Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay
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I will respond to questions and comments as time permits, but if you want to take issue with any position expressed here, you first have to answer this question:
What evidence would it take to prove your beliefs wrong?
I simply will not reply to challenges that do not address this question. Refutability is one of the classic determinants of whether a theory can be called scientific. Moreover, I have found it to be a great general-purpose cut-through-the-crap question to determine whether somebody is interested in serious intellectual inquiry or just playing mind games. It's easy to criticize science for being "closed-minded". Are you open-minded enough to consider whether your ideas might be wrong?
The ancient Greeks founded Western mathematics, but as ingenious as they were, they could not solve three problems:
It was not until the 19th century that mathematicians showed that these problems could not be solved using the methods specified by the Greeks. Any good draftsman can do all these constructions accurate to any desired limits of accuracy - but not to absolute accuracy. The Greeks themselves invented ways to solve the first two exactly, using tools other than a straightedge and compass. But under the conditions the Greeks specified, the problems are impossible.
Since we can do these tasks to any desired accuracy already, there is no practical use whatever for an exact geometrical solution of these problems. So if you think you'll get headlines, endorsement contracts and dates with supermodels for doing so, it is my sad duty to tell you otherwise.
Also, there are a few trivial special cases, like a right angle, where it is possible to "trisect" the angle. More specifically, it is possible to construct an angle one-third the given angle. For example, if you draw a diameter of a circle and mark off 60 degree intervals on the circle, you "trisect" the straight angle. This isn't trisection in any meaningful sense because it doesn't generalize to other angles.
"Impossible" is a welcome challenge to a lot of people. The problems are so easy to understand, but the impossibility proofs are so advanced, that many people flatly refuse to accept the problems are impossible. I am not out to persuade these people. Mathematicians have spent years corresponding with some of them, and many are absolutely immune to persuasion (The trisectors, I mean. The mathematicians are too, but they have reason to be). But if you want a (hopefully) intelligible explanation of why mathematicians regard the problems as impossible - as proven to be impossible - then this site might help you.
Warning: you need trigonometry and an understanding of polynomials - that is, the equivalent of a good high school math education - to follow this discussion.
One of the major problems people have with angle trisection is the very idea that something can be proven impossible. Many people flatly deny that anything can be proven to be impossible. But isn't that a contradiction? If nothing can be proven to be impossible, and you can prove it, then you've proven something to be impossible, and contradicted yourself. In fact, showing that something entails a contradiction is a powerful means of showing that some things are impossible. So before tackling the trisection problem, let's spend some time proving a few things impossible just to show that it can be done.
|Imagine you have a checkerboard and start covering it with dominoes so that each domino covers two squares. Obviously there are a vast number of ways to do it.|
|Now imagine you remove two opposite corner squares. Can you
still cover the checkerboard with dominoes?
No. Rather than try every possible way to place dominoes on the board, consider this: each domino covers both a red and a white square. If you remove the two corners shown, there will be 32 white squares but only 30 red squares. There's no way to cover the board with dominoes without leaving two unpaired white squares. So it can't be done. We have proven that something is impossible.
I expect some die-hard to ask what about coloring a white square red so there are 31 of each color. It doesn't matter. (Actually, it's easy to see that coloring a white square red will create a situation where you have to cover two red squares with a domino. Once you cover them, you have an unequal number of red and white squares remaining, and then we're back to square one, literally.) You can paint the board psychedelic if you like and it still can't be done. The traditional checkerboard coloring makes it easy to prove it, but if it can't be done on a traditional checkerboard pattern, it can't be done, period. In fact, a lot of proofs depend on marking, labeling, or grouping items in a certain way to show that some particular arrangement either is, or is not, possible.
Prime numbers like 2, 3, 5, 7, 11 .... are divisible only by themselves and one. As numbers get bigger, primes get more rare. Is there a largest one?
The ancient Greeks showed there is not. Imagine there is a largest prime p. Now calculate the number q, which is 2 x 3 x 5 x 7 x 11 x...(all the primes less than p) x p. It will be a huge number. Now consider q+1. It's not divisible by 2, because q is divisible by 2. Likewise, it's not divisible by 3, 5, 7, or any other prime up to p, because q is divisible by all those numbers. So there are only two possibilities. Either q+1 is prime, or it's divisible by primes bigger than p (q+1 will be vastly bigger than p - if p is only 19 than q is 9,699,690 - there's lots of room for bigger primes). Either way the initial assumption leads to a contradiction. Hence it must be wrong. There is no largest prime. It is impossible to find a largest prime. It's not that people have tried, failed, and given up. It's impossible because the idea itself leads to a contradiction. This method of proof - making an assumption and then showing that it leads to a contradiction - is called reductio ad absurdum.
