My printed tile map. It identifies the fat and skinny rhombus tiles so that I knew how many to make and how to place them. It was generated based on a simulation of an edge length of 250mm and a gap width of 6mm, adding up to a nice binary number.
I suspended this project in order to go on a roadtrip to capture pictures of the night sky in the beautiful deserts of the Southwest. I am currently working on them, and hope to share them soon, but the Penrose tile floor project carries a higher priority—we want our screen porch back while it is still summer!
Having prepared my tiles to the best accuracy I could coax from my woodworking tools, I now faced how to place them on the floor. As before, I considered the advice from Ken Adelman, who recommended “dis-aligning” the pattern from the rectangle of the room, to avoid difficult or awkward-looking tile fragments at the edges. He also recommended identifying a center point and creating reference lines radiating at angles that match the pentagonal symmetries of the tiling.
A timelapse of a Penrose tile installation by Ken Adelman
I am about to embark on a month-long road trip, and I am reluctant to start the next phase of this project—laying and setting the tiles, something I expect will take a considerable amount of time and attention. Instead, I need to make plans for this upcoming trip which involve excursions to remote areas of the Southwest for the purpose of making night timelapse sequences. I am skeptical that I can fully succeed at either, much less both, in the time remaining.
So for now, I will put my tiles aside, and will instead present a rendition of the tile-laying process made by my Facebook friend Ken Adelman, the person I referred to as having succeeded in Penrose-tiling his sunroom, and who has kindly counseled me in this project.
He made a timelapse of his installation that spanned several days. I have posted it to my Vimeo account and you can watch it here. I found the movement of the sun quite fascinating as the tiles were carefully placed and spaced, the shadows indicating the elapsed time involved.
Maybe I can make a similar movie, but it will have to wait until after I return from the Nightscaper Conference, where I hope to learn the modern tools of nighttime landscape photography. Technology has changed dramatically from when I embarked on my Nightscape Odyssey twenty years ago, and I am eager to keep up.
My hand-crafted P3 rhombuses, awaiting my return to install them.
The last cut on this fat rhombus tile. The blue arm at the top of the tile is a hold-down clamp. The fence is at an angle of 18 degrees and the blue gauge on the fence track is a stop that positions the tile at the exact required distance from the blade.
I placed an order for the Marmoleum planks that I intended to cut into my Penrose rhombus tiles. It is always a bit nerve racking, making calculations, optimizing the tile sizes, trying to minimize waste, and reaching a conclusion about how much raw material will be needed. What if I am off in my estimate for the sawblade kerf?
I learned late in the ordering process that the planks were not 1-foot by 3-foot; the sales rep contacted the factory at my request and reported back that they were 300 mm by 900 mm. Further, the ordering process called for the number of square feet, but the planks were packaged in bundles, and only full bundles were shipped, so the required square foot area was rounded up to the next bundle size.
For most flooring projects, this is probably just fine, but I needed to know just how many planks would be delivered so that I could ensure that I would be able to make all of the tiles in my design. In the end I learned that the bundles contained seven planks, 20 square feet of flooring, or should I say 1.89 square meters?
With the knowledge of the exact linoleum area of each plank, I could now partition them into rhombuses (rhombi?). I determined that I could get three fat or four skinny rhombuses from each plank. I counted the rhombuses in my pattern and ordered the exact number of planks required: 21 for the skinny rhombus, and 42 for the fat ones. These are nice multiples of seven; I was pleased. I placed the order, the moment of financial commitment to this project.
Later, I realized that I could have done better. If I had slightly tilted the skinny rhombus cookie cutter on the plank, they could have been a bit larger. The fat rhombuses would have been correspondingly larger, and I would have needed fewer planks and the waste would have been smaller. I contemplated revising my plan, but after discovering that the cutting complexity would be high (introducing opportunities for mistakes), and the gain was rather small– a few percent– coupled with the guidance of my partner who reminded me about false economies, I opted to stay with my original plan. I am glad that I did.
