Construction of a Regular Heptagon

[sdi]

The construction of a regular heptagon was unknown to the Greeks, and was only finally managed by Renaissance geometers.

Given points A and B, draw

1. circle AB,
2. circle BA intersecting circle AB at points C and D,
3. circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
4. circle EB intersecting circle BA at point G,
5. circle FG intersecting circle AB at point H;

then BH is the side of a regular heptagon, and may be copied around the edge of circle AB to form the other sides.

...just kidding! The regular heptagon is impossible to draw with a straightedge and compasses (or indeed with compasses alone), and this fact was known at least as early as Kepler. In fact, the regular heptagon wasn't even unknown to the ancients: Archimedes managed to construct one with only slightly more sophisticated tools. The one I've constructed here is just an approximation, though a very good one, and is related to a construction by Albrect Dürer.