platonic solids
TRANSCRIPT
A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, square, regular pentagon, regular hexagon, …
By a (convex) regular polyhedron we mean a polyhedron with the properties that All its faces are congruent regular polygons.
The arrangements of polygons about the vertices
are all alike.
The regular polyhedra are the best-known polyhedra that have connected numerous disciplines such as astronomy, philosophy, and art through the centuries.
They are known as the Platonic solids.
5
Regular Convex (Platonic) Solids
Tetrahedron(Triangular Pyramid)
Source: Wikapedia
6
Hexahedron(Cube)
Regular Convex (Platonic) Solids
7
Regular Convex (Platonic) Solids
Octahedron
8
Regular Convex (Platonic) Solids
Dodecahedron
9
Regular Convex (Platonic) Solids
Icosahedron
Platonic Solids
Cube
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
~~There are only There are only fivefive platonic solids platonic solids~~
Platonic solids were known to humans much earlier than the time of Plato. There are carved stones (dated approximately 2000 BC) that have been discovered in Scotland. Some of them are carved with lines corresponding to the edges of regular polyhedra.
Icosahedral dice were used by the ancient
Egyptians.
Evidence shows that Pythagoreans knew about the regular solids of cube, tetrahedron, and dodecahedron. A later Greek mathematician, Theatetus (415 - 369 BC) has been credited for developing a general theory of regular polyhedra and adding the octahedron and icosahedron to solids that were known earlier.
The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato (427 - 347 BC) who associated them with the “elements” and the cosmos in his book Timaeus.
“Elements,” in ancient beliefs, were the four objects that constructed the physical world; these elements are fire, air, earth, and water. Plato suggested that the geometric forms of the smallest particles of these elements are regular polyhedra.
Fire is represented by the tetrahedron, earth the octahedron, water the icosahedron, and the almost-spherical dodecahedron the universe.
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 4 6 4 4+4=6+2
Platonic Solids
Tetrahedron
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 4 6 4 4+4=6+2
4
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 4 6 4 4+4=6+2
4 8 12 6 6+8=12+2
Platonic Solids
Octahedron
Tetrahedron
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 4 6 4 4+4=6+2
4 8 12 6 6+8=12+2
5
Number of Triangles
About each Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 4 6 4 4+4=6+2
4 8 12 6 6+8=12+2
5 20 30 12 12+20=30+2
Platonic Solids
Octahedron
Tetrahedron
Icosahedron
Platonic Solids
Cube
Octahedron
Tetrahedron
Icosahedron
Number of Pentagons about each
Vertex
Number of Faces(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2
3 12 30 20 20+12=30+2
Platonic Solids
Cube
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
Platonic Solids
Cube
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
~~There are only There are only fivefive platonic solids platonic solids~~
We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by connecting the centers of the faces of the original polyhedron
The dual of the tetrahedron is the tetrahedron. Therefore, the tetrahedron is self-dual. The dual of the octahedron is the cube. The dual of the cube is the octahedron. The dual of the icosahedron is the dodecahedron. The dual of the dodecahedron is the icosahedron.
Polyhedron Schläfli Symbol The Dual
Number of Faces
The Shape of Each Face
Tetrahedron (3, 3) (3, 3) 4 Equilateral Triangle
Hexahedron (4, 3) (3,4) 6 Square
Octahedron (3,4) (4, 3) 8 Equilateral Triangle
Dodecahedron (5, 3) (3, 5) 12 Regular Pentagon
Icosahedron (3, 5) (5, 3) 20 Equilateral Triangle