Regular shapes The
Platonic solids are three-dimensional shapes with special, high,
symmetry. They are the next step up in dimension from the two-dimensional
regular polygons (squares, equilateral triangles, etc.). The five Platonic solids are the
tetrahedron (4 faces),
cube (6 faces),
octahedron (8 faces),
dodecahedron (12 faces), and
icosahedron (20 faces). They have been known since the time of the Ancient Greeks and valued for their aesthetic appeal and philosophical, even mystical, import. (See also the
Timaeus, a
dialogue of Plato.) In higher dimensions, the counterparts of the Platonic solids are the
regular polytopes. These shapes were first described in the mid-19th century by a Swiss mathematician,
Ludwig Schläfli. In four dimensions, there are
six of them: the pentachoron (
5-cell), tesseract (
8-cell), hexadecachoron (
16-cell), icositetrachoron (
24-cell, here called the octacube), hecatonicosachoron (
120-cell), and the hexacosichoron (
600-cell). The 24-cell consists of 24
octahedra, joined in 4-dimensional space. The 24-cell's
vertex figure (the 3-D shape formed when a 4-D corner is cut off) is a cube. Despite its suggestive name, the octacube is not the 4-D analog of either the octahedron or the cube. In fact, it is the only one of the six 4-D regular polytopes that lacks a corresponding Platonic solid.
Projections of the Earth Ocneanu explains the conceptual challenge in working in the fourth dimension: Part of Ocneanu's work is to build theoretical, and even physical, models of the symmetry features in QFT. Ocneanu cites the relationship of the inner and outer halves of the structure as analogous to the relationship of
spin 1/2 particles (e.g.
electrons) and
spin 1 particles (e.g.
photons). ==Memorial==