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Phantom Tiling

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.

Combining Tricurves

In July, guest author Tim Lexen wrote about his discovery of the tricurve, a shape made of arcs that has some interesting properties. He’s written a follow-up in which he explores them further. For a discussion of tiling with curve-sided shapes in general, see Tim’s MathBlog post.

Tricurves can be combined when the large, convex arc of one fills a concave space of another. A tricurve can be thought of as a shape that fills a concave arc with two smaller arcs of the same total length. In each case the new arcs stay within the boundaries of the original structure: touching the same bounding arc. This could go on repeatedly (see below) but we’ll focus here on joining two tricurves. Like the tricurves, assuming agreeable angles, the combined shape will often be able to tile the plane periodically, non-periodically, and radially with itself and related shapes.

Bending the Law of Sines


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.