Confession of a Weekend Trisector:
I was a crank for 48 hours
Thomas Whalen, Ph.D.
18 June 1999
Last Thursday, my eleven-year old daughter brought her homework to me for
help; she had to measure angles and had lost her protractor. Again. Eyes expecting
yet another in a long series of miracles, she waited patiently for me to draw a
protractor on a piece of paper. Everyone knows that it is easy to draw a right
angle. And most know that it is easy to bisect that angle. But the ancients in
their wisdom decreed that a right angle should be divided into 90 parts, not
64. You do not divide a ninety-degree angle into nine parts unless you can
trisect it. And everyone knows that you cannot trisect an angle with a
straightedge and compass. Mathematicians spent twenty centuries trying in vain
to trisect an angle with straightedge and compass until, in 1837, it was proved
that it could not be done. That proof has withstood attack for more than a
century and a half.
I executed a bit of careful origami on a scrap of paper, roughed out a serviceable protractor and sent her on her way.
Some men, if left alone with a full bottle on the table before them, will descend into alcoholism. Other men, if left alone with an angle drawn on the top of a stack of otherwise blank paper will fall into the depths of a different life-destroying addiction: trisectionism.
As I had folded that scrap of paper, I had been struck by the thought that there might be a loophole in the mathematical proof of the impossibility of angular trisection. Consider an easier case: trisecting a line. One obvious way to trisect a line is to construct a tripartite line parallel to the original and then project its partition onto the original line, as shown in Figure 1.
The construction proceeds as follows:
To trisect the line AB, erect a perpendicular on one end, BC. Then erect another line, at some point D, perpendicular to BC This line will be parallel to AB. Mark three short segments, DF, FG, and GH, on line DG. The line from A through G will intersect the line BC at some point H. Lines drawn from point H through the points F and G will trisect AB. This produces a flock of similar triangles and the proof is so obvious that any high school geometry student should be able to write it out without even breaking a sweat.
Figure 1: Trisecting a line by projection.
So why not play the same game with an angle? If you wish to trisect an
angle, just construct a new tripartite angle, then project that partition onto
the original angle? Maybe there is a loophole in the proof of the impossibility
of angular trisection because, with this construction, you would not have to
create the desired trisection solely from the original angle; you draw upon a
new, accurate trisection that you have created.
I consulted the Web. According to some Web sites, the reason that mathematicians consider the angle to be un-trisectable by Euclidean construction is that trisection requires the use of irrational numbers and there is no way to construct a line of a known irrational length using a compass and straightedge. I know barely enough about abstract algebra to imagine the general form of this proof. Points on a plane represent pairs of real numbers. A compass and straightedge let you add and subtract distances, allowing you to generate all integers; and to multiply and divide distances, as in Figure 1, allowing you to generate rational numbers; but do not give you enough operations to generate irrational numbers. Calculating an irrational number, such as pi, requires an infinite series of calculations. You do not have to think very hard to imagine a simple construction involving an infinite number of steps that would converge on the trisection of an angle; but that violates the rules. It is implied that an Euclidean construction only involves a finite number of operations.
But, not having read or tried to understand the proof, I could imagine a potential loophole if I try to trisect an arbitrary angle by reference to another angle that has already been trisected. If you create a line of random length, its length may be an irrational number. Or, if you draw two lines at a random angle, the size of the angle may be an irrational number. If the proof did not include the unary operations of creating a line of random length or an angle of random size, it might not be complete and might not apply to the construction that I intended.
By constructing a tripartite angle from an angle of random size, I may be basing my construction on an operation that has never been considered before. Maybe.
I faced a choice. I could either spend months learning a lot more about abstract algebra and then scour the mathematicians' proof looking for the shortcoming that I hoped to find; or I could start drawing Euclidean constructions. Acquiring a detailed understanding of abstract algebra would be a good thing; I always benefit from any improvement in my mathematical skills. But drawing diagrams would take a lot less time and be a lot more fun.
Rather than running to my basement to dig out my abstract algebra text, I began doodling.
After fiddling for an hour or so, I took the kids to the nearest big-box bookstore. While they looked at their books, I perused the math section, looking up "trisection" in the index of a variety of popular geometry books. Eureka! After only a few minutes, I found a reference to a mathematician named Frank Morely who almost solved the problem a few years ago. He showed that if all of the angles in a triangle are trisected (presumably with a protractor), then the intersection of those angles creates a little equilateral triangle, as shown in Figure 2.
