Platonic Solids

According to: <http://mathworld.wolfram.com/PlatonicSolid.html>, the Platonic solids (aka the regular solids or regular polyhedra) “are convex polyhedra with equivalent faces composed of congruent convex regular polygons.”  [Obviously, such a description is crystal clear -- which is why it’s better to see them than try to describe them in words, i.e. a picture is worth a thousand words!]  Meanwhile, there are only five such solids: the cube, dodecahedron, icosahedron, octahedron, and tetrahedron.  The idea that there are only five was proved by Euclid in his last proposition (chapter or book 13) of the classic, The Elements.  The Platonic solids are sometimes also called “cosmic figures”.

The Platonic solids were described by Plato in his Timaeus c. 350 B.C.E.  In this work, Plato equated the polyhedra with the “elements”:

                        the cube with earth,

                        tetrahedron with fire,

                        the octahedron with air,

                        the icosahedron with water, and

                        the dodecahedron with the stuff of which the constellations and heavens were made.

Circumscribed (enclosed within a circle), the five Platonic solids can be represented by:  

Or shown as plane figures, where the matched sides can be then be joined in 3 dimensions to complete the figure. 

An excellent example of the latter idea when plane figures are folded into solid ones is Buckminister Fuller’s Dymaxion [link to <http://www.bfi.org/map.htm> where he was able to accomplish a more accurate projection of the Earth’s continents using almost exclusively equilateral triangles.  The ICOSA triangles are roughly 63^o 26’ on an edge -- representing roughly 3,806 nautical miles.  In fact, the Fuller Projection provides for a much more accurate appearance than typical Mecator Projections used in maps throughout the world.  

The tetrahedron, octahedron, and icosahedron have a common ingredient of equilateral triangles, but the dodecahedron and icosahedron have an impressive connection as well.  Besides having either 20 vertices and 12 surfaces or vice versa, perpendiculars from the midpoints of the surfaces of one solid define the points of the other solid. 

The only pyramid that is a Platonic Solid is the tetrahedron.  The more famous pyramids -- such as The Great Pyramids of Giza and Teotihuacan -- are Square Pyramids (i.e. a square base, and four triangles meeting at the apex).  There are also Pentagonal Pyramids -- such as the alleged five-sided pyramid at Cydonia on Mars -- Polygon Pyramids, and variations on a theme, e.g. Interlocked Tetrahedra (known to Drunvalo Melchizedek as the Merkaba) and a “Pentagonal Rotunda”.

Sacred Geometry          Vesica Pisces         A Graphics Description

The Elements         Golden Mean         Golden Mean Mathematics

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