The regular heptagon (seven sided polygon) is the smallest polygon which can't be constructed used Euclidean compass and straight-edge. However, it can be constructed by using a marked ruler, and the first such construction was given by the the great Renaissance French mathematician François Viète (1540 – 1603).
Before we show how it can be done, we need a little algebra.
Constructing a heptagon is equivalent to constructing the point in the complex plane. This value satisfies and this equation can be factorized as
Since , it must satisfy the second term, and since we can divide through by to obtain
A bit of fiddling shows that this last equation can be written as
we have . However, with , then
\Thus if we can solve the cubic equation , we will have and from that we can construct the heptagon.
We have shown how an angle can be trisected using neusis constructions, and we also know that . Multiplying by 2 means that this equation can be written as
This means we can solve any equation
by putting , and then .
We now have to coerce the equation into the form . This is actually standard algebra, started by putting . Then
Now write so that
and the equation to be solved is
What this means is that if we can construct an angle for which , then the value will be . Then and so finally
This means that .
So to start we need to construct
\We will need a triangle like this:
Scaling by produces
And this second triangle is easily constructed from an equilateral triangle with unit sides:
So the construction starts as follows: let be the diameter of a unit circle (centred at ), and place an equilateral triangle , with being on the circle. Let , and join . Using Archimedes' neusis construction, trisect the angle . This will involve a circle centred at with radius , and a line which goes through , crosses this new circle at and the diameter at , and for which . It looks like this so far:
Now let be the mid point of and draw the perpendicular to through which crosses the original circle at and . Then and are two sides of a regular heptagon.
To see why this works, note that angle and that . Thus and so . But this is the value of from above which satisfies . Then which is the cosine of . And so the angle we need is obtained by the perpendicular.
Viète's construction is slightly different in details to this; what is remarkable is how he managed it without the tools of modern algebra (even though one of Viète's great innovations was to use letters for variables). You can read about his construction here.
One problem with the construction given above is that it requires the use a straight-edge with marks a distance apart. This is not insurmountable: is a constructible value, and so such marks can be generated by Euclidean methods. But it is more elegant, in some ways, to use a straight-edge on which the marks are a distance 1 apart. And in fact the above construction can easily be modified.
Start with the circle, the equilateral triangle, and the line . But instead of trisecting the angle , we first create a radius parallel to , and use Archimedes' trisection on the original circle to trisect angle : a line through which crosses the circle and the line at points and and with . But since angles and are equal, we can construct with a line through which is parallel to the new line. Like this:
This is pretty much what Viète did, which I think is a remarkable tour-de-force of geometric reasoning, especially given the mathematical tools he had available to him.