Pattern Puzzle: Escher + CAD = fun for all

Making the rounds on the intertubes of late (HT) are laser cut wooden floor tiles inspired by Escher (right).  You can buy these on Etsy.  I was sorely tempted to buy them for a future kitchen flooring project but for two things. The tiles are only 1/8″ thick and they are wooden -I wanted ceramic tile.

So after gazing longingly at them for a time, I wondered if I could make my own ceramic tiles. I don’t know if I can; I got a bit side tracked with the first part, that of making a pattern for the tile from which molds could be made.  The cheater quick step to making molds would be to buy some of the wooden tiles but that struck me as dishonest (the wooden tile producer created their own pattern) and besides, it’d take all the fun out of making the pattern myself.

Which brings us to fun ways to use CAD pattern making programs to make patterns for non-apparel or sewn product items. I still haven’t finished my project but thought I’d show you what I’ve done so far in StyleCAD and how I did it. As always, your mileage may vary.

First step was to find an image of Escher’s tessellated butterfly pattern (scroll down).

Second was to enlarge the image and print it out. I enlarged mine to about 8″ across the wing tips. While too large for floor tiles, I wanted something large enough to digitize with sufficient detail.

Third, I traced the outline of the image on the back side of the sheet and digitized it. Here is a pdf of the resulting pattern (I can post a dxf/gerber/stylecad file upon request).

Fourth…is a period of exploration which is where I’m at now. I’m discovering how the pattern interlocks (above right) so I can repair the pattern because it is not perfect. The math whizi in our midst would probably make short work of this and be able to articulate its features readily. To whit, Mr. Fashion-Incubator peered over my shoulder and said “oh, it’s a hexagon”.

My brain works with patterns in ways I can’t articulate, not having the context that a solid grounding in math provides. The first thing I saw was a circle (lower right). A series of them actually. My CAD program makes instant work of circle drafting and one can similarly quickly, assign however many divisions (points) one would want to the circle. I created a segmented circle pattern if you want to play with it yourself.

I took careful notes as to the circle sizes and the relationship to the butterfly shapes within them -and that’s where it is for now.

Currently, the figure is entirely too noisy to extrapolate anything definitive because the butterfly shapes aren’t wholly accurate (yet). I looked for a few hints on the web and in the Escher books I have but couldn’t find anything analyzing this particular design in the context I need it. So that’s where you come in if you’re so inclined to help. In return  I’ll share anything I come up with as the design evolves.  Thanks everybody!

Escher paver molds (I may get these for my patio).
How to make pavers
Open source lizard paver pattern.

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  1. margaret says:

    I’ve posted some very rough and ready diagrams which might explain why Mr FI saw hexagons.

    Kathleen – I’ve used my office blog to share this. (Occasional use of office resources for education/community reasons falls within our company code of ethics.) However, if the sketches are any use, please grab them and share them on your own site, it’s much more appropriate.

    Thank you for the interest and enjoyment I’ve had visiting here.

  2. Jessica says:

    A tessellation tile floor would be awesome!! My husband actually used to make tessellation art pieces out of plastic. He designed the patterns, created molds, and cast colorful pieces that he then interlocked to create intricate artwork. Lots of people call them puzzles and have never heard of a tessellation, so kudos Kathleen for using the proper vocabulary!

    I know he originally made the patterns by hand, but they weren’t accurate enough so he eventually made them with a computer….though I don’t think he used a CAD program to do it so I’m not sure if he’d have any relevant advice. This was all back before I met him, and it was something he did to utilize leftovers from his former plastics manufacturing business.

    Wouldn’t it be cool to do a fashion application and use leather or vinyl to make a tessellation bag or rug or some other product where the cut edges didn’t need to be finished and could be fused to a backing? Oh, the possibilities!

  3. Dana says:

    Once you get the design perfected, I bet it would be pretty easy to get a mold made to make the tiles. I friend of mine has had really good success with 3D printing companies and the cost was not astromonical.

