The Aperiodical 

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Small Sets of Arc-Sided Tiles

By Tim Lexen. Posted October 26, 2019

Tim has previously written guest posts here about tiling by tricurves, and is now looking at ways of tiling with other shapes.



In an earlier post elsewhere I covered some basic arc-sided shapes that tile by themselves. Lately I’ve been playing with groups of curved tiling shapes, asking a question common for me: how to get the most play value as an open-ended puzzle? This means getting the most interesting possibilities from the simplest set. “Interesting” includes variety, complexity, challenge and aesthetic appeal. “Simplest” covers not only size of set and the shapes, but also the least total information needed to describe or construct the shapes.

Phantom Tiling

By Tim Lexen. Posted October 31, 2018

Following on from his previous posts: Bending the Law of Sines, which introduced the idea of tricurves, and a further post on Combining Tricurves, Tim Lexen continues this series of guest posts by looking at some of the structures underlying tricurve tilings.

When we look at simple planar shapes for tiling, usually each shape’s properties and tiling structure are obvious. The framework for the tiling is usually defined by the shape in a straightforward manner. But here we’ll look at the uniquely useful arrangement of the tricurve’s arc centers—which is not obvious—and use this structure to add a dimension to the tiling.

Bending the Law of Sines

By Tim Lexen. Posted July 31, 2018

 



For me, the above shape emerged when playing with a drawing compass.  Of the two ancient tools, I preferred the compass over the straightedge.  I was fascinated with the classical geometric constructions, the intersecting circles and arcs. As a simple personality test, preferring a compass over a straightedge might mean something: maybe roundabout-holistic-intuitive more than straightforward-linear-realistic. At any rate, the pursuit of curves eventually led me to this topic, but to explain I need to start with straight lines and triangles.