Platonic Solids

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Platonic Solids And Zome System
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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,
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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.
Regular Polyhedra
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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.
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Platonic Solids
There are only five platonic solids

• Cube
• Octahedron
• Dodecahedron

• Tetrahedron
• Icosahedron

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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.

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Icosahedral dice were used by the ancient
Egyptians.
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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.
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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.
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Harmonices Mundi Johannes Kepler
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Construction of Regular Polyhedra
Using Equilateral Triangle
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Construction of Regular Polyhedra
Using Equilateral Triangle
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Platonic Solids

• Tetrahedron

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Construction of Regular Polyhedra
Using Equilateral Triangle
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Construction of Regular Polyhedra
Using Equilateral Triangle
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Platonic Solids

• Octahedron

• Tetrahedron

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Construction of Regular Polyhedra
Using Equilateral Triangle
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Construction of Regular Polyhedra
Using Equilateral Triangle
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Platonic Solids

• Octahedron

• Tetrahedron
• Icosahedron

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Construction of Regular Polyhedra
Using Equilateral Triangle
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Construction of Regular Polyhedra
Using Equilateral Triangle
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Construction of Regular Polyhedra Using Squre
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Construction of Regular Polyhedra Using Square
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Platonic Solids

• Cube
• Octahedron

• Tetrahedron
• Icosahedron

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Construction of Regular Polyhedra Using Square
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Construction of Regular Polyhedra Using Square
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Construction of Regular Polyhedra Using Regular
Pentagon
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Construction of Regular Polyhedra Using Regular
Pentagon
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Platonic Solids

• Cube
• Octahedron
• Dodecahedron

• Tetrahedron
• Icosahedron

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Construction of Regular Polyhedra Using Regular
Pentagon
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Construction of Regular Polyhedra Using Regular
Pentagon
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Construction of Regular Polyhedra Using Regular
Hexagon
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Construction of Regular Polyhedra Using Regular
Hexagon
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Platonic Solids
There are only five platonic solids

• Cube
• Octahedron
• Dodecahedron

• Tetrahedron
• Icosahedron

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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
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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.
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THE END!