Is there a set of interlocking building blocks that is only capable of filling space in patterns with the symmetry of a soccer ball (a.k.a. an icosahedron)?

For centuries, mathematicians and scientists thought this was impossible. The only known interlocking structures were periodic, consisting of a regular repetition of a single block or cluster of blocks, and it is mathematically impossible to have a periodic structure with even one five-fold symmetry axis.

An icosahedron was thought to be super-forbidden, since it contains six independent five-fold symmetry axes (e.g., through the pentagons on a soccer ball).

In the 1970s, Roger Penrose at Oxford University constructed interlocking tiles in two dimensions that form a five-fold symmetric pattern. Then, in 1984, Dov Levine and Paul Steinhardt, then at the University of Pennsylvania, showed that three-dimensional structures with five-fold symmetries are possible if the structure has two or more types of building blocks that repeat quasiperiodically (that is, with different repetition rates whose ratio is an irrational number).

The sculpture shows a cluster with four interlocking units that can be continued to fill Quark Park and beyond in a pattern with icosahedral symmetry. This geometrical construction led Levine and Steinhardt to propose the theoretical possibility of a new type of inon-crystalline, icosahedral solid matter, known as a "quasicrystal," which has actually been grown and studied in the laboratory and has led to the development of new materials, new theoretical concepts and new designs.

Paul Steinhardt

 “Forbidden Geometry” was conceived for Quark Park as a collaboration between Princeton physicist Paul Steinhardt and myself, a sculptor. The sculpture is influenced by Steinhardt’s work with Quasi Crystals and their underlying geometries, which were thought to be forbidden until Steinhardt’s discovery of their existence.

Using Steinhardt’s work as inspiration, I created a sculpture made of Indiana Limestone and glass. The work was produced at the non- profit Digital Stone Project. "Forbidden Geometry” is based on small 3- D models of 4 different zonohedra (famous geometric structures) which can be combined under certain rules developed by Steinhardt as building blocks for Quasi Crystals. I chose a configuration which combines 6 zonohedra, 3 of which are made of limestone and 3 of which are made of glass. The limestone elements were generated from a laserscan of the original small models. The scans were then digitally enlarged and cut in stone on a computer controlled milling machine at the Digital Stone Project. The configuration of the 6 zonohedra adheres to the rules for building Quasi Crystals developed by Steinhardt and demonstrates the starting point for a Quasi Crystal which could expand in all directions using the same 4 geometric shapes.

The use of glass and the reflection of light within alludes to the observation that Quasi Crystals have very unique properties interacting with photons, or units of light. Light seems to be reflected within the Quasi Crystal in a manner similar to the way in which electrons are reflected within quartz crystals. This suggests the possibility of using Quasi Crystals for photon based data transmission, which could revolutionize computing and data processing.

Cristoph Spath