Imatfaal's awesome post on Escher's tessellations on Polyhedra reminded me of some ornaments I made this summer. I made some of Escher's square tessellations onto cubes and then reprojected them onto spheres. I actually used a 60 sided Deltoidal hexecontahedron since that net is fairly easy to fold and looks pretty round.

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## 5 Comments

what do you mean, "projected"?

The same way that you talk about maps being projections since the only way to truly represent the surface of the earth is by actually putting it on a curved surface. To make maps we take that surface and project it onto a flat plane which will always lead to some inaccuracies. One of the simplest ways to perform this operation is to actually follow a path of light from the surface onto the new surface. That's what I did here. Imagine a cube with a sphere surrounding it. Now take a light source and put it at the center of the cube. The points on the cube will now be projected onto the sphere.

I'll have to do some Math Craft projects on globe projections...maybe we can do a quick one to make some polyhedral globe ornaments by projecting the points on a sphere onto a polyhedron.

Wow! Could you explain further how you actually projected the image - I presume a fancy graphics manipulation programme? Look forward to the project!

Yep. You're right though the math really wouldn't be that difficult and so it could be done pretty easy in environments like mathematica or mathlab. Here, I used a plugin for photoshop called flexify which has 250ish different projections you can use.

I wish I felt like I could use Escher's images in a blog post...but they are copyright protected and Escher copyrights seem to be enforced pretty strongly. The top fish image actually isn't an original Escher...so I probably could talk to that artist. But a good idea. I can probably make a post on reprojecting things that tile in squares using other images.

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