Math CentralQuandaries & Queries


Question from Nazrul, a teacher:
How can I trisect an angle?


For practical purposes, use a protractor!

For abstract mathematical purposes, there are many ways of trisecting an angle using aids such as special curves (eg, the "trisectrix of Maclaurin") or drawing implements. One particularly cute example is the "tomahawk":

The famously impossible problem is that of trisecting an angle using only a compass and [unmarked] straightedge. The reason that this is known to be impossible is that all constructions with these tools can be expressed in terms of rational operations and square roots. Thus, if you start inside the so-called "field of Hippasian numbers", those that can be obtained from the rationals by an arbitrary finite sequence of applications of +,-,*,/,√ , you cannot get out of it by "Euclidean constructions."

But the coordinates describing a 60 degree angle are in this field, and those of a 20 degree angle are not. Case closed.

If you want to understand this better [especially the proof of my statement above about 20 degree angles], I recommend Benjamin Bold's excellent and inexpensive little book Famous Problems of Geometry and How to Solve Them (Dover). If you want to go more deeply into the history of trisection attempts, Underwood Dudley's "Budget of Trisections" is the classic.

Good Hunting!


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