Crafting a Penrose Tiling

An array of wooden tiles assembled into an intriguing pattern flecked with stars forms a striking contrast to the regular arrangement of bricks making up the wall on which it hangs on the third floor of Avery Hall, home of the mathematics department at the University of Nebraska-Lincoln.

Constructed by Nebraska mathematician Earl S. Kramer from diamond-shaped cherry and maple tiles and installed in March 2005, the wall piece represents a patch of one of the infinite number of ways in which to arrange fat and skinny diamonds into an aperiodic pattern characteristic of a Penrose tiling.

The two types of tiles for assembling such a Penrose tiling are rhombs (each rhomb has four sides of equal length) with acute angles of 36 and 72 degrees. Matching rules specify the ways in which these rhombs must be assembled edge to edge to create an aperiodic tiling (one in which the tiling cannot be lifted and placed back onto itself with all points displaced but still looking the same).

The particular tiling pattern depicted in the wall piece is one of two Penrose rhomb arrangements that have the dihedral automorphism group d5, featuring rotations of order five and reflections across a line, readily apparent in the design.

For another artistic representation of a Penrose tiling, see "Tessellation Tango."

Reference:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

Plaster Models of Mathematical Surfaces

The shapes of surfaces captured the imagination and attention of many mathematicians during the nineteenth century. To unfold the visual secrets compressed and hidden within the shorthand of algebraic expressions, geometers drew pictures, fashioned models, and even wrote manuals on how to visualize or construct geometric forms.

These models and drawings were not only a source of pleasure but also a valuable tool for probing a slew of exotic geometric structures. Indeed, during the latter part of the nineteenth century, no university mathematics department could count itself at the forefront of research and pedagogy without owning a set of plaster models depicting an array of geometric forms. Firms, particularly in Germany, specialized in crafting these teaching aids.

Rudolf Clebsch's diagonal cubic surface.

Over the years, the models largely disappeared from view, supplanted by more abstract representations and new directions in mathematical research and pedagogical approach. More often than not, they ended up gathering dust in closets or simply in the trash.

I was reminded of this history on a recent visit to the mathematics department at the University of Nebraska in Lincoln. Retrieved from storage and exhibited in illuminated display cases, these models now serve as a link between the rich early history of geometric visualization and modern, computer-based approaches to representing the intricacies of geometric forms.

You can find collections and displays of plaster mathematical models at a number of universities, including the University of Illinois at Urbana-Champaign, University of Arizona, HarvardUniversity, Hebrew University of Jerusalem, and others. The original, hand-crafted models are also being recreated using digital fabrication technology (see http://vimeo.com/18819673) and inspiring the work of artists (see http://www.sugimotohiroshi.com/MathModel.html).

These graceful plaster models bring together the logically abstract and the visually concrete in mathematics. They are vivid testimonials to the work of nineteenth-century geometers and to the beauty of mathematical forms. They represent elegant milestones in the struggle by mathematicians to elucidate the fundamental principles of geometry.

References:

Fischer, G. 1986. Mathematische Modelle: Mathematical Models from the Collections of Universities and Museums , Vol. 1. Vieweg & Sohn.

Peterson, I. 1990. Islands of Truth: A Mathematical Mystery Cruise. W.H. Freeman.

Photos by I. Peterson