Pentagon construction

Question

What is the simplest way to construct a regular pentagon using Euclid's Elements? Using the compass and straight edge is easy to get one side but how should the second side begin?

Answer 1

I'm usually not in favor a straight-up hyperlink answer, but the Math Open Reference website has a host of information about constructing geometrical figures with clear, step-by-step illustrations.

See, in particular, the "Constructing a pentagon inscribed in a circle" entry.

Answer 2

Here is a construction that requires only a single compass radius (rusty compasses).

1. First construct a pair of perpendicular lines, an operation easily done using only one compass radius (which hereafter is called one unit of length). Label their point of intersection A.

2. Draw Circle A (all circles are labeled by their centers, which admits uniqueness since the radius is fixed). Label one intersection with either line B and one intersection with the other line C.

3. Draw Circle B, which intersects line AB at points A and D.

4. Draw Circle C and line segment CD, which intersect at point E within this line segment (only one line-circle intersection lies within the line segment). DE measures $1/\phi=(\sqrt5-1)/2$.

5. Draw Circles D and E which intersect at F and G. For subsequent definiteness, F is considered to lie on the same side of line CD as A, with G on the opposite side. D, E, and G are now three vertices of the final pentagon.

At this point things get tricky. We know that the remaining vertices of the pentagon are one unit apart, but each of the needed circles passing through one of these vertices is centered on the other and both prospective centers are unknown. We basically have to solve for them simultaneously.

6. Draw line segment FG. This bisects line segment DE at at H.

7. Draw Circle H, extending line segment CD if necessary to identify the points of intersection I (closer to C of the original line segment) and J (closer to D).

8. Draw Circle I, which intersects Circle H at points K and L.

9. Draw line segment KL which intersects Circle D at a point M on the same side of CD as G. M is one-half unit from the mirror line FG and one unit from the opposing vertex D; these distance conditions match up with the requirements for a vertex of the pentagon whose diagonals will be one unit (the lengths of DG and EG).

10. Draw Circle J, which intersects Circle H at points N and O.

11. Draw line segment NO which intersects Circle E at a point P on the same side of CD as G. M and P are the vertices for which we solved simultaneously.

12. Draw pentagon DEMGP which is regular, with side length $(\sqrt5-1)/2$ and diagonal $1$.