The tile-making involved many cuts on my table saw. It was important to set up each cut with a particular jig and fixture, and then cut all of the raw material that needed that setup, all at the same time, before changing the saw for the next cut. This would guarantee that all of the pieces would be congruent, with the same dimensions and angles. It was important to make them the same, but it was even more important to make them correct. A hundred identical tiles, all of which are the wrong shape, was my greatest fear. So I embraced the expression “measure twice, cut once” and fell into a paranoid checking of dimensions and angles.
I ended up creating 84 skinny rhombus tiles and 126 fat ones. It took 462 passes through the table saw to make them. I worried about the psychological lulling of attention with repetitive tasks. I have encountered experienced woodworkers, with missing finger tips, who recounted the event that severed them. Invariably it was a lapse of attention, usually because of a trivial or repetitive cut that caused them to misjudge or ignore the spatial positions of their hands relative to the saw.
Aware of these stories, while shopping for a table saw, I learned about a model that detects human contact with the blade and fires an explosive brake to instantly stop it, analogous to an airbag in a car. They are expensive, but for an inexperienced woodworker like me, it seemed like a good investment. I am quite pleased with my SawStop table saw. It is a precision tool that I hope to never trigger.
I spent most of a week cutting tiles from the Marmoleum planks. I took it in stages, and today I cut the last of them. My fingers are intact and I am eager to start placing the tiles. I am also pleased that they seem to be dimensionally correct. My precision is not to the thousandth of an inch. I might be able to claim ½ millimeter, which would be 1/50th inch. We will see how that translates to tile placement with pentagonal symmetry!
My tile size was based on an edge dimension of 250 mm, almost 10-inches. The angle of the fat rhombus is 72 degrees. Here are my checks.
The length of each side of the tile is 250 mmThis angle on the fat rhombus should be 72 degreesSkinny rhombus tiles accumulating on the workbench behind me. Hopefully they are all identical, but more importantly, matching the right size and shape.
A rectangular section of a symmetric Penrose tiling. This will be the pattern for the porch floor.
I had seen examples on the web of Penrose tiles, but they were always rather high-end installations. I recently encountered someone who had successfully created a Penrose flooring in his sunroom. I was able to ask him about the details of his project and the recurring theme in the ensuing discussion was “accuracy”. His floor was made of ceramic tiles, rhombuses carved from 1’x2’ rectangular commercial tiles with a computer-controlled water-jet cutter to one-thousandth inch precision.
I was not prepared to go to this level, so I sought less expensive materials and tooling, settling on modern linoleum, “Marmoleum”, a materal that can be obtained laminated to a medium density fiber board substrate that I could cut myself. Any lack of machine precision would be hidden by the spacing and grout lines between the tiles. At least that is my plan.
Still, it was important for the angles and dimensions of the tiles to be as consistent and accurate as possible. I made a proof-of-concept trial with sheets of plywood, cutting them on my table saw using the fence and miter gauge at the prescribed angles. This exercise showed me that the standard methods would not work. I needed a more specialized jig, one that could result in many, many congruent tile shapes being cut to precise angles and lengths. I learned that such jigs for the table saw are common, at least among the skilled woodworkers that make fine furniture and other beautiful objects.
What I needed was a “crosscut sled”, a fixture that could be crafted using the saw it would ultimately supplement, and there were many YouTube instructions on how to make one. After watching several, I opted to skip the learning curve and purchase a commercial version. It had a wonderful angle fence, riding on a machined steel guided platform running parallel to the blade, equipped with a stop that could exactly position the material for its cut.
It was perfect. I created several identical copies of the first fat rhombus, and started making the second skinny rhombus when I discovered that the jig could not reach the required angle, 54 degrees. It stopped at 50.
A customization was required and I was able to extend the range by routing a slot in the sled, and calibrating it. I now have a Penrose-compatible crosscut sled for my table saw. On to actually making my linoleum tiles!
My routing skills are not high, but I was able to extend the angular range of the crosscut sled to 60 degrees!
Alan Holden in his studio with shelves of his creations, ca 1970
I have long been intrigued by geometric patterns. As a teenager I made models of various polygons and polyhedra and learned the rules for constructing geodesic domes. A book that held my fascination for years was “Shapes, Space, and Symmetry” by Alan Holden.
