PLATONIC SOLIDS - Mathematical sciences
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PLATONIC SOLIDS
Audrey Johnson
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
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
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
Cube
Octahedron
Dodecahedron
Icosahedron
4 faces 6 edges 4 vertices
6 faces 12 edges 8 vertices
8 faces 12 edges 6 vertices
12 faces 30 edges 20 vertices
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