    • Another possibility would be a large “cookie cutter” and stamp-cut the tiles out of rolled clay. The pieces would shrink in the first firing, and thus leave space between them in the final application for grout. Seems to me it would work.

      • Carrie Mather-Crowner says:

        The cookie cutter is a great idea. If you want space for grout you will have to subtract it from the edge of the pieces in the design process. If you do not and all the pieces shrink the same amount (around 5-10% for clay), the interlocking nature of the shapes will prevent them from being spaced out. The design will remain the same, only 5-10% smaller.

        I hope you post pictures!

  4. Quincunx says:

    Fascinating. I’m still seeing the edges of the individual tiles, the six triangles that make up the hexagon, and you’re seeing the relationships between the tiles.

    The grainlines on the first image are a stroke of luck. If the tips of each arrow in a cluster were connected with straight lines, they’d show the hexagon and triangles. . .actually it would be a trifle too small but it would at least cut across the wavy outer edge. The wavy edge would cut into the hexagon and make negative space, it would overlap the hexagon and make positive space, and the negative and positive space would be equal. By this time I’ve also noticed that these triangles aren’t making a neat hexagon, so either the nodes can be hauled into place by making certain that the distance between the same node on two adjacent tiles is equal, or (since I am not working in a vector program) I start over altogether.

  5. Tiles slip-cast of clay will shrink in the kiln (as do hand-built or thrown pieces). The loss should be proportional, but account for it in your initial pattern if your final size is critical.

    When porcelain figurines became the rage in Europe, this was how they easily could spot the knock-offs made from molds pulled from originals – they were significantly smaller.

  6. Jennifer says:
    There are various techniques for generating these tessellations, but here’s the method to clean up the pattern you’ve already made. From your explorations it is clear that these fit together like hexagons which leads to the 30-60-90 triangle. If you adjust the pattern points slightly so that they coincide with the angles of the triangle precisely, the pattern will then tile precisely. Start by drawing a guide line from the upper left wing tip of the butterfly (marked A) to the lower right wing tip (marked C). Then create a point B such that ABC is a 30-60-90 triangle. Next draw the blue contour following the left edge of the butterfly. The exact shape doesn’t matter but it must start at A. Then draw the pink contour along the bottom of the butterfly. Again the exact shape isn’t important but it must pass from the blue end point (marked D) to the triangle point C. Finally, make a third contour from point D to triangle point B which is a natural extension of the blue contour. Now you need to make the rest of the butterfly pattern by copying these three lines. Take the blue and the green contours together and rotate them about point A counterclockwise 60 degrees and make a copy. This forms most of the upper edge of the pattern. Next take the pink and green contours together and rotate them about point C clockwise 120 degrees and make a copy. This completes the contour. The final picture shows the corrected pattern over top the original. The left and bottom edges are unchanged but the top and right edges have been corrected and it now will tile nicely.

    I don’t know anything about fabricating tiles but I do hope you can make it work, it would be gorgeous.

  7. Cast concrete is more durable than tile (assuming your floor has enough oomph not to collapse), and there are some lovely stains.

    You can incise lines, etc. into the top of whatever you use as your template for your mother mold. Plaster is traditional, and green soap for the release. I haven’t seen green soap for dog’s years, but pharmacies might still carry it. Any soap (not detergent) would work.

    More as you go, please!

  8. Dia in MA says:

    Oh, Carol! The fantasies you’ve set off in my mind. I’m imagining a concrete patio of tesselated tiles. And you can dye concrete to make it more interesting! Make them bigger and it would reduce the need for exactness. What fun!

  9. Sarah_H. says:

    If you decided to go with concrete, a set of the pattern could be created and the concrete poured in place into the pattern, Then the pattern is removed and rotated to pour the next set and so forth. (There are existing molds like this to do sidewalks at your house. One pattern is basketweave, another is irregular looking flagstones.) I see no reason you could not make your own mold.

  10. Brass shim stock (from auto supply stores) is traditional for what Sarah_H. is talking about. You overlap and interlock the ends similar to some (metal, plastic is no help) cookie cutters. I had typed “like a flat-felled seam” but that has more wraps.

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