The Euler Characteristic and Dice

Polyhedra are solids (think dice) that have polygonal faces meeting at edges, and the edges meet at vertices. Let the number of faces, edges and vertices be denoted by F, E and V, respectively. Euler noted that

2 FEV

for all polyhedra. (You should check this for various polyhedra.)

In this note, we will consider polyhedra with faces that are all the same regular polygons (i.e., all triangles, all squares, etc.) An example is the tetrahedron with all sides being equilateral triangles. We will also assume that the same number of edges emanate from each vertex.

Let m be the number of sides each face has, and let n be the number of edges coming from each vertex.

We make the following observations.

•Each edge is shared by two faces.

•Each face has m edges and m vertices.

These two facts imply

.2

FmE

•Each vertex has n faces.

This, together with the fact that each face has m edges implies

.n

FmV

From the Euler characteristic, we have

.22

4

12

2

22

nnmm

nm

nm

F

FFm

n

Fm

Note that the denominator must be positive:

)2(

2

022

n

nm

nnmm

This last formula limits the number of edges each side can have. First note that at least three faces must meet at each vertex (i.e., n>2) and each face must have at least three edges (i.e., m>2). However, since

,2)2(

2lim

n

nm

n

there must be a maximum value of n. In particular,

.6)2(

23

nn

n

We deduce that under the hypotheses we have made, the number of faces meeting at a vertex must be less than or equal to six. Further, if n=3, we have

,6m

and so the faces can only be triangles, squares, and pentagons (m=3, 4, 5). We proceed in cases.

Case 1. (m=3) If m=3, we have

.6

)2(

23

n

n

n

This gives three possible polyhedra with triangular faces: n=3, 4, 5. The number of faces in each case can be computed from

.22

4

nnmm

nF

In particular,

)(2010156

20)5(

)(88126

16)4(

)(4696

12)3(

236

4

nIcosahedro

Octahedron

nTetrahedro

nF

nF

nF

nn

nF

Case 2. (m=4) If m=4, we have

.4

)2(

24

n

n

n

This gives one possible polyhedron with square faces: n=3. The number of faces is

).(66128

12Cube

F

Case 3. (m=5) If m=5, we have

.3/10

)2(

25

n

n

n

This gives one possible polyhedron with pentagonal faces: n=3. The number of faces is

).(1261510

12onDodecahedr

F

We have generated a list of all possible polyhedra with isometric faces that are regular polygons. It turns out that there are only five, and these are included in a standard set of gaming dice.

Gaming dice usually have a pair of D-10 (ten sided dice), though these are not in the class we considered.

There are many other polyhedra. Some examples follow.

snubdisphenoid Square pyramid

Pentagonal Dipyramid Gyroelongated

Pentagonal Cupolarotunda

Some other polyhedra that have been named are: Decagonal Prism, Pentagonal Hexecontahedron, Triangular Prism, Triangular Hebesphenorotunda, Triangular Orthobicupola, Augmented Truncated Dodecahedron, Elongated Square Cupola, Pentakis Dodecahedron, Rhombic Triacontahedron, Trapezoidal Hexecontahedron, Pentagonal Icositetrahedron, Trapezoidal Icositetrahedron, Square Antiprism, Octagonal Prism, Hexagonal Prism, dodecahedron, Trigyrate Rhombicosidodecahedron, Gyroelongated Square Pyramid, echidnahedron, bilunabirotunda, sphenocorona, Square Cupola, Snub Disphenoid, Square Pyramid, Elongated Pentagonal Pyramid, Elongated Triangular Pyramid, tetrahedron, Pentagonal Orthobicupola, Triaugmented Triangular Prism, Elongated Triangular Cupola, Great Dodecahedron, Elongated Triangular Dipyramid, octahedron, Gyroelongated Pentagonal Pyramid, Great Stellated Dodecahedron, Paragyrate Diminished Rhombicosidodecahedron, Bigyrate Diminished Rhombicosidodecahedron, Parabidiminished Rhombicosidodecahedron, Gyroelongated Pentagonal Birotunda, Pentagonal Gyrobicupola, Gyroelongated Square Bicupola, Pentagonal Prism, Pentagonal Orthobirotunda, Decagonal Antiprism, Elongated Pentagonal Gyrobirotunda, Octagonal Antiprism, Hexagonal Antiprism, Elongated Pentagonal Orthobirotunda, Elongated Triangular Orthobicupola, Pentagonal Gyrocupolarotunda, Pentagonal Antiprism, Gyroelongated Square Cupola, icosahedron, Great Icosahedron, Elongated Pentagonal Rotunds, hexahedron, Elongated Pentagonal Cupola, Hexakis Icosahedron, Triakis Icosahedron, Hexakis Octahedron, Elongated Pentagonal Orthocupolarotunda, Metagyrate Diminished Rhombicosidodecahedron, Tetrakis Hexahedron, Elongated Pentagonal Dipyramid, Triakis Octahedron, Rhombic Dodecahedron, Augmented Triangular Prism, Augmented Dodecahedron, Square Orthobicupola, Pentagonal Dipyramid, Triangular Dipyramid, Pentagonal Rotunda, Elongated Triangular Gyrobicupola, Gyroelongated Pentagonal Bicupola, Pentagonal Cupola, Metabiaugmented Truncated Dodecahedron, Triangular Cupola, Biaugmented Truncated Cube, Tridiminished Icosahedron, Elongated Pentagonal Gyrobicupola, Metabigyrate Rhombicosidodecahedron, Parabigyrate Rhombicosidodecahedron, Metabidiminished Icosahedron, Pentagonal Orthocupolarontunda, Gyroelongated Pentagonal Rotunda, Tridiminished Rhombicosidodecahedron, Triaugmented Dodecahedron, Pentagonal Pyramid, Elongated Square Dipyramid, Triaugmented Truncated Dodecahedron, Gyroelongated Pentagonal Cupola, Metabiaugmented Dodecahedron, Gyroelongated Triangular Cupola, Triaugmented Hexagonal Prism, Elongated Square Pyramid, Gyroelongated Pentagonal Cupolarotunda, Gyroelongated Square Dipyramid, hebesphenomegacorona, Augmented Truncated Cube, Parabiaugmented Truncated Dodecahedron, Biaugmented Pentagonal Prism, Parabiaugmented Dodecahedron, Biaugmented Triangular Prism, Small Stellated Dodecahedron, Gyroelongated Triangular Bicupola, Gyrate Bidiminished Rhombicosidodecahedron, Metabiaugmented Hexagonal Prism, Parabiaugmented Hexagonal Prism, Gyrate Rhombicosidodecahedron, Metabidiminished Rhombicosidodecahedron, Augmented Sphenocorona, Snub Square Antiprism, Augmented Hexagonal Prism, Augmented Pentagonal Prism, Elongated Pentagonal Orthobicupola, Elongated Square Gyrobicupola, sphenomegacorona, Square Gyrobicupola, octahemioctahedron, tetrahemihexahedron, disphenocingulum, Diminished Rhombicosidodecahedron, Augmented Truncated Tetrahedron

You can learn more about polyhedra at:

http://mathworld.wolfram.com/Polyhedron.html

You can plot polyhedra using Maple with commands such as :> with(plots):>polyhedraplot([0,0,0],polytype=dodecahedron, style=PATCH, scaling=CONSTRAINED, orientation=[71,66]);