Hexagons From Squares

Today at my local library, we switched from fall décor to winter décor. Down came the leaves and the gourds, and up went the paper snowflakes. Snowflakes, of course, have six-way symmetry (D6, for you algebraists), which comes from the structure of the water molecule (derived from the not-quite-tetrahedral arrangement of the oxygen’s electron pairs.) Holding the science for a bit, this means that the paper snowflakes kids make in school aren’t quite right. Square paper is easy to get, and squares naturally lend themselves to symmetries and divisions of 2n, (n a natural number). The “snowflakes” they make are based of squares and therefore have 8-fold symmetry, not 6.

This might be a nitpicky observation, but I like accuracy in my cheap paper snowflakes, so inspired by the library décor, I decided to learn how to make a proper paper snowflake.

In preparation to teach a small workshop on origami this winter, I spent some time looking around on the internet for instructions and came across Origami-make.com. Hyo Ahn, the creator of the website, has a page about making other regular shapes from square paper, including a page on making a regular hexagon.

Fresh with the energy of not having anything else to do for the moment, I clicked through the instructions. As it turns out, it’s actually pretty simple to make a hexagon from a square. After trying Hyo Ahn’s method on a piece of paper (and then reliving kindergarten by making a snowflake out of it), I wondered whether that hexagon was actually regular, or just close.

This is of course the sort of question only mathematicians bother to ask. After all, who could sleep at night knowing their hexagonal snowflake has slightly imperfect symmetry?

What I do with my free time. Observant readers may realize that some of the math on this scribble is incorrect, which just goes to show that checking one’s initial work thoroughly is important. © 2013 Lauren Ellenberg

The short answer is yes: Hyo Ahn’s method makes a regular snowflake. The long answer requires some geometry.

© 2013 Lauren Ellenberg

The problem with getting a hexagon from a square is that squares like 90 and 45 degree angles. Hexagons are made from 120 degree (2π/3 radian) angles. That angle can be constructed from half or one fourth that measure, namely 60 and 30 degree angles (π/3 and π/6, respectively).

Luckily, there is a triangle tailor-made for this, and it’s the 1:2:square-root-of-3 triangle. With one irrational side, it may look a little scary, but it's actually one of the nicest triangles out there. A 1:2:sqrt(3) triangle has two neat sides and three neat angles. Most importantly, its angles are 30 and 60 degrees, which are the building blocks of a hexagon. So the challenge is to construct a 1:2:sqrt(3) triangle from a square.

Hyo Ahn’s method has one genius step that introduces a 1:2:sqrt(3) triangle to the square, which is then repeated to get equilateral triangles. On a piece of paper, the triangle looks right. But mathematically, how do we know it actually makes a 1:2:sqrt(3) triangle and not just a close approximation?

Folding to make the first 1:2:√(3) triangle by Hyo Ahn’s method. © 2013 Lauren Ellenberg

How does this actually make the correct triangle? Let’s label some line segments. Segment CD swings down to CA, so they have the same length. Let’s call that 2 units. B is the midpoint of CD, so the length of CB is 1 unit. Segment BA is at a right angle to CD.

© 2013 Lauren Ellenberg

Therefore: Triangle ABC is a right triangle with short side length 1, hypotenuse length 2, and long side length (you guessed it) sqrt(3). This leads to some other relationships. Because angle DCA is 60 degrees and bisected by segment CE, triangle ACD is equilateral. Therefore, triangles CDE and CAE are both 1:2:sqrt(3) triangles. Most importantly, triangle CFG is equilateral and is one sixth of the hexagon we want. With clever folding, we can mirror this triangle around to get our nice, mathematically regular hexagon.

Thus, Hyo Ahn's method makes a regular hexagon, subject only to the tolerances of the paper and how carefully you measure and fold. And now you can make your hexagonal snowflakes, secure in the knowledge that they are, in fact, regular, and will survive surprise inspection by groups of marauding mathematicians.