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Design note #4: Koch snowflakes

The Koch snowflake is one of the first fractal curves to be described.

Like other fractal curves, it has an infinitely long boundary, and the self-similarity is obvious as you zoom in. One of the cool things about the Koch snowflake is that it can be built from six smaller snowflakes, leaving another snowflake in the middle. That of course can also be decomposed, recursively, giving you this:

So that led to one of our first puzzles, which uses two sizes of snowflakes. I put the box of pieces in the Mathematics Commons at the University of Michigan. Both of the patterns below were created there.

Putting it together, in holiday colors…

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Design note #3: Patches as wall plaques

Our son Max attends the United States Air Force Academy in Colorado Springs, CO. The Academy is divided into 40 squadrons of about 100 cadets each. Each squadron has its own patch, and many of the squadrons display a wall plaque version of the patch at the squadron’s CQ desk (Cadet in Charge of Quarters).

Max asked me if I could create a replica of his squadron’s patch in plywood, about 22″ across. I created the vector drawing in Affinity Designer, and cut it out of a combination of 1/2″ and 1/4″ birch ply to get a layered effect. Several of the cadets assembled it, and painted it:

They did a nice job! In cause you’re wondering, Squadron 2’s logo references the F-102 Delta Dagger, a 1960s era fighter-interceptor.

Next thing you know, a couple of his buddies from his water polo team wanted plaques for their CQ desks. Squadron 28 (featuring a stylized SR-71 Blackbird) opted for two tones, alternating stain with a pleasant natural tone.

Squadron 11 went for a subdued look, and selected a dark stain for the entire plaque. I haven’t stained birch before, but this Fine WoodWorking forum has some suggestions for getting the best results.

I must admit that I am partial to the bright colors of the original patch, so I’ve shown a Photoshopped version below.

There was interest in a Squadron 8 patch as well, based on the F-15 Eagle. I prototyped it in Affinity Designer based on a patch from the USAFA website:

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Design Note #2: Oriented Triangles

This is inspired by the Izzi puzzle, which is composed of squares. I learned about it from Professor Mark Saul of the The Center for Mathematical Talent at NYU, who developed beautiful mathematical content for i2camp.org. The Izzi puzzle consists of squares that have bisected edges that are combinations of black and white.

On my teaching blog, I explored the idea of using equilateral triangles. You need only 24 pieces to have one of each possible triangle, and they can be assembled into a hexagon. One challenge is to match all edges.

Below is a prototype we created in acrylic. The picture is a hexagon, but not a solution.

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Design Note #1: the Arbor Circle


An arbor is a small group of trees. I think of it as smaller than a forest, bigger than a grove. Ann Arbor (A2 to locals) loves its trees, and the co-founders of Ann Arbor named their new tree town after their wives, who shared a first name.

Like many of our designs, the Arbor Letter design has mathematical inspiration that has connections to nature. If you look carefully at the trees, you’ll note they are self-similar. Whenever a branch forms, the branch is a copy of the tree, reduced in scale. So the trees are fractals.

At first, we were taken with large (12”) diameter designs that we hung on the wall. And we have a pair of 20″ diameter with our surname initials for our front doors.  But reduced to 3.5″, with a loop added, it makes a beautiful, delicate ornament. We now make the design in a variety of sizes, and have written software for composing variations in the shape, positioning, and number of trees.