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
 V  E  F  2
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
Fm
E
.
2
•Each vertex has n faces.
This, together with the fact that each face
has m edges implies
Fm
V
.
n
From the Euler characteristic, we have
Fm Fm
2

F
n
2

2
4n
F

.
m m
2
m

nm

2
n
 1
n 2
Note that the denominator must be positive:
2m  nm  2n  0

2n
m
(n  2)
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
2n
m  lim
 2,
n  ( n  2)
there must be a maximum value of n. In
particular,
2n
3
 n  6.
(n  2)
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
m  6,
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
2n
3
( n  2)

n  6.
This gives three possible polyhedra with
triangular faces: n=3, 4, 5. The number of
faces in each case can be computed from
4n
F
.
2m  nm  2n
In particular,
4n
F
6  3n  2n

12
F (n  3) 
 4 ( Tetrahedron)
696
16
F ( n  4) 
 8 (Octahedron)
6  12  8
20
F (n  5) 
 20 (Icosahedron)
6  15  10
Case 2. (m=4) If m=4, we have
2n
4
( n  2)

n  4.
This gives one possible polyhedron with
square faces: n=3. The number of faces is
12
F
 6 (Cube ).
8  12  6
Case 3. (m=5) If m=5, we have
2n
5
( n  2)

n  10 / 3.
This gives one possible polyhedron with
pentagonal faces: n=3. The number of faces
is
12
F
 12 (Dodecahedron).
10  15  6
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=dodecahe
dron, style=PATCH,
scaling=CONSTRAINED,
orientation=[71,66]);