Friday, November 14, 2008

Compass Constructions (for Geometry students)

This is for my Geometry students, but anyone can do this and have fun with it.

At this point of the semester in Geometry, you've probably learned a few basic compass constructions. Things like reconstructing a line, copying an angle or bisecting an angle.

One thing that you might not have covered in class, though, is the history of these simple constructions. Ancient Greek philosophers and mathematicians played with compass constructions a lot. Their compasses were a little different than ours. Our compasses can be set to stay at a certain width, but the ones the Greeks used would collapse as soon as you lifted them off the paper.

This created some difficulties in some constructions. There are three classic problems in Geometry, originating with the Greeks, that have still not been solved with their tools: drawing a square with the same area as a circle (known as "squaring the circle"), drawing a cube that doubles the area of an original cube, and trisecting an angle.

There are ways to trisect an angle using modern tools, though, and it can be fun to try. Here is one solution, using a "tomahawk."

But sometimes, it is nice to just have fun. The Math Me Thinks blog currently has a couple of complex contructions made with a compass. One of them resembles a Celtic knot pattern. I think I might build one and color it. It might be like making a personal Mandala.

Can you make one? Would you scan it in and share it with me?

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