Penrose zeolite

I modified this nice script due to roly hudson (see here) for generating Penrose tilings to include a decoration of the tiling by triangles. The result is a non-periodic pattern on the plane by a system of corner-sharing triangles (i.e. there are two triangles at each vertex). This is sometimes called a 2-D combinatorial zeolite, see for example this talk. Sliders control the shapes of the triangles.

penroseZeolites.gh

zeolite_start_here.3dm

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Comment by MillieJordon on June 5, 2022 at 9:32pm: Penrose tiling is a mathematical concept that can be used to tile a plane with pentagons in such a way that the pattern looks like it is made of smaller copies of itself. Penrose tiles are based on the Penrose kite and dart construction, which consists of two equilateral triangles, one of which has been folded along its diagonal. The resulting shape can then be unfolded to produce an infinite set of tiles. One of my friends asked me to have a look at https://dentmaker.net/ site from where I could buy an essay about my topic from college.

Comment by mark zirinsky on May 29, 2015 at 6:37am: thanks Elisa, I have been a lifelong mineralogist but haven't looked at zeolite crystal structures (on the micro level) in a long time. this is a fascinating area for geometrical study, and once I have worn out arabic tiling patterns i will have a deeper look at the theory. At the moment I am experimenting with surface engravings over complex forms, and perhaps will try engraving the meshes, it hadn't occurred to me to do that until i saw your colors superimposed on the patterns. thank you

Comment by Elissa Ross on May 28, 2015 at 12:45pm: Yes Mark, exactly. The actual zeolite minerals have an interesting repetitive structure, typically a system of corner-sharing tetrahedra (the tetrahedra are composed of a single silicon atom bonded with four oxygen atoms). Mathematicians (and others) are interested in modelling these as so-called "mathematical zeolites", which are basically corner-sharing n-simplices... In other words, two triangles at each vertex in 2D, two tetrahedra at each vertex in 3D, and so on. There is a bit of an industry in developing hypothetical zeolites (both finite and infinite) based on these combinatorial restrictions.

I'll post the GH definition. I'm kind of a noob so it isn't a thing of beauty.

Comment by mark zirinsky on May 28, 2015 at 9:25am: very interesting patterns. Named after the class of minerals called Zeolites, as often found in water softeners and ion exchange columns? I would be interested to see your script modifications to produce these.

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