Old Methods

The marks of Archimedes

Many methods of trisection-by-cheating have been developed, but not using only a compass and unmarked ruler as I do. Some of them are simple and elegant. Others are complicated and cumbersome. Some of them use only the permitted geometrical tools in an illicit manner. Others use additional tools in an ingenious manner. I shall give you just one example of each category. You will find many others on the internet.

I am particularly fond of Archimedes’ method, partly because it is one of the many  I discovered independently before I chose to survey the work of others.

Angle ABC is the angle to trisect. Extend AB to the right as shown. Draw a semicircle with centre B and radius AB. With the compass still open to that radius, hold its legs against the ruler. These two points will become D and E. Adjust both the ruler and compass until the edge of the ruler passes through C while D lies on the semicircle and E lies on the extended baseline. Angle AEC is exactly one third of angle ABC.

Archimedes succeeds to the extent that he uses only a compass and ruler, but he either puts marks on the ruler or uses the compass as a measuring instrument — and consequently  fails to honour this explicit part of the challenge.

The Tomahawk trisector

The example I have chosen for demonstrating trisection by the use of specialised tools is also one that I developed independently only to discover that it was an old idea. The edge DF on the head of the blue Tomahawk is divided into three equal segments. Only segment DE is shown (an extension of the lower edge of the Tomahawk’s handle would pass through the missing point). The arc has its centre at E and passes through D.

Angle ABC is the angle to trisect. The Tomahawk is positioned so that its corner F lies on BC while the lower edge of its handle lies on B and its arc is tangential to AB. Angle ABE is exactly one third of angle ABC.

The Tomahawk method is ingenious, but it breaks the most basic rule of the challenge in that it introduces a specialised tool. Although it is interesting, it does nothing to advance the cause of angle trisection using only a compass and ruler. My own methods do advance this cause.