Polygon Dissections

Polygon dissections have been a source of amusement and wonder in recreational mathematics for centuries. Today we'll look at a few related activity threads.


Setup

This activity is based on the Wallace–Bolyai–Gerwien theorem, which states that given two polygons with the same area, it is always possible to cut one into pieces and reassemble the pieces (via rotations and translations) into the other.

While the threads of this activity go down rabbit holes that it's hard to make an all-purpose document for, I've created a starter document that you can download. Aside from that, you'll want scissors and something to color with like crayons or markers. (If you go deeper, you'll likely want a ruler and a calculator!)


Challenges

  • The low floor for this activity is as a coloring and puzzle activity. The first five pages of the document feature a collection of squares that can be cut into pieces along the lines and reassembled into different regular polygons. Students can color these and then practice rebuilding the square and other polygon. The designs for the triangle, hexagon, octagon, and dodecagon are well-known in the dissection world. I liked Steve Phelps' version of the square-to-pentagon dissection better than the standard, so I have included that version. Many of the classic dissections can be found on MathWorld, but be warned: the ones that I tested appeared to be merely rough approximations of the actual dissection, so printing out MathWorld's images can result in pieces that can't actually create the other shape. I have gone through and reconstructed the triangle, hexagon, octagon, and dodecagon squares for the document so that they work!

  • The intermediate level for this activity is as a cutting and reassembling activity. The next four pages of the document feature a collection of polygons that all have the same area. The intention is that the page of squares is to be used as a playmat and the students' task is to figure out how to cut each of the non-square shapes so that it can be turned into a square. Many of the early shapes are nice in the sense that they share some dimensions with the square, making it doable in one or two cuts. Others are a little more dastardly.

    While it was not the main focus, students might also like to think about how they can turn non-squares into other non-squares. Students might also prefer to take a square and fashion it into one of the other shapes intead of dissecting the other way.

  • The high ceiling activity is finding ways to turn one shape into another that uses a small number of pieces. For example, four pieces is the best one can hope for when turning an equilateral triangle into a square, while two pieces suffice for turning a right isosceles triangle into a square. In some well-trodden cases, the lease number is known, but there are still many open questions!


Notes

  • The proof of the Wallace–Bolyai–Gerwien theorem gives a constructive method for turning one polygon into another. Assuming both polygons have area \(1\), the procedure is as follows: triangulate the polygons, turn the triangles into rectangles, turn each rectangle into a rectangle with edge length 1, creating two \(1 \times 1\) squares. The cuts in each square can be replicated in the other and then the polygon-to-square assembly reversed. This page gives a nice overview of the procedure with pictures.

    While this procedure is typically inefficient, it gives a way to get one's foot in the door. For many of our early examples, the process of lopping off triangles or rectangles can lead to an efficient dissection.

  • Many constructions are found by superimposing one tessellation upon another. Gavin Theobald, someone who has come up with many such dissections, has a fascinating page detailing some techniques and the dissections they have been used to uncover. I have included a strip of squares and triangles in the document that can be used to find the square-to-triangle dissection from the first page.


Extra Challenges

Recreating the classic dissections so they were precise and scaleable was actually quite involved! Students with some advanced geometry and trigonometry chops might like to tackle coming up with the schematic for one of the decompositions.


This post was sponsored by the Julia Robinson Mathematics Festival

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