Figure 2: Morley's Triangle
Morley's triangle suggested to me that a line that bisects the angle at the
apex will pass through the center of the small, embedded triangle and provide a
simple mapping between the two trisections at the other corners.
Maybe Morely had failed to see the obvious implication of his theorem because he was blinded by his prior belief that angles were not trisectable by Euclidean constructions.
I drove home in a state of euphoria. I could barely see the road, my mind was so crowded with visions of triangles whose corners were bisected and trisected. In my mind, I trisected perfectly every angle that I could imagine.
There should be a law prohibiting driving while under the influence of trisectionism.
That night, I began to consider practical issues. If my construction trisected the angle, then I was going to be famous. Newspapers would publish my construction on the front page, though probably below the fold. Mathematics never makes the headline. My construction would be taught in every high school geometry class in the world for centuries to come. Every living mathematician in the world would envy me. I would deserve a Fields medal, though that would undoubtedly be denied to me, having earned all my degrees in psychology rather than mathematics. Not actually knowing enough mathematics to teach, I would never be offered a professorship in the mathematics department at a major university. But a minor university might offer me a professorship just so that they could exploit my fame to attract a better class of undergraduate. I would have to refuse those job offers. I already had a good job as a research scientist; a job that I am sure I would prefer even if tenure was offered as an inducement. It did not matter. The fame alone would be sufficient to satisfy me. And I could certainly use that fame to negotiate a promotion in my current job.
On Saturday morning, I hurried down to the nearest big-box office supply store and bought a quality compass, protractor, and straight edge. I should say, I bought twenty bucks worth of quality. No sense investing too much money before I had some assurance that my approach was going to work.
Drawing Euclidean constructions is a surprisingly pleasant way to spend a Saturday morning. For example, it is considerably less taxing than trying to sketch portraits of human faces. And far less taxing than working through other kinds of mathematical problems.
I drafted diagrams of the form shown in Figure 3a. Beginning with an angle AOB, I then created a small random angle, OAC opposing it at some reasonable distance. Duplicating angle OAC to form angle CAD and angle DAB, I created the tripartite angle OAB which will be mapped onto AOB. Then, I bisected angle OBA to get line BE which is intersected by the lines from point A at points F and G. My three-dollar protractor confirmed that lines drawn from point O through F and G trisect AOB.
As there are no right angles easily available, a trigonometric proof of the trisection would require erecting a few perpendiculars here and there. Or there might be a simple proof based on summing the various angles in the existing triangles. First, though, I constructed a couple of more extreme cases just to make sure that nothing bizarre was going to show up.
Figure 3a Attempted trisections of
by projection from a tripartite angle OAB
through a line BE which bisects OBA.
Of course, something bizarre did show up. By creating a narrow angle to
trisect and a much wider angle to project from, the angles that would be
projected through a bisector at the apex deviated visibly from a trisection.
Far more than could be waved away as measurement error. As can be seen in
Figure 3b, if both angles are trisected with a protractor, the points of
intersection corresponding to B, G, and F, do not fall on any straight line.
Undoubtedly this makes an important statement about the location and
orientation of Morley's triangle. Something that Morley undoubtedly knew
Obviously, there was no way to salvage the construction with any adjustment to the line BF. The points B, G and F do fall on a curve. Any three distinct points fall on some curve. But the centre of that curve does not seem to correspond to any point that could easily be constructed from the original figure. I have not bothered investigating, but I bet that the centre of that curve could be at a coordinate that involves an irrational number; especially for angles AOB that are known from the mathematical proof to be untrisectable, such as sixty degrees.
Figure 3b: Demonstration that
two tripartite angles need not intersect any
I was devastated. I saw my not-yet-realized fame dissipating into the air
like smoke from an opium pipe.
Maybe this was but a temporary setback. So my first simple mapping from the tripartite angle to the angle to be trisected did not work. So what. There are lots of possible mappings from one angle to another. Maybe another one would work. There was too much fame on the line to give up so easily.
I thought about this mapping and I thought about that mapping. I missed lunch to juggle diagrams in my head. I pierced many a sheet of snow white paper with the spike of my shiny new compass. Over the course of a couple of hours, my thoughts converged upon Figure 4; a construction which is not really as complicated as it looks at first glance.
