Squares and Circles (From the Pillowbook I)

In a previous post, I have discussed triangles with curved edges and what they can tile. One can do the same with squares, only that things get more interesting, because there are six different shapes:

Curvies

I have called them pillows, mainly because I want them as nice, big, colorful pillows. Hmm. The first problem I’d like to discuss is to tile curvy rectangles with them, like this curvy 3×3 square:

Curvysquares 01

It is pretty clear that all curvy rectangles have odd dimensions. The left example uses all six pillow types, the right only two, blue and yellow. To see what combinations of colors are possible, the following observation is useful: Each pillow has a number of edges that are convex (curve outwards) and other that are concave (curve inwards). For instance, orange and purple both have two convex and two concave edges. Yellow has just four convex edges. With that, we have a little

Theorem: In any curvy rectangle, there are four more convex then concave edges in all pillows together.

A picture should make this clear:

Edgecount

This helps to predict how many pillows of each color we need.
For instance, suppose we want to tile a curvy 3×3 square with y yellow, r red, and b blue pillows. We then need y+r+b=9, and, by the theorem, 4y+2r-4b =4. It’s easy to see that this forces y=2, r=4, b=3. Similarly, if we are only allowed to use yellow, purple, and green, the only possibilities are y=2, p=5, g=2 or y=3, p=2, g=4. Here they are:

Curvysquare2 01

That we found a solution in positive integers does not mean that there is a tiling that realizes this solution. For instance, suppose we want to use red, orange, and purple, then we need to have r=2, but for o and p  we can have any pair of positive integers that sum up to 7. However, only o=2, p=5 and o=3, p=4 can be realized. The solutions are not unique, here are two symmetric ones:

Curvysquare3 01

There are about a dozen little exercises like these. To be able to say something interesting about larger curvy rectangles, we will need to study ragged rectangles in a few weeks.

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