The Penrose Tiling in the Bachelor Hall Courtyard of Miami University

By the end of summer 2024, the Department of Mathematics will move out of Bachelor Hall after occupying it for over three decades. There is a strong connection to the building, and especially the beautiful Penrose tiles in the courtyard of the building.

The story behind the installation of these gorgeous mathematically shaped tiles was remarkable and accounted for by articles of the main persons behind the idea, math professor emeritus Prof. Milton Cox and former math department chair professor emeritus Prof. David Kullman. We have asked them to share their thoughts on the mathematics and the history behind the tiling.

Milton Cox: A Mathematical Puzzle

The Penrose Tiling in the Bachelor Hall Courtyard presents acclaimed art and an international discovery for a mathematical puzzle unsolved for many years. Professors and students in math classes have viewed this art to understand many applications of mathematics, including geometry, topology, abstract algebra, and real analysis. The tiling is ideal for a senior capstone course in mathematics. Here is my story as I served as systems convener of the production of this art and science.

In September of 1976 I attended John Conway’s surprise Penrose presentation at Miami’s annual regional math conference. I viewed the tiling as remarkable art that solved an important mathematical question. The kite and dart shapes had special meaning for me since my middle name is “Dart” and my favorite aunt’s name was “Kite.” In October my young family journeyed to Boston to install a chapter of Pi Mu Epsilon, the national math honorary, at Boston University. I took along diagrams of Penrose Tilings so I could consider which one might be an artistic contribution to the Bachelor Courtyard, at that time in the planning stage. In Boston we toured the Massachusetts' State House, and there I got ideas and inspiration from the colors of the floor tiles. My daughter Ellen, age 6, had brought along a full set of colored pencils, and I used them to try out colorations for the Penrose tiles. Of course, two of the colors were the brick red and trim cream colors of Miami buildings. I imagined white and wedgwood blue could be other colors of early Miami days.

In 1977 I saw a Penrose tiling on the cover of the January issue of the magazine, Scientific American, and read the corresponding article by Martin Gardner. This was the first announcement and description of Penrose Tiling in the popular press. This enhanced my conviction that a Penrose Tiling would be valuable artwork for Miami. I selected the Cartwheel version of the tiling because it was featured on the cover of Scientific American. Having made this decision, I proposed the Cartwheel design and my colors for the Bachelor courtyard to our math department chair, Elwood Bohn. We set up a meeting with President Shriver, Treasurer Lloyd Goggin, and colleagues from Alumni Development. At that meeting we decided to investigate possibilities. The cost of the project was estimated to be \$10,000, and the Development Office found the funding to do the project. As far as I know, no specific donor was credited with this funding, so no donor’s name has been connected to the project.

I planned my family's summer vacation making sure the tiling would not be started while I was gone. Unexpectedly, it was, and when I returned, I saw that the construction crew initiated the Cartwheel 90 degrees off the correct line of symmetry for the building. After negotiations and alternatives were explored (for example, rotating the entire building 90 degrees), the beginning construction was undone, and the project proceeded along the correct axis. Although they had to remove and rotate the forms one week into the construction process, the artisans constructing the tiling continued to be of good spirits.

When I retired, I left with the math department the information about the company that installed the tiling, the architect’s detailed drawing made from my sketches and the places around the world where chips of each of the four colors were mined. These may have been moved to the University Archives.

Those of us working to restore and relocate the Penrose Tiling have investigated various alternatives. Because the mathematics department is moving from Bachelor Hall, but not to a new building, there was little interest or opportunity of restoration regarding the Penrose Tiling among math faculty members. Other locations were discussed, such as under the Upham Arch (the math department may move to Upham), at the location of the new McVey Data Science Building, or on the grounds of the art museum. However, there was little interest in pursuing options. The University Architect declared the tiling unsalvageable, so at that point there was no hope of acquiring funds for its transfer, even though there were volunteers to contribute. Yet, unexplored questions remain as to whether the brass molding could have been dug out carefully and reused. Perhaps the terrazzo chips could be dug out, cleaned, and then poured into the saved molding. This could limit costs of restoration to labor while avoiding the cost of acquiring a new brass molding and terrazzo chips from around the word.

With respect to a new format for a Penrose tiling, there was no specific plan proposed except for a few comments about reproducing the design in another type of display such as a wall hanging.