Figure 4: The Whalen Trisection: Not.
The construction of Figure 4 proceeds as follows:
Begin with an angle AOB to be trisected. Draw a convenient small angle AOC adjacent to the original angle and use it to construct a tripartite angle AOE. Draw an arc FG with some convenient radius, centered at point O. Bisect the angle AOE to determine the point J that bisects the arc FH. Construct two concentric circles centred at point J, one of radius JR and one of radius JF. On the other side, bisect AOB to find the point L and construct a circle centred at point L with radius LG. Construct two lines, JK and LM, that are parallel to line OA; one through point L and one through point J. Draw a line from point J through point L and another line from point K through point M. If the angle AOB is a different size than angle AOE, these lines will converge at some point N. A line from point P to point N will intersect line LM at some point Q. Draw a circle centred at point L with a radius LQ. That circle will intersect the arc GH at some point S. Can it be proven that the line from point O to point S trisects the angle AOB?
The logic of this construction is:
We want to trisect angle AOB. We can construct a tripartite angle AOE by creating some random, small angle AOC and then replicating it twice more. Because line OC trisects AOE, we need only project the proportion FR:FH onto GH. The concentric circles about point J have radiuses which are have a proportion that is nearly FR:FH, but are distorted because they have morphed an arc into a line. The similar triangles NJK and NLM project the line JK onto the line LM and the similar triangles NJP and NLQ project the proportion onto the line segment LQ. The concentric circles centred at point L map this proportion back onto arc GH. And the key point: the arc GH has the same radius as the arc FH, so the mapping of the line LM onto the arc LG undistorts the distortion of the proportion that was introduced when the arc JH was mapped onto the line JK. The angle AOB is trisected!
Besides, it makes an aesthetically pleasing diagram.
Given that drawing all of those lines and circles will accumulate some error, especially at point N where angles get rather acute, my little plastic protractor confirmed that the angle is trisected to a reasonable degree of accuracy.
It was Saturday night and my sister-in-law had offered to baby-sit the kids so that my wife and I could spend the evening in a fine restaurant celebrating the completion of her two-hundred-page MBA thesis. Now, coincidentally, we could also celebrate my impending fame as the genius who solved a problem that had eluded brilliant mathematicians for twenty-two centuries. It is a pity that Alfred Nobel was the jealous type and decreed that there would never be a Noble prize in mathematics. My wife, relieved to finally complete her thesis, nodded indulgently as I explained the implications of my construction for our future. Our family life would soon become very much more complicated. She understood.
That evening, I began to worry about the practicalities of releasing my construction to the world. Obviously I could not just send it to a single mathematician. He might claim it as his own. Plagiarism happens. Especially when there is so much at stake.
Clearly the first step would be to ask a group of mathematicians to review the proof jointly so that no one of them could take the credit. I reminded myself that the proof was not written out yet. I would have to clean up that detail real soon now. But I had no doubt that it would be pretty easy. After all, my logic was clear. But how to present the proof? That was the harder problem. I work with some good mathematicians, so I could invite them to a seminar in our auditorium and present the proof there. PowerPoint presentation or transparencies? PowerPoint looks better, but I would probably need the better resolution of transparencies. And, of course, after the standing ovation, they would advise me about which prestigious mathematics journal most deserved to receive the scholarly paper. By registered mail, just to be safe. A smug paper with only one reference; to Euclid's Elements.
Sunday morning, I settled down to write out the proof. The only sticky point was showing that mapping the line LM to the arc LG was the inverse of the mapping of the arc JH to the line JK. I was disquieted to note that the arc GS on the smaller circle was a very close approximation to a straight line, but that the arc on the larger circle, FR, was very far from linear. For more extreme examples, say projecting a very acute angle onto a quarter circle, the differences between mapping a two-degree arc to a line and mapping a ninety-degree arc to a line would be huge. There was no way that one mapping would be the inverse of the other. My heart sank. Was there some other way to map an arc to a line? I drew a quarter circle. How long would the line be? Of course, its length would be a quarter of pi. Damn. An irrational number again!
By Sunday afternoon, I had no heart left for the pursuit. Any mapping from a circle to a line was obviously going to require some fraction of pi. I remembered enough from my undergraduate course on power series to know that pi ain't going to be calculated with no compass and straightedge.
