Platonic Solids

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Platonic Solids
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Greek concept of Symmetry

• Seen in their art, architecture and mathematics
• Greek Geometry
• 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
• Regular polygons exist for any number of sides
• Ex. Pentagon, Hexagon, Heptagon, etc.

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Whoa its in 3-D!!

• A polyhedron is regular if
• Its faces are congruent regular polygons
• Its vertices are similar
• A cube is a regular polyhedron

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Euclid says

• 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

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Tetrahedron

• 4 Faces
• Each face is a triangle
• Associated by Pythagoreans with the element fire

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Octahedron

• 8 faces
• Each face is a triangle
• Associated with air

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Icosahedron

• 20 faces associated with water
• Each face is
• Drumroll please..
• A TRIANGLE!!

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Hexahedron

• 6 faces- associated with earth
• The sides are squares, this is a cube
• Associated with water

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Dodecahedron

• 12 faces
• Each face is a pentagon
• To the Pythagoreans, it represents the universe

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Vertices

• At least three polygonal faces must meet at any
  vertex, forming a peak.
• The sum of these angles is lt 360 degrees
• 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

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Archimedean Solids
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Not so Regular

• 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

• 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.

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Kepler

• Johannes Kepler rediscovered all Archimedean
  solids
• Originally believed planetary orbits were in
  spheres
• 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.

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Examples in the Elements

• 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