On Tricurves

If the straight edges of these colorful triangles were made curved, they would become tricurves, a shape which Tim Lexin described in a recent blog post. Image courtesy of Solkoll via Wikimedia Commons.

Tim Lexen, a mechanical engineer in Cumberland, Wisconsin, wrote a post about tricurves for the Aperiodical. As their name implies, tricurves are sort-of triangle cousins which have three sides, but instead of having three straight edges, each of their sides are curved.

“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,” Lexen wrote.

He invites us to imagine a plane which is tiled with repeating identical triangles. The picture starts to get interesting when he begins adding curves.

“But something unexpected happens when we add curves. For tiling we need equal amounts of concave and convex arc. The only way to do that is with two shorter concave sides joining a longer convex side,” he wrote. His post shows a variety of tricurve constructions, from the 30-60-90 tricurve to the 45-135-180 one.  He shows beautiful examples of periodic, non-periodic and radial tiling of tricurves.

Last year, Paul Bourke wrote a post inspired by Tim Lexen on his website. “There are a number of ways one can define a tricurve, the one used here is to start with an arc of some angle, replicate two identical curves ard [sic] rotate each about some angle about the ends of the arc. The Tricurve is the enclosed area,” Bourke wrote. Besides showing more examples of tricurves, he also links to a 2017 paper by Lexen about the shape, it’s tiling and variations, a method for finding its area, and more. (“This is intended as an informal paper. I am freely presenting the idea, for what it is worth; and I am soliciting feedback from any interested readers,” Lexen noted, adding that he can be reached via email at novustcl@charter.net.)

In a different post for the Math ∞ Blog, Lexen wrote about these and other curved shapes from the perspective of designing a flat puzzle that’s more interesting than one “with dozens or hundreds of identical pieces [which] may sound a little dull and predictable.” In fact, Cherry Arbor Designs now offers a wooden tricurve puzzle.

(While a few of the above mentioned posts provided a link to the National Curve Bank entry on tricurves, I was disappointed to find the link broken. This left me wondering if another curve bank has taken its place or if we are now living in a curve-bank-less world…Anyone know of similar current resources? If so, please reach out in the comments below, on Twitter @writesRCrowell or via email RachelJCrowell@gmail.com.)

John Golden used GeoGebra to create a related applet for tiling lenses. Check it out here.

Circling back to the Lexin’s original statement about preferring the compass over the straight edge, my question to you is “Are you Team Compass or Team Straight Edge?” Let me know in the comments below or on Twitter.

As for myself, I’m Team Straight Edge. This position dates back to my preschool years when I told my sister “I may not be a good draw-er, but at least I can draw a straight line.” (My drawing skills have since improved, at least marginally.)