Platonic solid 397

12

1. Regular tetrahedron, with four triangular faces and

three edges meeting at each vertex

2. Cube, with six square faces and three edges meeting

at each vertex

3. Octahedron, with eight triangular faces and four

edges meeting at each vertex

4. Dodecahedron, with 12 pentagonal faces and three

edges meeting at each vertex

5. Icosahedron, with 20 triangular faces and five edges

meeting at each vertex

It is not difficult to see why there can be no other

Platonic solids. Notice that for any polyhedron, at least

three polygonal faces must meet at each vertex of the

solid, and that, in order to form a peak at that vertex,

the angles of the faces meeting at that vertex must sum

to less than 360°. (If the sum of the angles around a

vertex is precisely 360°, then the surface would be flat

at the vertex. If, on the other hand, the angles sum to

more than 360°, then the surface would cave in at that

vertex.) Now consider the various regular polygons

that might be used to construct a Platonic solid:

Equilateral Triangles: Each angle in an equilateral tri-

angle is 60°. Thus three, four, or five such triangles

could surround a vertex of a polyhedron, but not six

or more. (The angle sum would no longer be less

than 360°.) Each of these possibilities does in fact

occur. These are the tetrahedron, the octahedron,

and the icosahedron, respectively.

Squares: Each angle in a square equals 90°. Thus only

three squares could possibly surround a vertex in a

polyhedron. This does in fact occur. It is the cube.

Regular Pentagons: Each angle in a regular pentagon

equals 108°, and so only three pentagons could pos-

sibly surround a vertex in a polyhedron. This does in

fact occur. It is the dodecahedron.

Regular Hexagons and Higher: Each angle in a regular

polygon with six or more sides is 120°or greater.

No three of these figures could possibly surround a

vertex in a polyhedron.

That each Platonic solid can be placed inside a

SPHERE

—the most perfect three-dimensional figure,

according to the ancient Pythagorean sect of 500

B

.

C

.

E

., in such a way that each vertex of the solid just

touches the sphere—was deemed deeply significant in

times of antiquity. The Greeks associated each Platonic

solid with an “element” of the physical world. The

tetrahedron, with its minimal volume per unit surface

area, was fire, and the stable cube was solid earth. The

less stolid octahedron was air, and the icosahedron,

with the maximum volume per surface-area ratio, was

water. The Pythagoreans, at first, were not aware of

the existence of the dodecahedron, but they attributed

the meaning of the entire universe to this figure upon

its discovery. (Its 12 faces match the 12 signs of the

zodiac.) The word quintessence, which today means

the best or purest aspect of some nonmaterial thing,

derives from the phrase quinta essential meaning the

fifth element, namely, the dodecahedron representing

all. The philosopher P

LATO

(ca. 428–348

B

.

C

.

E

.) wrote

extensively about the five regular polyhedra and their

significance in his work Timaeus. This is the reason

why we call them Platonic solids today.

German astronomer and mathematician J

OHANNES

K

EPLER

(1571–1630) also tried to attribute special

meaning to the five Platonic solids. At his time, only

five planets were known, and Kepler worked to relate

the orbits of these planets in some way to the five spe-

cial solids. He proposed that the orbit of each planet

lay on a sphere and that the distance between succes-

sive spheres was precisely such that each of the Platonic

solids, in turn, fits snugly between the two spheres,

with the inner sphere just touching the faces of the

polyhedron, and the vertices of the polyhedron just

touching the outer sphere. Kepler later abandoned this

theory. (He, in fact, is noted for discovering that the

orbit of each planet is not a circle but an

ELLIPSE

.) It is

interesting to note, however, that, allowing for eccen-

tricity, his Platonic solid theory for the first five planets

is accurate to within approximately 5 percent.

If we assume that each edge of a Platonic solid is 1

unit in length, then the

VOLUME

Vof each solid is given

as follows: