If we had a winner for the challenge, that would be Jennifer, followed closely by Lynn who wrote me privately. I used the instructions Jennifer left in comments to perfectly tessellate the Escher pattern. At right is the proof (there is also a larger image).

Lynn’s approach was different and closer to what I would have done if I hadn’t have gotten the assist. I knew there would have been a faster way to do it mathematically but I didn’t know the secret to doing it. Lynn and I discussed that; she didn’t want to post a comment, leaving a link to her  sample (pdf) because she didn’t want to get chewed on for not doing it with math because she says she’s not spatially inclined. Here is how Lynn explained her approach:

I used the photo as my base layer and drew the butterfly outline on a layer above. I didn’t scan it. I get a better result if I just draw it. Then I copied, rotated, tweaked, rinse and repeat, until I got the original shape to fit in each position.

I copied the original drawing on layers to fill out the pattern. You only need one layer for your mold, but you can see how they nest together.

I didn’t use math, or concentric circles, I wouldn’t know how. Other than rotating the first drawing 60 deg to form a circle, it was all trial and error.

I don’t have an aptitude for engineering, or spatial relationships. (I have to hold a map upside down to travel south), so I can’t do the apparel pattern puzzles you present. But I can manage 2 dimensions and I like working in Illustrator. It was fun and I got some extra practice.

By the way, needing to hold a map upside down has nothing to do with spatial ability. Rather, it is common that people with dyspraxia need to do that. Among dyspraxics are many with autism of whatever flavor who are known to have autie super powers; literally incredible spatial abilities.  She insisted she couldn’t tell X from Y; I told her I can’t either but I have no complaints about my spatial skills and sent her a picture of my monitor. [If you wonder, the markings don’t help much but it hasn’t hurt me any.]

But back to Jennifer’s comment. It was difficult for me to follow it at first so I broke it up according to each step of the process. You’ll also need the pdf she created to follow along:

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. [I had to look that up.]

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 followed her instructions step by step and ended up with a perfectly tessellated pattern (full size with dimensions in case the pdf doesn’t scale right). I was so excited. The one I made is kind of ugly but I plan to remake it with prettier lines now that I understand how to do that. It’s great to be able to separate the artistic and engineering elements of the design to produce it most effectively.

At this juncture, I checked my email to make sure I hadn’t missed anything and found an email from Teijo who sent an Illustrator file.  His version is really cool -shown at right. While the file didn’t import well into my program, you get the idea. I haven’t checked it to see if it tessellates perfectly but knowing him, it probably does.

Speaking of, although the file didn’t import well for me it may work for you if you have the latest version. I think it is more than a tad annoying that CS5.5 isn’t backwards compatible to CS5. Really, one version is too much to expect? Oh my, trying to upload it I discover that WordPress won’t let me upload an ai file. That’s an easy fix. I upload it as a pdf and when you download, change the pdf file extension to ai and you’ll be in business.

Okay, gotta go! Sally is having a dinner party and me and Martha have to get cleaned up.