Category:Platonic solids

The Platonic solids are a set of five fundamental regular solids. These five solids are the only fully-regular, convex polyhedra that can be constructed.

Historical significance

Described in Euclid's Elements (circa 300 B.C.) and in Plato's Timaeus (circa ~360 B.C.), both texts state that these are the only five possibilities. The fact that these shapes are fundamental and that there are only five of them captured some philosophers' imaginations, and Plato associated the first four with the "four elements", Earth, Air, Fire and Water, with the final shape (the dodacahedron) representing "the arrangement of the heavens". Aristotle appears to have picked up on this and declared that the "fifth element" was the substance of space (quintessence, or aether).

Other attempts to try to use the solids to "explain" aspects of nature included attempts to relate the sizes of planetary orbits in our solar system to the relative sizes of the Platonic solids when nested. Although this sometimes gave a fair approximation, the idea became less credible as more planets were discovered.

Conditions

The criteria for a solid being a member of this set is that it must have all its faces identical, its faces must be regular polygons (with all sides equal-lengthened, and all corners equal-angled), and that every junction of faces must be identical.

Members

The five members are:

• Tetrahedron – four triangular faces
• Octahedron – eight triangular faces
• Cube – six square faces
• Dodecahedron – twelve pentagonal faces
• Icosahedron – twenty triangular faces

Variations

Truncation

We can use the "Platonics" as the basis of a further family of solids by chopping off their corners to produce new regular-polygon faces. If we remove the corners of a cube to trim the six square faces into regular octagons, the resulting shape with six octagonal faces and eight triangular faces (where the corners were), is a truncated cube. If we make the cuts deeper so that they touch, the original square faces again become squares (rotated through 45 degrees), and we have a shape with six square faces and eight triangular faces, a cuboctahedron. Similar operations can be applied to the other Platonic solids.

Stellation

Another way of creating further shapes from the Platonic solids is stellation, which usually involves extrapolating the edges and planes of the existing faces to produce "star-like" shapes.

Others

There is a further extended family of not-quite-regular polyhedra, such as the rhombic dodecahedron, which has twelve identical diamond-shaped faces. While this shape has every face identical, and every face junction identical, it fails to be classed as a Platonic solid because its diamond-shaped facets look fairly regular, and all its edges are the same length, the diamond shape has two different types of corner, with two different angles.

Deliberately-irregular strictly-convex solids with regular faces are Johnson solids, after Norman Johnson, who proved that there were only 92 of them (after which any further variations have to involve flat or concave junctions between faces).

External links

• Platonic solids (mathsisfun.com)
• Platonic Solids in All Dimensions, by John Baez (math.ucr.edu/home/baez)
• Platonic Solids and Plato's Theory of Everything (mathpages.com)
• The PLATONIC SOLIDS, in art (dartmouth.edu)
• Platonic solid (britannica.com)
• Platonic solid (mathworld.wolfram.com)
• Platonic solid (wikipedia.org)