Of course we all love triangles - both the shape and, if you remember elementary school - the musical instrument. The vast majority of the time the triangles you come across are the ones we’re generally familiar with: three straight sides, with acute, obtuse, or right angles. But more than just a bunch of boring old three-sided polygons, the world of triangles is wild, weird, and wonderful! And if you venture out in the geometry hinterlands, you may also encounter some of these:
We’ll begin with the seemingly oxymoronically named circular triangle. These are triangles that have arcs (curves) for sides, instead of straight lines. They can be constructed in a few different ways: among the more common are convex circular triangles, called Reuleaux triangles, and concave ones, called arbeloses. The arbelos was first discovered (or written about, anyway) by Archimedes; he named it after the Greek word for a shoemaker’s knife. The arbelos also has a closely related shape called the salinon, but that’s not quite a triangle, so we’ll skip that and move on to….
...the Schwarz triangle! The Schwarz is a spherical triangle tesselation tile. That means it’s essentially a 3D circular triangle that’s made from a space along the outside of a sphere, and if you have enough identical Schwarz triangles and fit them together (the process of tesselation), they’ll completely cover the entire outside of the sphere, with no gaps.
On the subject of unusual triangles that reference their surfaces, we also run into the hyperbolic triangle, which as you’d guess, is a triangle made up of three points connected by hyperbolic line segments that lie on a hyperbolic plane (that’s a surface where the space curves away from itself at every point). Or perhaps a hyperbolic triangle can also be one that tells a lot of wildly exaggerated stories? But that’s linguistics, not geometry.
There are also some “standard” Euclidean three-sided straight triangles with unique properties. Remember the circumscribed circle on your shirt? You can circumscribe other spaces inside a triangle, too - like a square; that is, find the largest square you can fit inside a given triangle. In most regular triangles, the square can be placed inside only one way, but in a Calabi triangle, that square will fit in three different ways:
Finally, we’ve got a triangle filled with other triangles: the Sierpiński! A Sierpiński triangle is a triangular fractal pattern; a triangle composed of triangles, that are composed of smaller triangles, that are composed of smaller triangles, etc., in a recursive pattern. A three-dimensional version also exists, the Sierpiński pyramid.
Hopefully we’ve sated your curiosity for fantastic triangles! But we encourage you to go out and look for more. As the poet Isidore Lucien Ducasse wrote: “Grandiose trinity! Luminous triangle! Whoever has not known you is a fool!”*
*original: “Trinité grandiose! Triangle lumineux! Celui qui ne vous a pas connues est un insensé!”