It is impossible to find a largest prime, but it is possible to find arbitrarily long stretches of numbers without them. Our number q is composite - it is the product of smaller numbers. It's easy to see that q+2 must be composite, as well as q+3, q+4 (divisible by 2) q+5, q+6 (divisible by 2 and 3) and so on up to q+p. We can find so-called "prime deserts" of any desired length, but there are always primes after them. Indeed, we keep finding pairs of primes, like 5 and 7, or 101 and 103, however high we go, though nobody has yet shown there is an infinite number of them.
Can you represent the square root of two as a fraction? The ancient Greeks also found out that this is impossible. Imagine that there is a fraction p/q, where p and q are whole numbers and p/q is in lowest terms (that qualifier is important), that equals the square root of two. We can conclude:
The sect called the Pythagoreans believed that everything was ultimately based on whole numbers. They were horrified by this discovery. Tradition has it that they decreed death to any member who divulged the secret. They called numbers that could not be represented as ratios as irrational, and to this day irrational carries a negative connotation. Actually this proof is related to the proofs that trisecting the angle, doubling the cube and squaring the circle are impossible.
Reductio ad absurdum is a powerful way of showing that some things are impossible, but not the only way. If we can succeed in showing all the things that are possible, then anything else must be impossible.
In simple cases we can list the possibilities by brute force. For example, what kinds of three dimensional shapes have regular polygon faces, with all faces and vertices identical? Clearly you have to have at least three faces meeting at a vertex, and the sum of all the angles meeting at a vertex can't equal or exceed 360 degrees. For triangles, we can have three, four or five meeting at a vertex (six would equal 360 degrees). For squares we can have only three, same for pentagons, and for hexagons and above, it can't be done (three hexagons add up to 360 degrees). So that's it. There are five shapes, and no others, shown below. (If you allow faces and edges to cross through each other, it gets a bit more interesting.)
When there are a potentially infinite number of possibilities, we have to use the general properties of the problem to devise a proof. Here's a fairly simple example. What kinds of repeating patterns can we have on a plane, for example, wallpaper? We say something has n-fold symmetry if, when you rotate it 360 degrees, there are n positions where it looks the same. For example, a honeycomb has six-fold symmetry - you can rotate it 360 degrees and there will be six positions where the pattern looks the same. A sheet of square graph paper has four-fold symmetry. So does a checkerboard if we ignore the colors, but if we include the colors, then it only has two-fold symmetry. Something with no symmetry at all has one-fold symmetry.
|Now a single object like a flower or propeller can have any kind of symmetry at all. But what about a pattern that repeats? Look at the pattern at right.||
o o o v q o o \ / a\ /a o o u-----x-----p o o o o o o o o
The pattern consists of points, some of which are lettered. All the points are identical (that's what a repeating pattern means). So if we can rotate the pattern around one point and see symmetry, we can rotate it around every point. If we rotate the pattern by angle a so it still looks the same, then a point originally at p will be rotated to the same position as some other point (labeled q). Since we can rotate in either direction, a point originally at u would be rotated in the other direction to some other point v. The pattern shown here is the familiar honeycomb pattern, but notice we have not made any assumptions about the angles in the pattern at all. The pattern shown is just for illustration.
Let's assume each point is distance one from its neighbor: ux = xp = xv = xq = 1.
Distance vq has to be a whole number, but it need not be 1. If a = 90 degrees (4-fold
symmetry), it could be zero. But it has to be a whole number, because it's the distance
between two points on a row of the pattern. From elementary trigonometry, we can also see
vq = 2 cos a.
Now the cosine can only have values between -1 and 1, so 2 cos a, which must be a whole number, can only take on values -2, -1, 0, 1 and 2. Thus cos a can only be -1, -1/2, 0, 1/2 and 1, or a = 180, 120, 90, 60 and 0 (or 360), respectively. Since the rotations are symmetry rotations, a = 360/n, so n = 2, 3, 4, 6, or 1.
Thus, repeating patterns in the plane can only have 1, 2, 3, 4 or 6-fold symmetry. In particular, repeating patterns in the plane cannot have five-fold symmetry.
