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Platonic Solids (see also Archimedes Solids)

Platonic solids are what some refer to as perfectly symmetrical polyhedra as they have identical vertices, faces and edges.  In addition, each face is a regular polygon.  There are only five Platonic solids - the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. 

Each platonic solid has a dual: The faces of the cube correspond to the vertices of the octagon and vice versa.  Likewise, the dodecahedron is the dual of the icosahedron. The tetrahedron is its own dual.

• Each vertex in a tetrahedron is the point where 3 equilateral triangles meet without overlap or space in between.
• 4 equilateral triangles meet at each vertex of an octahedron.
• In an icosahedron, 5 equilateral triangles meet at each vertex.  
• This trend does not continue because 6 equilateral triangles meet in a plane.  So we move onto squares.
• In a cube, each vertex is where 3 squares meet.  Again this stops as 4 squares meet to form a flat surface.
• In a dodecahedron, 3 regular pentagons meet at each vertex.  
• 4 pentagons meet to form a plane, as do 3 regular hexagons.  And so there are only 5 platonic solids

Each platonic solid also obeys the relationship

# of faces + # of vertices = # of edges + 2

There are a number of books and web sites (e.g. Hands-on Math) describing how to construct platonic solids out of a variety of different materials.  We will attempt to do something original (or so we think) by describing how we designed these polyhedral entities in SolidWorks.  Presumably, the same principles apply in most other 3D CAD packages.

Have you ever tackled problems involving construction of geometric entities using a limited set of tools (e.g. a ruler and a compass)?  Often these problems can be easily solved by cheating - e.g. using a protractor; however, staying within the given constraints gives rise to elegant and simple solutions, not to mention immense gratification.

We imposed similar constraints in constructing the Platonic solids in SolidWorks:

• Throughout the design process, specify only one basic dimension (akin to translating a given length using a compass - the two-legged kind).  Other dimensions are considered derived from this basic length - e.g. multiples or halves, etc.
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Once we got past the tetrahedron and octahedron (constructing a cube is rather obvious and so we will not insult anyone's geometric intelligence by delving into how we extruded a square), it was touch-and-go whether we could design an icosahedron period, never mind adhering to the above constraints.  We did not surrender easily, however, and managed to come up with the interesting solutions:

• Designing a tetrahedron
• Designing an octahedron
• Designing an icosahedron
• Designing a dodecahedron