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Platonic Solids are regular polyhedra but unlike regular polygons which are infinite, there are only five Platonic Solids. The Triangle, Square, Pentagon, Hexagon, etc. are all regular polygons and the list goes on infinitely.
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Platonic Solids have the same amount of regular polygons as sides and each vertex has the same amount of polygons around them. Also each of this sides join each other with the same angle.
The five Platonic Solids are: Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron.
It's uncertain who
first described the Platonic solids but sources indicate that Plato's friend,
Theaetetus, first aknowledged them.
The Platonic solids constituted the concluding Book XIII of Euclid's Elements.
Plato equated the solids with different elements:
Tetrahedron = Fire
Cube = Earth
Octahedron = Air
Dodecahedron = Universe
Icosahedron = Water
Proof for the existance of five Platonic Solids:
Euler's formula states that for a given polyhedron, not necesarily regular, the number of Faces minus the number of Edges plus the number of Vertives equals two:
| F
- E + V = 2 |
Each polyhedron has n faces, each face is a k-gon and at each vertex, p faces touch. Now we can substitute into Euler's formula as follows:
| n-(nk/2)+(nk/p)=2 |
| n(1-(k/2)+(k/p))=2 |
| n>0 |
| so:
1-(k/2)+(k/p)>0 |
| (k/2)+(k/p)>1 |
We know that we need at least n to be 4 since we can't create a closed polyhedron with only 3 plane. K has to also be 3 since there is not a 2-side polygon. Finally we know that p>2. Putting everything toguether:
| K |
P |
Polyhedron |
| 3 |
3 |
Tetrahedron |
| 3 |
4 |
Octahedron |
| 3 |
5 |
Dodecahedron |
| 4 |
3 |
Cube |
| 5 |
3 |
Icosahedron |
| Faces: |
4 |
|
Interactive 3D: Click and drag on top of the polyhedron to rotate it. Drag and let go to let it rotate by itself.SHIFT+Dragging up/down will act as a zoom. SHIFT+Dragging right/left will rotate sideways. |
| Edges: |
6 |
||
| Vertices: |
4 |
|
Faces: |
6 |
|
Interactive 3D: Click and drag on top of the polyhedron to rotate it. Drag and let go to let it rotate by itself.SHIFT+Dragging up/down will act as a zoom. SHIFT+Dragging right/left will rotate sideways. |
|
Edges: |
12 |
||
|
Vertices: |
8 |
| Faces: |
8 |
|
Interactive 3D: Click and drag on top of the polyhedron to rotate it. Drag and let go to let it rotate by itself.SHIFT+Dragging up/down will act as a zoom. SHIFT+Dragging right/left will rotate sideways. |
| Edges: |
12 |
||
| Vertices: |
6 |
| Faces: |
12 |
|
Interactive 3D: Click and drag on top of the polyhedron to rotate it. Drag and let go to let it rotate by itself.SHIFT+Dragging up/down will act as a zoom. SHIFT+Dragging right/left will rotate sideways. |
| Edges: |
30 |
||
| Vertices: |
20 |
| Faces: |
20 |
|
Interactive 3D: Click and drag on top of the polyhedron to rotate it. Drag and let go to let it rotate by itself.SHIFT+Dragging up/down will act as a zoom. SHIFT+Dragging right/left will rotate sideways. |
| Edges: |
30 |
||
| Vertices: |
12 |
http://www.mathpages.com/home/kmath096.htm
http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/
http://www.sbu.ac.uk/water/platonic.html
http://www.klingenstein.org/Additional_Resources/projects/2000/courant/courant.html
In order to create 3D interactive animations you need to use LiveGraphics3D, a non-commercial Java applet by Martin Kraus. The link can be found above.
The objects to animate
will need to be created with Mathematica.
Included below there is a link to the
mathematica file with all the platonic solids: