APPENDIX III

The Construction of the Perfect Heptagon

 

Modern mathematics has produced proof of the impossibility of geometrically constructing a perfect heptagon.  Nevertheless, there are two constructions that produce a heptagon so close to perfection that in practice it is often difficult to discover any disparity, for the accuracy depends upon the thickness of a line.

The Ad Quadratum Construction

1.    Within the circle to be divided into seven equal arcs, an octogram is inscribed consisting of a right square UVQT and a turned square MONP.

2.    Within the turned square a right square YZCA is inscribed.

3.    The vertical diameter MN is drawn cutting AC at B.

4.    AC is extended to meet the circumference at E.  (The accuracy of this operation can be checked by drawing the arc BD down to E.)

5.    From centre E and radius EB an arc is drawn to cut the circumference at F and K.

6.    EF and EK are then rotated around the circumference to create the polygon EFGHLJK.

The Ad Triangulum Construction

1.    Within the circle to be divided into seven equal arcs a hexagram AECFBD is inscribed consisting of two triangles ABC and DEF.

2.    The vertical diameter AF is drawn, bisecting BC at G.

3.    From centre C and radius CG an arc is drawn to cut the circumference at H and J.

4.    CH and CJ are rotated round the circumference to create the polygon CHKLMNJ.

This construction is much the simplest, but both produce an identical result, for GC and BE are identical.  The AD Triangulum, in simplified form, was the construction used by medieval masons if it was ever required.