1. Platonic Solids
And
Zome System

2. Regular Polygons
A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, square, regular pentagon, regular hexagon, …

3. Regular Polyhedra
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.

4. 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. ~There are only five platonic solids~
Cube
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
Euclid proved this in the last proposition of the Elements.

6. 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.

7. Icosahedral dice were used by the ancient Egyptians.

8. Evidence shows that Pythagoreans knew about the regular solids of cube, tetrahedron, and dodecahedron. A later Greek mathematician, Theatetus ( BC) has been credited for developing a general theory of regular polyhedra and adding the octahedron and icosahedron to solids that were known earlier.

9. The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato ( 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.

10. Harmonices Mundi
Johannes Kepler

11. Construction of Regular
Polyhedra Using
Equilateral Triangle
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

12. Construction of Regular
Polyhedra Using
Equilateral Triangle
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

13. Platonic Solids Tetrahedron
Euclid proved this in the last proposition of the Elements.

14. Construction of Regular
Polyhedra Using
Equilateral Triangle
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

15. Construction of Regular
Polyhedra Using
Equilateral Triangle
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 8 
 12
6+8=12+2

16. Platonic Solids Tetrahedron Octahedron
Euclid proved this in the last proposition of the Elements.

17. Construction of Regular
Polyhedra Using
Equilateral Triangle
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 8 
 12
6+8=12+2 5

18. Construction of Regular
Polyhedra Using
Equilateral Triangle
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 8 
 12
6+8=12+2 5
 20
 30
12+20=30+2

19. Platonic Solids Tetrahedron Icosahedron Octahedron
Euclid proved this in the last proposition of the Elements.

20. Construction of Regular
Polyhedra Using
Equilateral Triangle
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 8 
 12
6+8=12+2 5
 20
 30
12+20=30+2 6

21. Construction of Regular
Polyhedra Using
Equilateral Triangle
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 8 
 12
6+8=12+2 5
 20
 30
12+20=30+2 6

22. Construction of Regular
Polyhedra Using
Squre
Number of Squares
About each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3

23. Construction of Regular
Polyhedra Using
Square
Number of Squares
About each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3 6 12 8
8+6=12+2

24. Platonic Solids Tetrahedron Icosahedron Cube Octahedron
Euclid proved this in the last proposition of the Elements.

25. Number of Squares about each Vertex
Construction of Regular
Polyhedra Using
Square
Number of Squares about each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3 6 12 8
8+6=12+2 4

26. Number of Squares about each Vertex
Construction of Regular
Polyhedra Using
Square
Number of Squares about each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3 6 12 8
8+6=12+2 4

27. Construction of Regular
Polyhedra Using
Regular Pentagon
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

28. Number of Pentagons about each Vertex
Construction of Regular
Polyhedra Using
Regular Pentagon
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

29. Platonic Solids Tetrahedron Icosahedron Dodecahedron Cube Octahedron
Euclid proved this in the last proposition of the Elements.

30. Number of Pentagons about each Vertex
Construction of Regular
Polyhedra Using
Regular Pentagon
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 4

31. Number of Pentagons about each Vertex
Construction of Regular
Polyhedra Using
Regular Pentagon
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 4

32. Construction of Regular
Polyhedra Using
Regular Hexagon
Number of Hexagons
about each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3

33. Number of Hexagons about each Vertex
Construction of Regular
Polyhedra Using
Regular Hexagon
Number of Hexagons about each Vertex
Number of Faces
(F)
Number of Edges
(E)
Number of Vertices
(V)
Euler Formula
V + F = E + 2 3

34. ~There are only five platonic solids~
Cube
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
Euclid proved this in the last proposition of the Elements.

35. Dual of a Regular Polyhedron
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

37. The dual of the tetrahedron is the tetrahedron
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.

38. THE END! Polyhedron The Dual Number of Faces The Shape of Each Face
Schläfli Symbol
The Dual
Number of Faces
The Shape of Each Face
Tetrahedron
(3, 3) 4
Equilateral Triangle
Hexahedron
(4, 3)
(3,4) 6
Square
Octahedron 8
Dodecahedron
(5, 3)
(3, 5) 12
Regular Pentagon
Icosahedron 20
THE END!