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Escher-Style Tessellating Lizard /

by seanmichaelragan Dec 12, 2011
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Opened curve is not a problem, while I tried to use it for creating tile mosaic I stumbled on curve tiling imperfection.
Three lizards combine perfect, but when we copy that cluster of three — there is errors. See picture embedded.
Maybe I'm a crazy perfectionist, the biggest tile I wanted to make is 34mm... But we're talking Escher here! Let us all try to make it perfect!
I'm onto fixing that. It will be stupid to call it "remix" so, I wanted to update existing one.

I am not persuaded that you have actually discovered errors because I don't know anything about the methods you're using to rotate, translate and align these shapes for tiling. For all I know it's your vector software that's causing the problem. That said, it's possible that the vectors as drawn have irrational-number components that cannot be perfectly resolved at arbitrary scale without introducing small rounding errors. I'm getting out of my depth here, but if that is in fact the case it seems like there should be a mathematical rather than geometrical way to demonstrate it. And to correct it. I'll think more on this.

So, as I see it, there is only one critical polyline here, which has six-fold rotational symmetry about the center of the ideal hexagon. For this shape to be truly ideal, that is to be arbitrarily transformable without the introduction of rounding errors, we'd have to develop a coordinate system in which every component in that polyline is a rational number. We could then present an .SVG or other appropriate vector file containing the idealized coordinates of that polyline with instructions to create the tile perimeter mathematically by applying the appropriate symmetry operations upon it. This is not how the files are currently presented, which is my roundabout way of saying that I think you are correct: As currently presented, this shape will indeed accumulate rounding errors that increase in significance with the degree of scaling. I also feel that we could do better. Even if we can't develop an entirely rational-number presentation of the critical polyline, we could at least present the coordinates of that polyline to a much larger number of significant digits and encode the total tile perimeter as an ideal six-fold rotation / reflection, rather than as raw coordinates. I hope sometime in the not-too-distant future I can find time to do that. If not, hopefully these comments will be helpful to anyone else who wants to try.

Hi I printed five of these to prototype for a museum classroom project. They all seem to be just a little too large to fit one another. I haven't adjusted the size at all. Is there something else that I should consider? Thanks.

Had the same problem. Was thinking of altering in Blender, but have started looking at other similar models.

These have ideal geometry, meaning there is no allowance for slip. If you're cutting them from a continuous panel with a laser, this is no problem because the laser kerf provides the necessary clearance so they'll fit together cleanly. But if you're printing them, especially on low-precision consumer equipment, you'll have to either modify the file or tweak your print settings to provide a bit of play at the edges. And the results you get on one model of printer are unlikely to give a satisfying fit on another model printer, or even another unit of the same model. TL;DR: These are fine for laser cutting as-is, but you shouldn't expect to be able to print them and get a satisfying tiling action without a bit of trial-and-error.

That was my guess/excuse for requesting a laser cutter in the studio. Thanks for the quick and clear repsonse!

That is so freaking awesome:)

When I open the dwg file it
’s not a closed line. There’s a gap in the right side of the tail when the head is pointing up. I simply joined the lines by moving the end of one of them but when you rotate and join the finished piece they no longer fit perfectly. Can you correct this? I’m not sure how to do it myself.


I have vectorized a similar lizard pattern, but with 90 dgr period..

Sean, very cool to see these posted. The black and white set that Angus sent were printed on my TOM, glad you liked them. :)

I'm tempted to tile a kitchen like this.

I'm converting this into a printable tile. What is the layername of the outline of the tile?

Hi Link-

Sorry I didn't get to this earlier. I just checked the .AI file, and both outline and inner details are squashed into one layer called "Layer 3". You may also want to look at the .CDR and .STL files from Angus that I just uploaded. Apologies if we duplicated your efforts.

I've had the same thought. The pattern is pretty busy, so I think you'd want subdued colors; if you tiled a floor with colors like Angus's tiles no one would be able to stand to look at it. I think for flooring all the tiles should be one color, or a very narrow range of hues, randomly distributed. I would love to get these in Saltillo tile, for instance.

Maybe a backsplash for a bathroom sink.