4.5 Platonic Solids
Wednesday, February 25, 2009
Symmetry in 3-D
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Sphere – looks the same from any
vantage point
Other symmetric solids?
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CONSIDER REGULAR POLYGONS
Start in The Plane
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Two-dimensional symmetry
Circle is most symmetrical
Regular polygons – most
symmetrical with straight sides
2D to 3D
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Planes to solids
Sphere – same from all directions
Platonic solids
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Made up of flat sides to be as symmetric
as possible
Faces are identical regular polygons
Number of edges coming out of any
vertex should be the same for all vertices
Five Platonic Solids

Cube

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Most familiar
Tetrahedron
Octahedron
Dodecahedron
Icosahedron
Powerful?
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
Named after Plato
Euclid wrote about them
Pythagoreans held them in awe
Vertices
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Edges
Faces
Faces at
each
vertex
Sides of
each face
Vertices
V
Edges
E
Faces
F
Faces at
each
vertex
Sides of
each face
Tetrahedron
4
6
4
3
3
Cube
8
12
6
3
4
Octahedron
6
12
8
4
3
Dodecahedron
20
30
12
3
5
Icosahedron
12
30
20
5
3
Vertices
V
Edges
E
Faces
F
Faces at
each
vertex
Sides of
each face
Tetrahedron
4
6
4
3
3
Cube
8
12
6
3
4
Octahedron
6
12
8
4
3
Dodecahedron
20
30
12
3
5
Icosahedron
12
30
20
5
3
Some Relationships
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Faces of cube = Vertices of
Octahedron
Vertices of cube = Faces of
Octahedron
Duality
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Process of creating one solid from
another
Faces - - - Vertices
Euler's polyhedron theorem

V+F-E=2
Archimedean Solids
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Allow more than one kind of regular
polygon to be used for the faces
13 Archimedean Solids (semiregular
solids)
Seven of the Archimedean solids are
derived from the Platonic solids by the
process of "truncation", literally cutting
off the corners
All are roughly ball-shaped
Truncated Cube
Archimedean Solids
Soccer Ball –
12 pentagons, 20 hexagons
Solid
(pretruncating)
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Truncated
Vertices
Edges
Faces
Solid
(pretruncating)
Truncated
Vertices
Edges
Faces
Tetrahedron
12
18
8
Cube
14
36
24
Octahedron
14
36
24
Dodecahedron
32
90
60
Icosahedron
32
90
60
Solid
(post-truncating)
Tetrahedron
Truncated Edges
Vertices
Faces
8
18
12
Cube
24
36
14
Octahedron
24
36
14
Dodecahedron
60
90
32
Icosahedron
60
90
32
Some Relationships
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New F = Old F + Old V
New E = Old E + Old V x number of
faces that meet at a vertex
New V = Old V x number of faces
that meet at a vertex
Stellating
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Stellation is a process that allows
us to derive a new polyhedron from
an existing one by extending the
faces until they re-intersect
Two Dimensions: The Pentagon
Octagon
How Many Stellations?
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Triangle and Square
Pentagon and Hexagon
Heptagon and Octagon
N-gon?
Problem of the Day

How can a woman living in New
Jersey legally marry 3 men,
without ever getting a divorce,
be widowed, or becoming
legally separated?