PLATONIC SOLIDS Audrey Johnson. Characteristics of Platonic Solids zThey are regular polyhedra zA...
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Transcript of PLATONIC SOLIDS Audrey Johnson. Characteristics of Platonic Solids zThey are regular polyhedra zA...
PLATONIC SOLIDS
Audrey Johnson
Characteristics of Platonic Solids
They are regular polyhedraA polyhedron is a three dimensional
figure composed of polygonsThere are exactly five platonic solidsAll the faces are the same regular
polygonThe same number of polygons meet
at each vertex
To Be a Platonic Solid…
At least three faces must meetthe 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
More Examples:
Octahedron made up of four equilateral triangles 4*60=240 < 360 degrees
Icosahedron made up of five equilateral triangles 5*60=300 < 360 degrees
As a result there cannot be 6 equilateral triangles since 6*60=360. If this was so the triangles would form a single-planed figure and not a solid
The cube: Made up of three squares 3*90=270 < 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
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
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
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