July 4, 2011

Mobile of the Fourth Dimension

An intricate geometric framework of linked pentagons hangs in the atrium of the Fields Institute for Research in Mathematical Sciences in Toronto. The five-foot-diameter construction resembles a giant soccer ball stripped of its skin to reveal an elaborate supporting structure.

Created by Marc Pelletier, the stainless-steel sculpture represents a mathematical object known as the 120-cell. It is a three-dimensional shadow, or projection, of a four-dimensional dodecahedron.

A regular dodecahedron has 30 edges and 12 faces, each of which is a regular pentagon. Its four-dimensional analog—a polydodecahedron or hyperdodecahedron—contains 120 dodecahedra, three to an edge. The resulting 120-cell consists of 720 pentagons and has 600 vertices and 1200 edges.

Pelletier's sculpture embodies one possible, particularly symmetric projection of this four-dimensional object in three dimensions. In this projection, not all of the 120 dodecahedra of the 120-cell are visible explicitly. As it slowly rotates, it shows off its various symmetries.

The sculpture features an undistorted dodecahedron at its center. This dodecahedron is surrounded by 12 others, which are only slightly distorted by foreshortening. Proceeding outward, the next layer has 20 dodecahedra, then 12 more that are considerably flattened by foreshortening. The final layer consists of 30 dodecahedra that are seen edge-on and so appear flat, delineating the sculpture's outer surface. Steel rods define the edges.

Installed and dedicated in 2002, the Fields Institute sculpture honors geometer H.S.M. Coxeter, who described the 120-cell in his classic book Regular Polytopes. Coxeter died in 2003 at the age of 96.

Pelletier later produced a copy of this sculpture for Princeton mathematician John H. Conway. It was on display in 2006 and 2007 at a temporary outdoor art space known as Quark Park, in Princeton, N.J. (see "Quark Park").

Photos by I. Peterson

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