The ability for computers to represent 3D objects and to realistically render them, to interact with them, and to display them in 3D was years in the future; in the 1970s, physical models were essential for teaching geometric principles and understanding crystal structures. The author, a physicist and chemist, had crafted a lifetime of such models and described them in his book. I could not match the patience and skill required to make his beautiful and complex cardboard models.
Also in the 1970s, Roger Penrose, the British mathematician and scientific colleague of Stephen Hawking, investigated an arcane branch of geometry to answer the question of whether an infinite plane can be tiled with a set of shapes that did not overlap, have gaps, or repeat. He found that the answer was yes, it was possible, and he discovered several sets of shapes that could do it. The first set included pentagons, stars, “boats” and diamonds. The second set was simpler, it needed only two shapes called kites and darts. The third was simpler yet, a pair of rhombuses, skinny and fat parallelograms with equal length sides. These sets of shapes, P1, P2, and P3 are known as “Penrose tiles”.
I’m not going to explain the mathematical concepts behind them, even if I could, but I will call attention to their esthetic beauty, which you can find by a simple Google search. Penrose tiles became popularized by Martin Gardner’s famous Mathematical Games column in Scientific American in 1977, and suddenly everybody was making Penrose patterns out of them. Penrose was able to patent the shapes, which were subsequently licensed for games and puzzles. He famously won a lawsuit against a company that used their non-repeating feature to prevent their embossed toilet paper from sticking on the roll!
When presented with the “canvas” of the floor in my newly renovated screen porch, I immediately wondered how best to cover it. I really liked the idea of ceramic tiles, impervious to rain and snow, and then realized this floor could be a host to the mathematical beauty of Penrose tiles!
I researched the idea. There were Penrose tilings in public spaces, famously at Texas A&M and more locally, Carleton College. I also encountered individuals who had made such patterns in their private homes. I learned that this was not a project for the wing-it, make-it-fit crowd; the tile shapes needed to be cut to thousandth-inch precision. I considered what was needed to cut ceramic tiles to this precision and decided to look at other options, the first of which was to make precision cuts of plywood panels with beautiful wood veneers. Perhaps a laser cutter could mark and cut wood tiles to the necessary precision.
A chorus of my technically and construction-astute friends warned me against this plan—the plywood edges would respond to the outdoor conditions by curling up or down in response to temperature and moisture changes.
So I went on to evaluate other materials, and re-discovered the appeal of “luxury vinyl tile”, a heavy duty version of vinyl flooring. Within this category was “marmoleum” a natural mix of linseed oil and other natural ingredients, a modern linoleum. I found that it was offered in tile plank sizes that could be trimmed into Penrose tile shapes!
I now had a medium, but needed a pattern. Penrose tiling is not quite as simple as laying the tiles down wherever they fit. In order to tile the plane, with no overlaps and no gaps, one must follow the “edge matching rules”. By matching the edges, the tiling that results will ensure that the plane will be perfectly covered. To help accomplish this, tiles are marked in such a way that adjacent tiles will be placed according to the edge matching rules.
I wanted to make a scale mock-up of the floor pattern. I tried some of the online Penrose tile patterns, printed them out, and cut multiple copies. The paper-thin substrate, scissors-cut by hand, were not very successful. They didn’t lay flat or align well and were easily disturbed by any slight breeze or sneeze.
I discovered an alternative. An Etsy store of homemade wooden toys that included among their catalog of rocking horses and train cars, a set of laser-engraved, wooden Penrose tiles, beautifully crafted, sanded and finished, in either P2 or P3 shapes, all in a handy home-sewn carrying bag! I ordered one set each from Wooden Giraffe Toys and had them within a few days! I immediately started making aperiodic five-fold symmetric tilings from them to get a feel of what the floor might look like.
The front of the tiles had the edge-matching rule markings, but the backsides were a solid contrasting color. Once the pattern was confirmed by the front markings, individual tiles could be flipped to create the visual pattern I sought.
Now that I have a Penrose tile pattern to my liking, I need to figure out how to actually make the tiles and install them.
A pattern of wooden Penrose rhombuses made from the set offered by Wooden Giraffe Toys. I don’t consider them toys; they are tools!
