**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
AOB

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
decades ago.

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
projection of

two tripartite angles need not intersect any

straight line.

I was devastated. I saw my not-yet-realized fame dissipating into the air
like smoke from an opium pipe.

But yet...

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 Goëdel'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...