Sacred Geometry

Geometryfor fun (click here to play Skin-on-Skin on YouTube)The Platonic solids feature prominently in the philosophy of Plato for whom they are named. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid.

Earth was associated with the hexahedron/cube, Air with the octahedron, Water with the icosahedron, and Fire with the tetrahedron.

There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water.

The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

They are 3dimensional convex regular polyhedrons, the 3D analogs of the convex regular polygons of 2D. There are PRECISELY 5. They are unique in that the faces, edges and angles are all congruent.

1http://en.wikipedia.org/wiki/List_of_regular_polytopes#Tessellations

In geometry, 2 sets are congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).

The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron.

A convex polyhedron is a Platonic solid if and only if all its faces are congruent convex regular polygons, none of its faces intersect except at their edges, and the same number of faces meet at each of its vertices.

Each Platonic solid can therefore be denoted by a symbol {p, q} where p = the number of sides of each face or the number of vertices of each face) and q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).

The symbol {p, q}, called the Schlfli symbol, gives a combinatorial description of the polyhedron. The Schlfli symbols of the five Platonic solids are given in the table below.All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have: pF = 2E - qV The other relationship between these values is given by Euler's formula V E + F = 2. This nontrivial fact can be proved in a great variety of ways (in algebraic topology it follows from the fact that the Euler characteristic of the sphere is 2). Together these three relationships completely determine V, E, and F. Note that swapping p and q interchanges F and V while leaving E unchanged.

PolyhedronVerticesEdgesFacesSchlfli symbolVertexconfigurationTetrahedronSelf dual4643,33.3.3CubeOctahedron81264,34.4.4OctahedronCube61283,43.3.3.3DodecahedronIcosahedron2030125,35.5.5IcosahedronDodecahedron1230203,53.3.3.3.3

Platonic Solids

Platonic solids are perfectly regular solids with the following conditions: all sides are equal and all angles are the same and all faces are identical. In each corner of such a solid the same number of surfaces collide.

Only five Platonic solids exist: tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron.

The solids and their regularities were discovered by the Pythagoreans and were called originally Pythagorean solids. The Greek philosopher Plato described the solids in detail later in his book "Timaeus" and assigned the items to the Platonic conception of the world, hence today they are well-known under the name "Platonic Solids."

The tetrahedron is bounded by four equilateral triangles. It has the smallest volume for its surface and represents the property of dryness. It corresponds to fire.

The hexahedron is bounded by six squares. The hexadedron, standing firmly on its base, corresponds to the stable earth.

The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices and corresponds to air.

The dodecahedron is bounded by twelve equilateral pentagons. It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.

The icosahedron is bounded by twenty equilateral triangles. It has the largest volume for its surface area and represents the property of wetness. The icosahedron corresponds to water.

http://www.3quarks.com/GIF-Animations/PlatonicSolids/Platonic Solids

TetrahedronThe heat of fire feels sharp and stabbing like little tetrahedra. The tetrahedron is bounded by four equilateral triangles. It has the smallest volume for its surface and represents the property of dryness.4D hypercubic4D hyperspherical

4 faces4 points6 edges

Hexahedron

The highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. 4D hypercubic4D hyperspherical6 faces8 points12 edges

Other Views of the Cube 3D

Hexahedron and its Dual: Octahedron

Truncated Hexahedron

Space Filling with Polyhedrons -cube

Planar SpaceHyperbolic SpaceCube /Hexahedron

Other Views of 4D Cube - Tesseract

4D Hypercube A hypercube projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is inside-out representing the space outside the cubic boundary.

Also called the Tesseract.

Other Views of Cube 5D

5D Hypercube : PenteractA penteract is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells. The name penteract is derived from combining the name tesseract (the 4-cube) with pente for five (dimensions) in Greek. It can also be called a regular deca-5-tope or decateron, being made of 10 regular facets.It is a part of an infinite family of polytopes, called hypercubes. The dual of a penteract can be called a pentacross, of the infinite family of cross-polytopes.Penteract dual : Pentacross

Penteract orthogonal views

OctahedronThe Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices and corresponds to air.4D hypercubic4D hyperspherical

8 faces6 points12 edges

Other Objects Octahedron 4D4D Octahedron : 24-cell.24 Cell Orthogonal and Net views

IcosahedronWater, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little ballsThe icosahedron is bounded by twenty equilateral triangles. It has the largest volume for its surface area and represents the property of wetness.

Icosahedral Prism4D hypersphericalAKA 600-cell

20 faces12 points30 edges

Space Filling with Polyhedronsicosahedron

Planar SpaceHyperbolic SpaceCannot fill planar space with this object

120-cell

DodecahedronThe dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element. The dodecahedron is bounded by twelve equilateral pentagons. It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.

4D hypercubic4D hypersphericalAKA 120-cell

12 faces20 points30 edges

Space Filling with PolyhedronsDodecahedronPlanar SpaceHyperbolic SpaceCannot fill planar space with this objectHowever, was Plato on to something?

The Poincar homology sphere (also known as Poincar dodecahedral space) is a particular example of a homology sphere. It is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120.

Based on analyses of the WMAP data, cosmologists during 2004-2006 focused on the Poincar dodecahedral space (PDS), but also considered horn topologies (which are hyperbolic) to be compatible with the data. [from wiki Shape of the Universe and Poincare pages]

Wiki links to image-packed pages

List of Regular PolytopesRegular Polytopes basic geometryConvex Uniform Honeycomb