The impetus was twitter’s #makevember and my repair of a VHF speaker grill. I started this work a few years ago and have finally got round to putting the code in Github in a form I’m happy with. Lacking the mathematical skills for the purely analytic approach of Ed Pegg, my approach is largely computational, using a variety of techniques including one- and two-variable optimisation. I’ve also developed (and it continues to grow) a library of functions and modules which operate on descriptions of tiles by their perimeter or as a polygon, and on lists and nested lists of tiles. I also chose to use OpenSCAD, not because of the CSG operations it supports (although I use these in generating the final forms) but because I find it a good (but not perfect) functional language with a responsive IDE for working with geometry. So I set about doing this from first principles, creating code which is based directly on the constaints as detailed in the Wikipedia article. However this means that the OpenSCAD code contains large amounts of mathematical ‘assembler code’ which makes no sense in itself and is removed from the constraints on the tile perimeter which define the shape. Laura’s code was based on very impressive work using Mathematica by Ed Pegg. This code is a great piece of work and generates a wide variety of forms using all 15 types. I first came across this problem when Laura Taalman posted her OpenSCAD code on Thingiverse. Research is ongoing and so far 15 distinct types of pentagon and their tiling unit have been discovered. Indeed it is, as the following example shows.Regular pentagons do not tesselate so interest has fallen on tiling with irregular pentagons. Is it possible to extend a 5.5.10 vertex figure to a full tiling using the simpler 5-rhomb and 10-rhomb shapes from Dürer? But it also includes other complex shapes such as five-pointed stars and combined decagons. Kepler's Aa tiling contains vertices with type 5.5.10 and indeed an entire 5.5.10 rose with a decagon surrounded by a ring of pentagons. It does not appear in Harmonices Mundi, where Kepler described the shape much more prosaically as a "combined decagon". Kaplan says that Kepler called his fused double decagon shapes "monsters" but does not say where the term comes from. The tiling was more complex than any of Dürer's and Kepler himself noted that the "structure is very elaborate and intricate".Ĭraig Kaplan points out in The trouble with five that several mathematicians have proved that Kepler was correct and that his Aa diagram can be continued into a tiling of the plane. His solution, which he illustrated as diagram Aa in Harmonices Mundi, involved five pointed stars and a peculiar fused double decagon shape. Like Dürer, Kepler could not resist looking beyond regular polygons to see if he could find plane tilings involving pentagons. In Harmonices Mundi he also discussed the fact that pentagons can appear in tilings of the sphere, where they form the basis of the dodecahedron. Kepler knew that pentagons could not occur in edge-to-edge plane tilings of regular polygons. Kepler also experimented with pentagon tilings. You can view Dürer's third and fourth tilings on Wikisource. It is possible that these empty spaces could be filled with rhombs but without more information it is unclear what Dürer had in mind. In his interesting article on pentagon tilings, The trouble with five, Craig Kaplan points out a variant of this third tiling with a central rose (a decagon surrounded by pentagons) that is continued by a spiral pattern of pentagons and 10-rhombs.ĭürer does not show how to complete his fourth tiling, which also has a central pentagon, but merely remarks that the "leftover areas you can then fill with whatever you like". You can view the thick tiling at the bottom right of this Wikisource page.ĭürer's third tiling contains a unique central pentagon and so cannot be periodic. The 5-rhomb and 10-rhomb play an important role in the modern theory of Penrose tilings, where they are often informally called the thick and thin rhombs. We could call this the "thick" tiling because of the thicker appearance of the 5-rhomb. The second tiling also contains pentagon and 10-rhomb prototiles but changes the tiling pattern to add gaps that can be filled by 5-rhombs. The first and simplest combines pentagon and 10-rhomb prototiles and could perhaps be called the "thin" tiling. The "thin" tiling with pentagon and 10-rhomb prototiles is the simplest of Dürer's pentagon tilings.ĭürer illustrates four pentagon tilings in the Painter's Manual.
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