = [lim_R®0] area{ (interior of  DABCD) Ç S } / R² ,

where S is any small 2-sphere with center at A whose radius, R, is smaller than the distance from A to each of the other vertices and to each of the edges and faces not containing A.

Show that the measures of the solid and dihedral angles of a tetrahedron satisfy the following relationship:

m(ÐA) = m(ÐAB) + m(ÐAC) + m(ÐAD) - p.

Show that two solid angles with the same measure are not necessarily congruent.

Suggestions

Solid angles, whether in Euclidean 3-space or a 3-sphere or a hyperbolic 3-space, are closely related to spherical triangles on a small sphere around the vertex. You can think of starting with a sphere, S, and creating a solid angle by extending three sticks out from the center of the sphere. If you connect the ends of these sticks, you will have a tetrahedron. The important thing to notice is how the sticks intersect the sphere. They will obviously intersect the sphere at three points, and you can draw in the great circle arcs connecting these points. Look at the planes in which the great circles lie. In this problem you need to figure out the relationships between the angles of the spherical triangle and the dihedral angles.

The formula given above for the definition of the solid angle uses the intersection of the interior of the solid angle with any small sphere S. This intersection is the small triangle that you just drew, and the area of the intersection is the area of the triangle. Since the measure of a solid angle is defined in terms of an area, it is possible for two solid angles to have the same measure without being congruent — they can have the same area without having the same shape.

What you are asked to prove here is the relationship between the measure of the solid angle and the measures of the dihedral angles. Since they are closely related to spherical triangles on the small sphere, you can use everything you know about small triangles on a sphere.

We will study congruence theorems for tetrahedra which can be thought of as the three-dimensional analogue of triangles. A tetrahedron has 4 vertices, 4 faces, and 6 edges and we can denote it by DABCD where A, B, C, D are the vertices.

Problem 19.2. Edges and Face Angles

If DABCD and DA'B'C'D' are two tetrahedra such that

ÐBAC @ ÐB'A'C', ÐCAD @ ÐC'A'D', ÐBAD @ ÐB'A'D',

CA @ C'A', BA @ B'A', DA @ D'A' 

then 

DABCD @ DA'B'C'D'.

Figure 19.2. Edges and faces.

Part of your proof must be to show that the solid angles ÐA and ÐA' are congruent and not merely that they have the same measure.

Suggestions

If S is a small sphere with center at A and radius R, then

S Ç (interior of  DABCD)

is a spherical triangle whose sides have lengths

R x ÐBAC, R x ÐCAD, R x ÐBAD.

In the last problem, you saw how solid angles are related to spherical triangles. This problem asks you to prove the congruence of tetrahedra based on certain angle and length measurements. (Note that the angles shown above are not the dihedral angles of the tetrahedron.) So, since you can use spherical triangles to relate solid and dihedral angle measurements, why not use them to prove tetrahedra congruencies? Use the hint given to see what measurements of the spherical triangle are defined by measurements of the tetrahedron. Then see if the measurements given do in fact show congruence, and show why.

Problem 19.3. Edges and Dihedral Angles

If 

AB @ A'B', ÐAB @ ÐA'B', AC @ A'C',

ÐAC  @ ÐA'C', AD @ A'D', ÐAD @ ÐA'D', 

then 

DABCD @ DA'B'C'D'.

Figure 19.3. Edges and dihedral angles.

This is very similar to the previous problem, but uses different measurements — here we have the dihedral angles instead of the angles on the faces of tetrahedron. Look at this problem the same way you looked at the previous one — see how the measurements given relate to a spherical triangle, and then prove the congruence.

Problem 19.4. Other Tetrahedra Congruence Theorems

Make up your own congruence theorems! Find and prove at least two other sets of conditions that will imply congruence for tetrahedra, that is, make up and prove other theorems like those in Problems 19.2 and 19.3.

It is important that you make sure your conditions are sufficient to prove that the solid angles are congruent, not just that they have the same measure.

Problem 19.5. The Five Regular Polyhedra

A regular polygon is a polygon lying in a plane or 2-sphere or hyperbolic plane such that all of its edges are congruent and all of its angles are congruent. For example on the plane a regular quadrilateral is a square. On a 2-sphere and a hyperbolic plane a regular quadrilateral is constructed as follows as in Figure 19.4. See also Figure 11.20 for a picture of a regular octagon on a hyperbolic plane.

Figure 19.4. Regular quadrilaterals.

Note that half of a regular quadrilateral is a Khayyam quadrilateral. On 2-spheres and hyperbolic planes there are no similar polygons; for example, a regular quadrilateral (congruent sides and congruent angles) will have the same angles as another regular quadrilateral if and only if they have the same area. (Do you see why?)

A polyhedron in 3-space [or in a 3-sphere or in a hyperbolic 3-space] is regular if all of its edges are congruent, all of its face angles are congruent, all of its dihedral angles are congruent, and all of its solid angles are congruent. The faces of a polyhedron are assumed to be polygons that lie on a plane [a great 2-sphere, a great hemisphere].

Show that there are only five regular polyhedra. In Euclidean 3-space, to say "there are only five regular polyhedra" is to mean that any regular polyhedra is similar (same shape, but not necessarily the same size) to one of the five. It still makes sense on a 3-sphere and a hyperbolic 3-space to say that "there are only five regular polyhedra," but you need to make clear what you mean by this phrase.

These polyhedra are often called "the Platonic Solids," and are described by Greek philosopher Plato (429-348 b.c.) as "forms of bodies which excel in beauty;"² but there is considerable evidence that they were known well before Plato's time.^†3 The regular polyhedra are the subject of Euclid's thirteenth (and last) book.

Suggestions

Your argument should be essentially the same whether you are considering 3-space, or a 3-sphere, or a hyperbolic 3-space. There are many widely different ways to do this problem. Here we suggest one approach. First note that the faces of a regular polyhedron must be regular polygons. Then focus on the vertices of regular polyhedra. Show that if the faces are regular quadrilaterals or regular pentagons, then there must be precisely three faces intersecting at each vertex.

Show that it is impossible for regular hexagons to intersect at a vertex to form the solid angle of a regular polyhedron. If the faces are regular (equilateral) triangles, then show that there are three possibilities at the vertices.

You may find it helpful to review the chapters on holonomy and area, isometry and patterns, 3-spheres in 4-space, and the first part of this chapter. Also, note the connections between this problem and Problem 18.6b which can also be used to solve this problem.

The five regular polyhedra are usually named the Tetrahedron, the Cube, the Octahedron, the Dodecahedron, and the Icosahedron. (See Figure 19.5.) There is a duality (related to but not exactly the same as the duality in Chapter 16 on duality and trigonometry) among regular polyhedra: If you pick the centers of the faces of a regular polyhedron, then these points are the vertices of a regular polyhedron which is called the dual of the original polyhedron. You can see that the cube is dual to the octahedron (and vice versa), that the icosahedron is dual to the dodecahedron (and vice versa), and that the tetrahedron is dual to itself.

Figure 19.5. The five Platonic solids.