Transcript PLATONIC SOLIDS - Mathematical sciences

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