Notice, this is not a matter of people trying to find five-fold repeating patterns and failing, then concluding it must be impossible. We devised a method to find all the patterns that are possible. We showed that only certain patterns are possible; therefore all the others are impossible. Many proofs of impossibility are based on the fact that we show what is possible, thus ruling out everything else.
More (much more) on symmetry and patterns
Plato's famous analogy of the Cave gives a lot of insight into the Greek mathematical mind. In this analogy, a prisoner in a cave knows the outside world only from shadows cast on the wall of the cave. Needless to say, his knowledge of the world is pretty imperfect. To many Greeks, this world, or at least our sensory picture of it, were crude images of a perfect, real world. Thus, to Greek mathematicians, the lines we drew were crude approximations of real lines, which were infinitely long, infinitely sharp, infinitesimally narrow, and perfectly straight. In principle, lines intersected in infinitely tiny, precise points. In geometry, "the truth is out there", the answer already exists, and they sought constructions that would find the answer infallibly and with infinite precision. Thus the Greeks rejected any technique that smacked of approximation or trial and error. The problem with trial and error is that, regardless how closely the construction seems to fit, we can never be sure there isn't some microscopic error below the limits of our visibility. The Greeks wanted to find the correct point directly, the first try, with absolute precision.
|It is possible to build tools that will solve the problem, using a ruler and compass. One of the simplest, at right, was dubbed the "hatchet." It consists of a right-angled T as shown with a semicircle on one side. Points A, B, C and D are all equally spaced. Place A on one side of the angle with the long arm of the T passing through O, and slide the hatchet so that the semicircle just touches the other side of the angle at E. Lines OB and OC trisect the angle. The upper diagram shows how it works, the lower diagram shows how one might be constructed from cardboard.|
Proof: Since AB=BC and angles ABO and CBO are right angles, triangles AOB and BOC are identical and angles AOB and BOC must be equal. Also BC=CE and angle CEO is a right angle, so triangles BOC and COE are also identical and angles BOC and COE are identical. Thus angle AOB = BOC = COE.
So what's wrong with this device? It can be laid out with ruler and compass, but not physically made. A real hatchet, however perfect, would have imperfections. Even if it were absolutely perfect, the process of lining the device up would be one of trial and error. So this device was unsatisfactory to the Greeks. It's tempting to try to use a straightedge and compass to lay out the hatchet right on the desired angle, but it also can't be done without trial and error.
It is possible to trisect the angle using a marked straightedge, but that's not allowed by the ancient Greek rules, since a mark can't be lined up against another point, line or arc without trial and error, and without some inherent error of alignment. Likewise, you can solve the problem with two carpenters' squares, but that's also not allowed since it also involves trial and error and since no carpenters' square will have an absolutely perfect right angle.
There are subterfuges people have developed for seeming to get around these restrictions. You can hold the compass against the ruler to measure distance; technically you haven't "marked" the ruler but the effect is the same. There are a variety of constructions that involve sliding the pivot point of the compass along a line or arc; these all involve some measure of trial and error as well (though the people who devise these constructions sometimes hotly deny it). Using the compass for anything but drawing an arc around a fixed center is forbidden.
I have modified this page several times as people come up with evasions of the classical restrictions. So let's clarify the general principle: all loopholes violate the rules.
|This is perhaps the simplest trisection using a marked straightedge. It
was discovered by Archimedes. Given the angle AOX, draw a circle of arbitrary radius
centered at O. Extend one side of the angle through the opposite side of the circle at D
Mark off interval BC on the straightedge. BC = OX = radius of the circle.
Slide the straightedge so that B lies on line DOX, C lies on the circle and the straightedge passes through A. Angle CBD is one-third of AOX. Note the element of trial and error inherent in positioning the straightedge.
You can eliminate the marking by setting a compass with radius BC and sliding the pivot along the line until B, C, and A are on a straight line. However, if you do, there are two additional points to consider: where the radius of the compass hits the circle (call it C') and where the radius hits the ruler (call it C"). You can never be certain that C' = C" = C, regardless of how well the points seem to agree. Even if the same pencil line covers all three points, if they are different in the most microscopic degree, the construction is not exact.
