 You can use the compass and straightedge to construct a regular polygon with 3, 4, 5, or 6 sides. But what about a heptagon? For many centuries the best that could be done was an approximate construction. Several exist, but probably the simplest appears in his book of what the artisan requires of geometric construction by Abulwafa who lived in the 10th century. Abulwafa gave several geometric constructions suitable for artisans. His approximate heptagon construction may be the simplest that exists. Begin with an equilateral triangle inscribed in a circle, bisect one side, and the bisected side approximates one side of a regular heptagon inscribed in the circle. And this is a remarkably good approximation, and only a very close look will reveal that this is not a closed polygon. But if you've ever been wondering where the little symbol at the beginning of these videos appears, this is actually the origin of it. And while this is a very good approximation, it's still just an approximation. How about an exact construction? Archimedes showed that the construction of the regular heptagon relied on the construction of a figure with very specific properties. But his actual solution has been lost, so we don't know if he actually constructed the heptagon or identified the figure that would need to be constructed in order to construct the heptagon. And so we turn to Abulzal who lived in the latter part of the 10th century. Now Abulzal had an interesting career for a mathematician. He was a marketplace juggler with a talent for mathematics. And he was the first we know of to construct a regular heptagon. Abulzal's construction had two important parts. He showed that the required angle could be found by constructing a triangle whose angles had a ratio of 1 to 2 to 4. We'll leave the connection between this triangle and the regular heptagon as an exercise for the reader or viewer. The important thing is that he then showed how to construct the triangle. So let's take a look at that. Let A, G, B be a triangle where angles A, G and B have a ratio of 1 to 2 to 4. Now we'll extend this base B, G so that B, E is equal to AB and G, D is equal to AG. And to avoid the somewhat complicated geometric formulation we'll let B, E, B equal to X and G, D equal to Y. And for convenience we'll let the base of our triangle have length 1. Now it's worth noting a few useful things about our triangle. If angle G, AB is theta, then 7th theta is 180 degrees. Now let's join AE and AD to make a couple more triangles. And because we have a bunch of isosceles triangles, we can actually find all of these angles. And the important ones are angle DGA, which is 5th theta. And so angles G, D, A and G, A, D are both theta. So angle A, B, E is 3th theta, and so that gives us angles BE, A, B, A, E as 2th theta. And this gives us a couple of similar triangles. AG, B is similar to DA, B and AE, B is similar to EGA. Now by using proportionalities we find that X squared is equal to 1 plus Y and Y squared is equal to X times X plus 1. And Abusal recognized that the first curve defines a parabola and the second a hyperbola. So that means we can graph this parabola and the hyperbola, and the ordinate and abscissa of their intersection points gives X and Y, which allows us to construct the required triangle. Now the three vertices of the triangle are actually points on a circle. So if we draw that circle, this side BG is actually one side of a regular heptagon inscribed in the circle. And so what we have is what may be one of the most remarkable geometric constructions ever made.