In 1997 physicist Roger Penrose sued Kimberly-Clark Corporation for infringing on his “Penrose patent” with their Kleenex-Quilted toilet paper. He won the lawsuit but fortunately for lavaphiles the patent has expired leaving much room for aperiodic creativity in the bathroom.

Math is involved in many aspects of house design (two years ago I wrote about how math is even related to something as mundane as the roof), but it is especially important in the design of bathroom floors. The most examined floors in houses are those of bathrooms, as they are stared at for hours on end by pensive thinkers sitting on toilet seats. The best bathroom floors present beautiful tessellations not as mathematical artifact but mathematical artwork, and with this in mind I designed a three colored Penrose tiling for our bathroom a few years ago. This is its story:

Roger Penrose published a series of aperiodic tilings of the plane in the 1970s, famously describing a triplet of related tilings now termed P1, P2 and P3. These tilings turn out to be closely related to tilings in medieval islamic architecture and thus perhaps ought to be called “Iranian tilings” but to be consistent with convention I have decided to stick with the standard “Penrose tilings” in this post.

The tiling P3 is made from two types of rhombic tiles, matched together as desired according to the matching rules (indicated by the colors, or triangle/circle bumps) below:

The result is an aperiodic tiling of the plane, i.e. one without translation symmetry (for those interested, a formal definition is provided here). Such tilings have many interesting and beautiful properties, although a not so-well-known one is that they are 3-colorable. What this means is that each tile can be colored with one of three colors, so that any two adjacent tiles are always colored differently. The proof of the theorem, by Tom Sibley and Stan Wagon, doesn’t really have much to do with the aperiodicity of the rhombic Penrose tiling, but rather with the fact that it is constructed from parallelograms that are arranged so that any pair meet either at a point or along an edge (they call such tilings “tidy”). In fact, they prove that any tidy plane map whose countries are parallelograms is 3-colorable.

The theorem is illustrated below:

This photo is from our guest bathroom. I designed the tiling and the coloring to fit the bathroom space and sent a plan in the form of a figure to Hank Saxe from Saxe-Patterson Inc. in Taos New Mexico who cut and baked the tiles:

Hank mailed me the tiles in groups of “super-tiles”. These were groups of tiles glued together to an easily removable mat to simplify the installation. The tiles were then installed by my friend Robert Kertsman (at the time a general contractor) and his crew.

The final result is a bathroom for thought:

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February 12, 2017 at 10:07 pm

Stan WagonVery nice to see our 3-coloring realized physically like this. I had a carpet made using three colors (light red, light blue, white) in exactly this way…..

Stan Wagon

February 20, 2017 at 6:50 pm

LeeIs the house for sale now that you’re moving to LA?

June 16, 2017 at 11:38 am

ianagolWhen I learned about the 3-colorability of Penrose tilings (from visiting your house) and read the proof, I realized that it is a form of small-cancellation theory. Anyway, one can give a similar proof of 4-colorability of maps in the plane in which every country is connected and adjacent to at least 6 countries (with trivalent vertices wlog). The point is that for any disk in such a map, there must be a country on the boundary which is adjacent to at most 3 countries, and hence after coloring (by induction) the region with one fewer countries, there is at least one color available to color this country. Such maps can’t exist on a sphere, since a map on a sphere must contain a country with at most 5 neighbors (in fact, several such countries). So this doesn’t seem to yield any insight into the 4-color theorem.

August 3, 2017 at 11:05 pm

recessionconeLove this! You might enjoy seeing a Penrose tiling on my bedroom wall: http://recessioncone.blogspot.com/2011/11/penrose-wall.html?m=1

August 31, 2017 at 5:23 am

BeytugNever seen these type of tiles before but they fit perfectly with the bathroom.

September 16, 2017 at 4:50 am

Bathrrom Lights DirectI love these tiles, they add a touch of class to the bathroom.