GIF by: TokyoJunkie at the English Wikipedia https://commons.wikimedia.org/w/index.php?curid=829215

This week I am working on scripting a podcast episode for pi day, and it’s lead me down the road of Geometric Constructions, and more particularly, “straight-edge and compass” or “classical” constructions. This is a method of drawing shapes and connecting points using only a straight-edge and a compass. (In this context, a straight-edge is considered a ruler of infinite length that has no markings on it, nor the ability to have markings drawn on it.)

Perhaps you think this is an esoteric and weird thing to be interested in. Well you’re not wrong, but geometric constructions are also an important step in a classical mathematics education. To help explain, here is an answer  by Matthew Hast to the question “Why should kids learn how to use a compass and straightedge” on the Math Educators stack exchange:

“The point of compass-and-straightedge constructions is for the students to get experience reasoning about axiomatic systems. The accuracy of the actual drawings is basically irrelevant (as long as it’s not so sloppy that it impedes visualization) — the point is that they can prove the accuracy of the idealized constructions.

This method of axiomatic construction was first studied for historical reasons that perhaps aren’t as relevant now — having to do with the “synthetic” style of axiomatic geometry used by the Ancient Greeks, as opposed to the “analytic” style more popular in modern mathematics (e.g., Cartesian coordinates). However, geometric constructions can still serve useful educational purposes now, because it’s a simple axiomatic system that can be a good way for students to learn to connect geometric intuition with careful, precise reasoning.

For example, once students have learned how to construct bisections of angles, one can ask them to try to trisect an angle, and use this to introduce the concept of an impossibility proof. This could also be used to illustrate how subtle changes to axioms can have a major impact on what’s provable/constructible (like how doubling the cube is impossible with compass and straightedge, but becomes possible as soon as you can mark lengths on the straightedge).

These are just a few specific ideas that came to mind — the general idea is to use the constructions as a tool to teach logical and geometric reasoning.

For a case study in how not to teach geometric constructions — which may highlight some pitfalls to avoid — see Schoenfeld 1988, ‘When Good Teaching Leads to Bad Results: The Disasters of “Well-Taught” Mathematics Courses’.”

At this point, it would seem constructions like this are more of a teaching tool and perverse hobby. However, a different type of construction has become an increasingly hot topic in the math world. Origami!

By Robert Salazar – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=47457085

Origami is the Japanese art of paper folding. Paper cranes are a classic example of origami, but a quick Google dive will show you how incredibly life-like and non-angular expertly crafted origami pieces can be. In the world of math, origami constructions are a lot like the compass and straight-edge constructions. Origami – or traditional, fold-only origami – is just one branch of a larger subject of paper folding constructions which can include making cuts, wetting the paper, etc. In math terms, these are just different constraints on the problem. Some make a given problem easier, some make a problem harder, some just make it different. For example, some difficult or unsolvable problems in constructions can be solved rather easily with paper folding (see “trisecting the angle“)

So why are mathematicians interested in origami all of a sudden? Unlike straight-edge and compass constructions, origami has some immediately interesting and useful applications. From wiki:

“Also in 1980, Koryo Miura and Masamori Sakamaki demonstrated a novel map-folding technique whereby the folds are made in a prescribed parallelogram pattern, which allows the map to be expandable without any right-angle folds in the conventional manner. Their pattern allows the fold lines to be interdependent, and hence the map can be unpacked in one motion by pulling on its opposite ends, and likewise folded by pushing the two ends together. No unduly complicated series of movements are required, and folded Miura-ori can be packed into a very compact shape.^[7] In 1985 Miura reported a method of packaging and deployment of large membranes in outer space,^[8] and as late as 2012 this technique had become standard operating procedure for orbital vehicles.^[9][10]“

Want to see what this looks like? Here is a GIF from a video by the engineers at Brigham Young University. The full video can be found HERE.

So… I’m in. I’m going to start playing with constructions in my spare time. Who’s with me?