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Platonic Solids
Greek concept of Symmetry
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Seen in their art, architecture and mathematics
Greek Geometry
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The most symmetric polygons are regular
Regular polygons have all sides and angles congruent
 Ex. Regular triangle is an equilateral triangle
 Ex. Regular quadrilateral is a square
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Regular polygons exist for any number of sides
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Ex. Pentagon, Hexagon, Heptagon, etc.
Whoa its in 3-D!!
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A polyhedron is regular if
Its faces are congruent regular polygons
 Its vertices are similar
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A cube is a regular polyhedron
Euclid says…
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Only a few regular polyhedra exist
In contrast there are infinite regular polygons
Actually there are exactly five types of regular
polyhedra
And they are…
Tetrahedron
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4 Faces
Each face is a triangle
Associated by Pythagoreans with the element fire
Octahedron
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8 faces
Each face is a triangle
Associated with air
Icosahedron
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20 faces – associated
with water
Each face is
Drumroll
please……………..
A TRIANGLE!!
Hexahedron
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6 faces- associated with
earth
The sides are squares,
this is a cube
Associated with water
Dodecahedron
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12 faces
Each face is a pentagon
To the Pythagoreans, it represents the universe
Vertices
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At least three polygonal faces must meet at any
vertex, forming a peak.
The sum of these angles is < 360 degrees
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Otherwise the faces overlap or form a flat surface
In a regular polyhedra the situation at one vertex
is the same as at any other
6 triangles, 3 hexagons = 360 degrees
Archimedean Solids
Not so Regular
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Allow different types of regular solids to be
used to create a 3-D object
Archimedes discovered 13 such solids
Best known example
The truncated icosahedron
 In Europe they use it to play football
 In America it is called a soccer ball
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Misnomers
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Plato used this knowledge to found an elaborate
theory where all things are composed of right
triangles
Renaissance mathematicians learn the regular
solids from Plato, but must relearn Archimedean
solids.
Kepler
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Johannes Kepler rediscovered all Archimedean
solids
Originally believed planetary orbits were in
spheres
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Later realized planetary obits were elliptical
“If one inscribed a cube into the sphere of
Saturn the faces of the cube would be tangent
to the sphere of Jupiter.”
Examples in the Elements
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Hexahedron: shape of the crystalline structures of lead
ore and rock salt
Octahedron: shape of crystals formed by fluorite
Dodecahedron: shape of crystals formed by garnet
All three of above: shape of crystals formed by iron
pyrite
Tetrahedron: shape of basic crystalline form of the
silicates
Truncated Icosahedron: outlines the vertices where
sixty carbon atoms form the molecule known as
“buckyball”