Unlike some less fortunate wretches, it had only taken forty-eight hours for me to admit that I wasn't going to find any loophole in the proof of the untrisectability of an angle.
There was pain in my withdrawal from trisectionism. It was more than a little embarrassing to have to confess to my wife that the completion of her thesis had been the most significant accomplishment in our household last weekend after all. By a long shot. She understood.
Twice in two days I had plummeted from a peak of ecstasy to utter despair. Like any chemical addition, trisectionism forces dramatic mood swings on the addict that wrenches the emotions unbearably.
But, at least, now my daughter has a protractor that she can use for her homework. And a new compass, too.
On Monday, I sought solace in the mathematics section of the local university library. There, I found a book by Underwood Dudley called The Trisectors that clearly labeled me as a mathematical crank. Some solace. I did note that his inventory of failed trisections did not seem to include my approach of mapping a tripartite angle onto the target angle. At least I was an original crank. Some consolation.
I can reinforce Dudley's observations about trisectors with my own personal experience as a crank for a weekend. Crankness arises from an alignment of a number of factors.
The crank starts with a problem that is unsolved by the acknowledged experts in a field. In the case of trisectors, the problem is the trisection of the angle, accompanied by squaring the circle and doubling the cube. However, other fields have their own similar problems. Archeology has the problem of how the Egyptian Pyramids were built; biology has the search for Bigfoot and the Loch Ness Monster; physics has the unified field theory. My own field, experimental psychology, is rife with unsolved problems that are easy to state and whose resolution is still being debated by the experts, from the validity of recovered memories to the mechanisms that underlie hypnosis to the impact of subliminal stimuli.
Apart from the fact that they are unsolved by experts, these problems all have three common features.
First, they all seem to have a simple solution - from drawing a few lines and circles to camping out in the Scottish Highlands with a camera - that is accessible without extensive formal training. In particular, they seem to have solutions that can be arrived at by tinkering, by thought experiments, or by making strange analogies with phenomena in other fields.
Second, they can all be "solved" with an approximate solution. You can devise a procedure that will trisect a sixty-degree angle to an accuracy that is smaller than the resolution of a protractor. You can speculate about telekinetic powers that would levitate two-ton stone blocks. You can take a blurry photograph of something sticking out of a lake. You can hypothesize about all kinds of ill-defined subconscious psychological processes.
Third, cranks require that their problem and solution will be of interest to a very large number of people. Few cranks pursue highly-specialized problems that will only interest a dozen experts in the world. A trained entomologist may be fascinated by the differences in the number of hairs on the legs of two different varieties of wolf spiders, but no crank ever will. It has no media appeal.
Cranks prefer famous problems, but this does not seem to be one of the defining characteristics of their problems. There are numerous cranks who have devised their own ill-conceived problem, devised an equally ill-conceived solution, and then spent the rest of their lives trying to convince established authorities that they have accomplished something earth-shaking.
Dudley, in his description of trisectors, discusses several of their personal characteristics. The characteristics probably apply to cranks in other fields as well.
First, cranks lack formal training in the field in which they are working. This does not necessarily mean that they are uneducated. Cranks may be well-educated in other fields. Medical doctors, engineers, and university professors may become cranks. In my own personal experience, I have found several people holding doctorates in physics or master's degrees in engineering who believe that they have a special instinctive approach to psychological phenomena that was somehow never considered by psychologists during their years of education.
Second, even outside of formal training, cranks are resistant to learning about the field in which they are cranks. They will produce elaborate rationalizations to justify remaining ignorant. "Mathematicians do not know how to trisect an angle, so why should I waste my time doing the same thing that they have been doing?" "If I start studying physics, my creative spirit will be as stunted." "Psychology is not really a science, and I am a scientist, so I already know more than they do." "Biologists spend so much time in the lab that they forget that the most important phenomena are discovered in the field." "The really important knowledge is innate. I was born with a special understanding of triangles."
