## You're reading: Posts Tagged: tiling

### Mathematical Objects: Aperiodic monotile with Chaim Goodman-Strauss

A conversation about mathematics inspired by the new aperiodic monotile. Presented by Katie Steckles and Peter Rowlett, with special guest Chaim Goodman-Strauss.

The paper announcing the discovery is An aperiodic monotile by David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss.

Chaim was recording from MoMath in New York, which will be running a creative artwork competition based on the monotile with UKMT. Chaim also mentioned a meeting in Oxford: Hatfest: celebrating the discovery of an Aperiodic Monotile.

Note: This podcast was recorded after the discovery of the ‘hat’ and ‘turtle’ monotiles but before the announcement of the ‘spectre’ monotile. Confused? Don’t worry, we explain in the episode!

### Not Mentioned on The Aperiodical, Summer 2020

Since we’re all busy people, sometimes news and other interesting bits of maths don’t get reported quite as they happen. Here’s a few stories that slipped through the cracks over the summer.

### Small Sets of Arc-Sided Tiles

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

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.

### Tiling a finite plane

One of the many jobs we’re gradually getting round to in our new flat is that of tiling a small section of the kitchen surface, which for some reason was left blank by the original builders and all intervening owners. And what better thing to tile it with than binary numbers?