Octacube (sculpture)

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.)

The Platonic solids
Tetrahedron	Cube	Octahedron	Dodecahedron	Icosahedron

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), octacube (24-cell), hecatonicosachoron (120-cell), and the hexacosichoron (600-cell).

The 24-cell consists of 24 octahedrons, 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.[note 1]

Attempts to picture the 24-cell
Schlegel diagram	4-dimensional rotation

Stereographic projection of the Earth

Ocneanu explains the conceptual challenge in working in the fourth dimension:[1] "Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize."

Although it is impossible to see or make 4-dimensional objects, it is possible to map them into lower dimensions to get some impressions of them. An analogy for converting the 4-D 24-cell into its 3-D sculpture is cartographic projection, where the surface of the 3-D Earth (or a globe) is reduced to a flat 2-D plane (a portable map). This is done either with light 'casting a shadow' from the globe onto the map or with some mathematical transformation. Many different types of map projection exist: the familiar rectangular Mercator (used for navigation), the circular gnomonic (first projection invented), and several others. All of them have limitations in that they show some features in a distorted manner—'you can't flatten an orange peel without damaging it'—but they are useful visual aids and convenient references.

Stereographic projection of a 24-cell

In the same manner that the exterior of the Earth is a 2-D skin (bent into the third dimension), the exterior of a 4-dimensional shape is a 3-D space (but folded through hyperspace, the fourth dimension). However, just as the surface of Earth's globe cannot be mapped onto a plane without some distortions, neither can the exterior 3-D shape of the 24-cell 4-D hyper-shape. In the image on the right a 24-cell is shown projected into space as a 3-D object (and then the image is a 2-D rendering of it, with perspective to aid the eye). Some of the distortions:

• Curving edge lines: these are straight in four dimensions, but the projection into a lower dimension makes them appear to curve (similar effects occur when mapping the Earth).
• It is necessary to use semi-transparent faces because of the complexity of the object, so the many "boxes" (octahedral cells) are seen.
• Only 23 cells are clearly seen. The 24th cell is the "outside in", the whole exterior space around the object as seen in three dimensions.

To map the 24-cell, Ocneanu uses a related projection which he calls windowed radial stereographic projection. As with the stereographic projection, there are curved lines shown in 3-D space. Instead of using semitransparent surfaces, "windows" are cut into the faces of the cells so that interior cells can be seen. Also, only 23 vertices are physically present. The 24th vertex "occurs at infinity" because of the projection; what one sees is the 8 legs and arms of the sculpture diverging outwards from the center of the 3-D sculpture.[1]

Octahedral symmetry diagram showing mirror planes as great circles (6 red, 3 blue). Rotation axes are also shown: 2-fold (pink diamonds), 3-fold (red triangles), and four-fold (blue squares).

The Octacube sculpture has very high symmetry. The stainless steel structure has the same amount of symmetry as a cube or an octahedron. The artwork can be visualized as related to a cube: the arms and legs of the structure extend to the corners. Imagining an octahedron is more difficult; it involves thinking of the faces of the visualized cube forming the corners of an octahedron. The cube and octahedron have the same amount and type of symmetry: octahedral symmetry, called O_h (order 48) in mathematical notation. Some, but not all, of the symmetry elements are

• 3 different four-fold rotation axes (one through each pair of opposing faces of the visualized cube): up/down, in/out and left/right as seen in the photograph
• 4 different three-fold rotation axes (one through each pair of opposing corners of the cube [along each of the opposing arm/leg pairs])
• 6 different two-fold rotation axes (one through the midpoint of each opposing edge of the visualized cube)
• 9 mirror planes that bisect the visualized cube
  • 3 that cut it top/bottom, left/right and front/back. These mirrors represent its reflective dihedral subsymmetry D_2h, order 8 (a subordinate symmetry of any object with octahedral symmetry)
  • 6 that go along the diagonals of opposing faces of the visualized cube (these go along double sets of arm-leg pairs). These mirrors represent its reflective tetrahedral subsymmetry T_d, order 24 (a subordinate symmetry of any object with octahedral symmetry).

Using the mid room points, the sculpture represents the root systems of type D4, B4=C4 and F4, that is all 4d ones other than A4. It can visualize the projection of D4 to B3 and D4 to G2.

Many molecules have the same symmetry as the Octacube sculpture. The organic molecule, cubane (C₈H₈) is one example. The arms and legs of the sculpture are similar to the outward projecting hydrogen atoms. Sulfur hexafluoride (or any molecule with exact octahedral molecular geometry) also shares the same symmetry although the resemblance is not as similar.

Molecules with the same symmetry

Cubane	Sulfur hexafluoride

The Octacube also shows parallels to concepts in theoretical physics. Creator Ocneanu researches mathematical aspects of quantum field theory (QFT). The subject has been described by a Fields medal winner, Ed Witten, as the most difficult area in physics.[2] 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).[1]