Third, cranks do not think critically about their own solutions. As soon as they get a solution that looks good, they stop analyzing the solution and start promoting their idea. When they do think about their solution again, it is never critically. Rather they concentrate on ways to extend their solution to new problems in new areas. If they have trisected the angle, then, rather than re-examining their construction with a more rigourous eye, they launch into doubling the cube. Even well-trained academics will not bring the appropriate critical approach to the topic of their crankhood. They fail to recognize that, in addition to the general techniques of critical thinking, each area of science has its own methods of critical evaluation which are optimized for that discipline. Mathematics has the deductive proof. Theoretical physics considers the interdependence of different theories. Biology demands controlled experiments. Psychological theories must stand up to convergent measures. Each of these procedures provides a different way of making a very close, detailed examination of a phenomenon. Cranks use none of these because any solution that looks good at first glance is adequate. Their goal is not to demonstrate the absolute truth to themselves, but to try to convince someone else that they have special knowledge.
Dudley, in his book on trisectors, alludes to the social and psychological forces that propel someone to crankhood, but, in my humble opinion, does not sufficiently acknowledge the tremendous power of those forces. If you succeed in accomplishing the impossible and find an Euclidean construction that trisects an arbitrary angle, you really will become famous. The trisection may be a fantasy, but the fame is not. You will be interviewed on radio and television. You will be able to walk into any bookstore anywhere in the industrialized world for the rest of your life and find your work described. You will be more famous than if you won a lottery or if you murdered your family. How many people remember who won the Nobel Prize in physics five years ago? But every high school student will be taught about the person who trisects the angle. Even established scientists in the first rank should be impressed by the amount of fame that will be awarded to anyone who can trisect the angle.
And, to the extent that the fame is real, the danger of plagiarism is equally real. Especially considering that most attempts to trisect an angle result in relatively simple constructions. Cranks may write volumes about their work, but the fundamental solution that they are describing is never as complex as, say, the solution to Fermat's conjecture or even the proof of GoŽdel's completeness theorem. Any trained mathematician could look at my Figure 4 for a few minutes and then reproduce it next week. The idea embedded in it is simply not rocket science. It is entirely possible for an unscrupulous academic who has even a brief exposure to the construction to claim it as his own and make a much better case than the unknown crank. Even if nine out of ten academics are too honourable to do that, who wants to trust to luck that their sole claim to scientific sainthood will not be ripped off by one of the few dishonourable people that they may encounter.
And, finally, because the crank's solution is invariably a trivial approximation at best and completely wrong at worst; because he is unwilling to either trust expert advice or become a true expert himself; and because he demands far more time and attention than any expert will give to him; the crank will be ignored and dismissed by the very experts that he is most desperate to impress. The wages of crankness is ignoble anonymity. The crank's final reward is unbearable frustration.
The glorious dream, the desperate paranoia, and the ultimate frustration are not delusions. All are real products of the real world.
Finally, I should comment that the identification of cranks is a difficult practical problem. Government agencies, charities, and industry are approached regularly by people who claim that they have a brilliant new idea that will benefit all mankind if only they can obtain sufficient funding. Invariably they target the most senior executives that they can find. Senior executives are generalists. They do not have specific training in every specific discipline required to dismiss every crank out of hand. Worse yet, they are loath to dismiss someone who might be the saviour of their organization. History has given us enough examples of important discoveries dismissed too quickly by senior executives. So, when approached by a crank, they often make some diplomatic, neutral promise that they will look into the matter. But, in the frustrated mind of the crank, any small acknowledgement is elevated to the status of total acceptance. The most successful cranks will then approach the next senior executive in the next organization with the claim that the last organization showed significant interest in their idea. And, once encouraged no matter how obliquely, they never give up.
Bureaucracy needs simple rules of thumb that can be used to filter out the worst cranks.
The rule of thumb that I use most often is to note whether the person is able to place his work in the context of the work of others that have preceded him. No one has ever created a new paradigm in science without understanding the previous paradigm. Copernicus understood Ptolomy's mathematics. Einstein understood Newtonian mechanics and Heisenburg understood Einstein's theories. Anyone who expects to trisect an angle will have to learn abstract algebra first because, one way or another, they will be changing our understanding of Galois' contributions.
Personally, I cannot be bothered going down to the basement and digging out that abstract algebra text.
But another weekend starts tomorrow. And I still have that pile of paper and my shiny new compass sitting on the dining room table.
When I think about it, I'm sure that I can map arbitrary angles onto each another if I place them in a three dimensional space. If I can do that, then all I have to do is project that three-dimensional mapping onto an Euclidean plane...