PLATONIC SOLIDS

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PLATONIC SOLIDS

• Audrey Johnson

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Characteristics of Platonic Solids

• They are regular polyhedra
• A polyhedron is a three dimensional figure
  composed of polygons
• There are exactly five platonic solids
• All the faces are the same regular polygon
• The same number of polygons meet at each vertex

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To Be a Platonic Solid

• At least three faces must meet
• the sum of the interior angles of the sides
  meeting at each vertex must be less than 360
  degrees
• For example, the tetrahedron is made up of
  equilateral triangles which consists of three 60
  degree angles
• 3 equilateral triangles meet at each vertex so
  the sum of the interior angles is 180 degrees
  which is less than 360 degrees

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More Examples

• Octahedron
• made up of four equilateral triangles
• 460240 lt 360 degrees
• Icosahedron
• made up of five equilateral triangles
• 560300 lt 360 degrees
• As a result there cannot be 6 equilateral
  triangles since 660360. If this was so the
  triangles would form a single-planed figure and
  not a solid

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• The cube
• Made up of three squares
• 390270 lt 360
• As a result, if four squares met at a vertex then
  the interior angles would equal 360 and would
  form a plane and not a solid

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Unique Numbers

• Tetrahedron 4 faces 6 edges 4 vertices
• Cube 6 faces 12 edges 8 vertices
• Octahedron 8 faces 12 edges 6 vertices
• Dodecahedron 12 faces 30 edges 20 vertices
• Icosahedron 20 faces 30 edges 12 vertices

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Unique Relationship

• Duality
• The cube and octahedron are duals
• The icosahedron and dodecahedron are duals
• The tetrahedron is a dual to itself
• This means that one can be created by connecting
  the midpoints of the faces of the other

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In Real Life

• The five platonic solids are ideal models of
  crystal patterns that occur throughout the world
  of minerals in numerous variations
• To the Greeks, these solids symbolized fire,
  earth, air, spirit, and water