#Pentagon tessellation for free#
#Pentagon tessellation download#

Shapeways is a 3D printing service from which you can order prints and have them mailed to you. We’re also going to create an STL file for a multi-pentagon translation-tessellation unit like this for each type of pentagon: This 8-piece unit is the key to the tessellation pattern for the Type 11 pentagon. Here’s what that process looked like when we scaled the Type 11 pentagon model:Įight copies of the Type 11 pentagon can be put together to make a larger shape that tiles the plane only by translation, without needing any rotations. We made the area calculations in OpenSCAD by marking the vertices of each pentagon with different size spheres to tell them apart, and adding up areas of triangles to get the total area. Second, we manually adjusted the scale of each pentagon so that all 15 pentagons would have the same area and therefore look and feel like they are the same relative size, even though they have a range of different shapes. First, we chose representatives in each family to look as much as possible like the representative pentagons shown on the Wikipedia page on Pentagonal Tilings. We’ve made individual pentagon pieces before, but for this project we decided to remake new STL files for all of them in a more consistent way.

#Pentagon tessellation for free#

Finally, we’ll make a home-printable set of these same pentagons that we can give away for free at Thingiverse. 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.

#Pentagon tessellation download#

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: Irregular pentagon tessellations, or tilings should not be mistaken with semi regular variants like the Cairo Tessellation, named after the tiling in the streets of Cairo.Īnd so far only three types of identical irregular convex hexagons have been discovered that can tile a plane, and no convex heptagons, or octagons.Some years back we created the Pentomizer, a customizable model on Thingiverse that could generate all possible irregular pentagons that tile the plane. Pentagons having one side more than a square are quite difficult shapes to tile, and since 1918 only 15 different irregular shaped variants have been identified.Īnd only as recently as last week, Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell announced that they had discovered this 15th irregular variant. Its intuitive to understand how the more sides to a shape, the more difficult it is to tile without leaving gaps, or overlays. In this quick and nerdy post I would like to share and I suppose also celebrate an interesting shape that I learned about today, that is not only beautiful and dynamic, but can also be tiled without leaving gaps, or overlays.