The Euler Characteristic and Dice

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The Euler Characteristic and Dice
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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
for all polyhedra. (You should check this for
various polyhedra.)
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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.
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• 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

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• Each vertex has n faces.
• This, together with the fact that each face has m
  edges implies

From the Euler characteristic, we have
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Note that the denominator must be positive
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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., ngt2) and
each face must have at least three edges (i.e.,
mgt2). However, since
there must be a maximum value of n. In
particular,
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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 n3, we
have
and so the faces can only be triangles, squares,
and pentagons (m3, 4, 5). We proceed in cases.
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Case 1. (m3) If m3, we have
This gives three possible polyhedra with
triangular faces n3, 4, 5. The number of faces
in each case can be computed from
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In particular,
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Case 2. (m4) If m4, we have
This gives one possible polyhedron with square
faces n3. The number of faces is
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Case 3. (m5) If m5, we have
This gives one possible polyhedron with
pentagonal faces n3. The number of faces is
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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.
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There are many other polyhedra. Some examples
follow.
snubdisphenoid
Square pyramid
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Pentagonal Dipyramid
Gyroelongated Pentagonal Cupolarotunda
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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 Rhombicosidodecahed
ron, 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
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You can learn more about polyhedra
at http//mathworld.wolfram.com/Polyhedron.html
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You can plot polyhedra using Maple with commands
such as gt with(plots) gtpolyhedraplot(0,0,0,po
lytypedodecahedron, stylePATCH,
scalingCONSTRAINED, orientation71,66)