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Making Tricurves

Tim Lexen has written a series of posts on the topic of Tricurves: Bending the Law of Sines, Combining Tricurves and Phantom Tiling. In this latest post, Tim has been working with our own Katie Steckles to turn Tricurves into real objects to play with.

When you discover an interesting mathematical shape or object, there’s a strong instinct to play with it – maybe by drawing sketches and doodles to test the limits of the idea. But in the case of Tricurves, drawing an accurate shape takes a little time, and it doesn’t lend itself well to idle experimentation.

Producing a physical version of a shape, in enough quantity to allow for experimentation, makes it much more tangible. In our own respective locations, we’ve each made use of laser cutting facilities to produce wooden Tricurve tiles to play with, and we encourage you to join in.

Euler in maths and engineering

Inspired by Katherine Johnson’s character in the film Hidden Figures and her use of Euler’s Method, engineer Natalie-Claire Luwisha has written this guest post about Euler’s contribution to engineering.

I thoroughly enjoyed Hidden Figures because of the overall message and inspiration it generated for all women, especially women of colour. Even today in the 21st century, most of the STEM (science, technology, engineering and mathematics) industries still have a very low percentage of women and even fewer women of colour. One major factor in this is the lack of visible role models for young girls and women to aspire to, so this story based on real-life events was ideal to help tackle the issue.

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.

The Big Internet Math-Off: Cheering from the sidelines

Photos of all the Big Math-Off competitors

Benjamin Leis has been giving some top commentary on the Math-Off matches as they happen, on Twitter. We asked him to share some of his thoughts in more detail

We’re almost at the final of the Big Math-Off and I, your humble and slightly quirky commentator, thought I’d take a look back at the highlights so far. First of all, the format itself is genius. Why go to the beach when you could instead be rooting on for your favorite theorem or mathematical phenomenon? But if you somehow have missed any of the rounds so far, the posts live on and you should take the time to read them all.

Rather than exhaustively survey all the entries so far I thought I would highlight a few that struck a chord with me and what about them was interesting. So first up, since I’m involved in running a middle school math club, I’m always looking for ideas that will transfer to that setting. That generally means: ones only requiring inexpensive easy to procure materials, an idea that requires no more than Algebra and Geometry and which also has a good hook and most importantly, but which rules out a few otherwise excellent entries, it needs to be something that I haven’t done already.