Artwork exposed to outdoor elements over several years (over 45 years in this case) requires special attention and care. The loss of the Penrose Tiling illustrates the damages that artifacts can suffer when their well-being is not preserved for the future. The Penrose Tiling in the courtyard of Bachelor Hall, a unique work of art at Miami University featuring an international historic and educational breakthrough, appears to be lost. For me, the memory of my experiences joining art and mathematics, colleagues and students, and abstract and real world, will preserve one of the highlights of my life.

David Kullman: Periodic vs. Nonperiodic

This Penrose Tiling is one answer to a long-unsolved problem about nonperiodic tilings. A tiling is periodic if its design can be repeated by sliding, without rotating or reflecting the shapes. If this is not possible, then the tiling is said to be nonperiodic. Most tilings that we encounter are periodic.

A two-dimensional version of David Hilbert’s 18th problem, posed in 1900, asks for the smallest set of tile shapes that will tile the plane only nonperiodically. In 1964 a Harvard University doctoral student discovered such a set that used more than 20,000 differently shaped tiles. Later he was able to reduce the number to 104. A much smaller set, made up of only six distinct tiles, was discovered by Raphael M. Robinson in 1971. Then, in 1974 the British mathematical physicist, Sir Roger Penrose, found a set needing only two tiles. Penrose began with a rhombus having vertex angles of 72 and 108 degrees. He divided the long diagonal at a point called the golden section. This gives a ratio of (1+Ö5)/2, or approximately 1.618, for the lengths of the two segments of the diagonal. He joined this point on the diagonal to the two vertices with largest angles, forming two new polygons – a kite and a dart. A Penrose pattern is formed by starting with darts and kites surrounding one vertex and then expanding radially.

In September 1976, Miami University held its Fourth Annual Mathematics and Statistics Conference. The theme that year was Recreational Mathematics, and one of the featured speakers was the British mathematician, John Conway. On the spur of the moment, to fill in for a speaker who had cancelled at the last minute, Conway offered to give a talk about “Penrose’s Puzzle Pieces.” (He just happened to have some transparencies with him.) The following January an article about Penrose tiles appeared in Scientific American.

At the time Bachelor Hall was still on the drawing board, and some of the mathematics faculty hoped that the design would include something uniquely mathematical. Milt Cox proposed to the University administration that a Penrose tiling would be a fitting decoration for the Bachelor Hall courtyard. Lloyd Goggin and the architect agreed, and Milt was appointed to come up with a specific design. He decided that Penrose’s cartwheel design was the most interesting, and he chose the colors wedgewood blue, cream, red, and white to complement the red brick exterior of the Bachelor Hall.

The architect selected terrazzo, which is made by embedding small chips of marble or granite in mortar and then polishing the surface after it has dried, as the tiling medium. A brass framework was fabricated in the outline of the tiling, and the colored tiles were poured individually. Work on the terrazzo began in the summer of 1979, while both Milt and our department chair, Elwood Bohn, were on vacation. When they returned, it was discovered that the brass frame had been rotated 90° so that the tiling’s axis of symmetry did not coincide with the axis of symmetry of the building.

Work was immediately stopped while the powers that be scratched their heads. One idea that was rejected was to rotate the building 90 degrees. Fortunately, only about 25% of the tiles had actually been poured, so it was decided to chisel them out and start over, with the frame properly oriented. A minor flaw is the drain at the very center of the tiling. Here the beauty of pure mathematics must be tempered by constraints of the real world.

A quasicrystal connection: A quasi-periodic crystal (quasicrystal) is a crystal-like structure that is ordered but nonperiodic. That is, it can fill space but is not repeated by sliding. A two-dimensional slice of a quasicrystal will have properties similar to those of a Penrose tiling. According to the classical crystallographic restriction theorem, crystals can possess rotation symmetries involving only multiples of 180, 120, 90, or 60 degrees. Diffraction patterns of quasicrystals show other rotation symmetries, such as 72 degrees (as in a pentagon). The first experimental observations of what came to be known as quasicrystals were made by Daniel Shechtman and his coworkers in 1984. Shechtman received the Nobel Prize in Chemistry in 2011 for his discovery.

Department of Mathematics

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