We’ll also make some larger multi-pentagon units for each type of tessellation. What we’ll be creating today is fifteen bulk sets of tiles suitable for printing at Shapeways, one for each of these fifteen types. Here’s what they look like (image credit to Ed Pegg): In 2015, mathematicians discovered the fifteenth type of tessellating pentagon, and not long afterwards it was shown that it was the last - in other words, there are exactly fifteen families of tessellating irregular convex pentagons. For example, here is a particularly beautiful pentagonal tiling from Deke’s Techniques: A regular pentagon (with all sides and angles the same) can’t be used as bathroom tiles, but some irregular pentagons (with possibly different sides and angles) can. ![]() If we say that a 2D shape “tiles”, or “tessellates”, the plane, we basically mean that you could use that shape to make bathroom or kitchen tiles that fit together exactly on the wall. If you want to see what it takes for me to prep models for posting on Shapeways and Thingiverse, scroll down. I made some pentagons! You can buy pre-printed sets at Shapeways or download free files from Thingiverse and 3D print or lasercut them yourselves. ![]() Today, we finally get it done in this post we’ll walk through the surprisingly lengthly process of turning a model into a released product/design on Shapeways and Thingiverse, using these pentagon sets as an example. Second, because a lot of work goes into creating, packaging, creating media for, and setting up 15 different products on Shapeways. First, because they are expensive to order as prints. We’ve been avoiding doing so for a long time, and you’re about to see why. One thing we hadn’t done until today is create bulk sets of all fifteen types of tessellating pentagons that people could order from Shapeways. At the time, we went a little crazy with the pentagons, using them to make thermoform bracelets, 3D wall paper, desk organizers, fancy silver bracelets, stainless steel desk toys, and wall art: In the above video you can see how easy it is to create a tessellation with these types of regular polygons.Some years back we created the Pentomizer, a customizable model on Thingiverse that could generate all possible irregular pentagons that tile the plane. Did you know that you cannot create a tessellation with regular pentagons? Or with regular octagons? As a matter of fact there are only three types of regular polygons that can be used to make regular tessellations. The equilateral triangle, the square, the regular pentagon are all examples of regular polygons. Regular polygons have all sides and all angles congruent. This type of tessellation can not be achieved with any type of polygons. If you want to cover the plane with regular congruent polygons, you are trying to create a regular tessellation. Let's talk a little about the math in tessellations. ![]() The tiles in the kitchen and the puzzle you have solved are nothing but tessellations. I'm sure that you have already seen many tessellations in real life. A tessellation is a collection of figures that can be put together to fill a plane surface without overlaps or gaps. Not all polygons are tessellation shapes.
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