The Problem

What will you find on this site?

Things you will find on no other trisection site: a journal-published construction fully endorsed by the international mathematical fraternity, and many other methods of trisecting angles exactly using only a compass and an unmarked ruler — a challenge that has exercised the minds of geometry lovers for two and a half thousand years. Although my simple methods work perfectly, have been accepted by prominent mathematicians, and were published in the March 2007 issue of the Mathematical Association of America’s College Mathematics Journal, Euclidean trisection with only a compass and ruler is — quite correctly — still considered impossible. Puzzled? All will be explained. Rest assured in the meantime that practical compass-and-ruler non-Euclidean trisection does not depend upon making marks on the side of the ruler in the manner made famous by the great Archimedes of Syracuse.

I do not, Archimedes, know why
You put marks on your ruler. Sir, I
Know that trisection tricks
Give a trisector kicks,
But why fudge when it's easy as pi?

The impossibility of trisection

As early as the fifth century B.C. Greek geometricians searched for a universal method for trisecting any arbitrary angle exactly. They had already found various ways to trisect many specific angles such as 45 degrees and 90 degrees. However, despite prodigious efforts spread over many centuries, they failed utterly in their quest to find a method that would work with any arbitrary angle.

For over two thousand years countless others faced similar frustration. Finally, in 1837 P.L. Wantzel explained their inevitable frustration by publishing the first completely rigorous proof that compass-and-ruler trisection within the classical constraints is impossible. This did little to discourage the world’s would-be trisectors: claims were still frequently made that the ancient challenge had finally been met. And they are still made today. The claimants, unfortunately, are deluded — every last one of them. Some are deluded because their methods simply don’t work. Others are deluded because they have broken the unwritten rules of what is essentially a mathematical game. These rules are really quite simple. In practical lay terms they can perhaps be summed up in three short sentences...

Many readers will know that things are a little more complicated than that. It is necessary to define just what compasses and straightedges are, and to establish the simple rules governing arc-drawing and line-drawing and the finding of construction points. But it is enough for our purposes here to say that the classical constraints demand extreme simplicity — and the absolute accuracy that comes with it.

Needless to say, I don't abide by all the constraints imposed by the demands of Euclidean geometry, for they do indeed render the challenge impossible. Nor did the great Archimedes abide by them. He cheated by marking his ruler. I cheat too — but without having to mark it. You may wonder if there is anything noteworthy in being able to trisect by violating the strict rules of this ancient challenge. The answer is simple: yes, there is. Even in non-Euclidean constructions, it has widely been considered impossible, unless one used marks, to trisect using only a compass and ruler, the standard tools that are available to would-be trisectors in the real world. As the classical constraints demand the use of a straightedge (a single edge of infinite length), and I have found this instrument rather difficult to come by, my methods conveniently employ the two real-world tools usually stipulated when it is said that trisection is impossible — and they employ them successfully. That is why they are noteworthy.

Striving to achieve the impossible

Although it has been known for nearly 200 years that compass-and-ruler trisection within the classical constraints is impossible, thousands have continued stubbornly to embark on the ancient quest of finding a way to do it. I am one of them — sort of. I have two excuses for such an apparent waste of time. Neither involves the traditional trisector's disdain for the pronouncements of mathematics, which is akin to the perpetual motion machine designer's disdain for the pronouncements of science. Although I am neither a mathematician nor a scientist, I have the utmost respect for both disciplines — so much so that I believe our threatened world would be a very much better place if the scientific approach to solving problems were universally applied by both institutions and individuals. Science works. It works because its entire focus is the pursuit of truth by observation and reason. It does not object to being challenged. In contrast to many other areas of human thought, it demands that it should be, for its respect for the truth is so steadfast that it welcomes the exposure of its own inadvertent falsehoods.

So my first excuse is that science and its related disciplines do make mistakes. It is quite sensible for fresh minds to revisit their pronouncements in case they have misconstrued or overlooked anything.

My second excuse is that the intellectual challenge of wrestling with the impossible can be a rewarding one — as long as it is not done out of ignorance and contempt for the realities of mathematics and science and logic. One invariably develops a deep understanding of the truths underlying the insoluble problem confronted. And occasionally — just very occasionally — one might glean an original and perhaps useful insight. Over the years I have gleaned a number of such insights in various fields, both technical and artistic. My first super-simple method of non-Euclidean trisection is one of them — as are the new ones that have followed.

Another method soon to be announced

Although you will find most of my methods on this site, the latest — which takes the challenge a surprising step closer in one important respect to meeting the impossible aims of the ancient quest — is still under wraps (August 2009). It will be added in due course, but probably only after it has also been published in an academic journal. This process is presently under way, for it has already gained the stamp of approval of prominent mathematicians, and is being discussed with the editor of a respected British mathematical journal.

Like its namesake once swung by the Sioux,
The old Tomahawk trisector’s through,
For it seems it’s been matched
By the ruler unscratched,
Which can paddle its own brave canoe.