The Universe of Discourse

Fri, 22 Dec 2017

Orthogonal polygons

A couple of years ago I wrote here about some interesting projects I had not finished. One of these was to enumerate and draw orthogonal polygons.

An orthogonal polygon is simply one whose angles are all right angles. All rectangles are orthogonal polygons, but there are many other types. For example, here are examples of orthogonal decagons:

orthogonal decagons

If you ignore the lengths of the edges, and pay attention only to the direction that the corners turn, the orthogonal polygons fall into types. The rectangle is the only type with four sides. There is also only one type with six sides; it is an L-shaped hexagon. There are four types with eight sides, and the illustration shows the eight types with ten sides.

Contributing to OEIS was a life goal of mine and I was thrilled when I was able to contribute the sequence of the number of types of orthogonal !!2n!!-gons.

Enumerating the types is not hard. For !!2n!!-gons, there is one type for each “bracelet” of !!n-2!! numbers whose sum is !!n+2!!.[1]

In the illustration above, !!n=5!! and each type is annotated with its !!5-2=3!! numbers whose sum is !!n+2=7!!. But the number of types increases rapidly with the number of sides, and it soons becomes infeasible to draw them by hand as I did above. I had wanted to write a computer program that would take a description of a type (the sequence) and render a drawing of one of the polygons of that type.

The tricky part is how to keep the edges from crossing, which is not allowed. I had ideas for how to do this, but it seemed troublesome, and also it seemed likely to produce ugly, lopsided examples, so I did not implement it. And eventually I forgot about the problem.

But Brent Yorgey did not forget, and he had a completely different idea. He wrote a program to convert a type description to a set of constraints on the !!x!! and !!y!! coordinates of the vertices, and fed the constraints to an SMT solver, which is a system for finding solutions to general sets of constraints. The outcome is as handsome as I could have hoped. Here is M. Yorgey's program's version of the hand-drawn diagram above:

M. Yorgey rendered beautiful pictures of all types of orthogonal polygons up to 12 sides. Check it out on his blog.

[1] “Bracelet” is combinatorist jargon for a sequence of things where the ends are joined together: you can start in the middle, run off one end and come back to the other end, and that is considered the same bracelet. So for example ABCD and BCDA and CDAB are all the same bracelet. And also, it doesn't matter which direction you go, so that DCBA, ADCB, and BADC are also the same bracelet again. Every bracelet made from from A, B, C, and D is equivalent to exactly one of ABCD, ACDB, or ADBC.

[ Addendum 20180202: M. Yorgey has more to say about it on his blog. ]

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