It also doesn't do to indulge in arm waving and say "slide the compass and ruler until B, C, and A are on a straight line." We know point C exists somewhere. You have to prove that you have actually found it. You have to be able to prove that C' = C" = C.
|There are other curves besides circles that can be drawn, and can be used to trisect the angle. The simplest, shown at right, is the Archimedean Spiral. The radius of the spiral is proportional to azimuth. Obviously an angle can be trisected by drawing an Archimedean Spiral over the angle, finding the radius where the other side of the angle intersects the spiral, then trisecting the radius. Draw arcs from the trisection points of the radius to the spiral, then draw radii through those points on the spiral. In fact, an angle can be divided into any number of equal parts using this method.|
The problem with the Archimedean Spiral, or any other curve used to trisect the angle, is that although you can construct an infinite number of points on the curve using a straightedge and compass, you can't construct every point. However closely spaced the points are, there will always be tiny gaps between them, so that any curve we draw will always be approximate, not exact.
If you think about it, this whole business is actually pretty artificial. The Greek compass, unlike modern ones, had a spring so that it snapped shut once it was lifted off the page. Anyone who has ever had a compass change radius while drawing a circle can picture the potential for error in such a device. Of course, we can do all the constructions the Greeks did, in many cases a lot more simply, by using a compass that stays fixed. To use something as error-prone as a spring-loaded compass, then worry about possible imperfections in constructing a tool like the hatchet, or positioning a marked straightedge, is completely arbitrary.
Also, if you think about it, the business of lining a straightedge up through two points to draw a line also has a lot of trial and error about it - as much as the hatchet tool. You line the straightedge up against one point, then position it against the other, then go back and correct any shifting at the first point, and so on. Then, actually to draw the line, you need to take into account the fact that any marking tool has finite width, so that as often as not the drawn line doesn't pass exactly through the two points.
Renaissance instrument makers soon discovered this problem. They found out that markings plotted using only compasses were more accurate than those made using straightedges. They began devising alternative constructions that eliminated use of the straightedge. Although the constructions were often more complex, they were still more accurate than those that required straightedges. Mathematicians finally showed that every construction that can be done with a compass and straightedge can be done with a compass alone. The only qualification is that we define constructing two points on a given line as equivalent to constructing the line itself.
Many people who "solve" the angle trisection problem inadvertently violate the rules:
The triple angle formula in trigonometry for the sine is: sin 3a = 3 sin a - 4
We can rewrite this to 4 sin3a - 3 sin a + sin 3a = 0
In other words, trisecting an angle amounts to solving a cubic equation. That's why nothing has been said about doubling the cube. Doubling the cube amounts to finding the cube root of two, that is, also solving a cubic equation. So algebraically, the two constructions are equivalent. Squaring the circle is a bit more complicated.
Recall how the Pythagoreans, to their horror, found out that there are other kinds of numbers than integers (whole numbers) and rational fractions. The process of discovering all the types of numbers that exist turns out to be directly related to the proof that the three classic problems are unsolvable. Since the time of Pythagoras, mathematicians have discovered that there are many types of numbers:
|The final solution to the three famous problems of antiquity came from studying classes and properties of numbers. Geometrical constructions can be considered the equivalent of mathematical operations. For example, using only a straightedge, you can add, subtract, multiply and divide:|
|If you can draw circles, you can also construct square roots.
Proof: Triangle ACD is a right triangle since all angles inscribed in a semicircle are right angles. So if one angle is p, q = 90 - p and the remaining angles have the values shown. So triangles ABD and DBC are similar. Thus we can write BD/1 = X/BD, or BD squared = X. Hence BD equals the square root of X.
One of the first fruits of these studies was the discovery by the young Karl Friedrich Gauss that it was possible, using a ruler and straightedge, to construct a polygon of 17 sides. This was something completely unsuspected. He also found that polygons of 257 and 65537 sides could be constructed. (Strictly speaking, Gauss only discovered it was possible; other mathematicians devised constructions and showed that no other polygons were possible, but Gauss made the pivotal discovery.) A number of books give constructions for the 17-sided polygon; constructions for the other two have been devised but are hardly worth the effort.
Numbers that can be expressed as combinations of rational numbers and square roots, however complicated the combination, are called surds. Only surds can be constructed using a compass and straightedge. We can now pose (and answer without proof) the following questions:
Gauss did more than just find new polygons. Building on his results, other mathematicians showed that polygons of 2, 3, 5, 17, 257 ... sides are the only ones with prime numbers of sides that can be constructed. (Of course you can also construct polygons by repeatedly bisecting the angles of these polygons to construct polygons with 4, 6, 8, 10, 12, 16, etc. sides. You can also construct polygons of 15 sides by combining the constructions for 3 and 5 sides, etc.) By enumerating what was possible, he ruled out many other things as impossible. In particular, 7 and 9-sided polygons cannot be constructed using straightedge and compass. Constructing a 9-sided polygon requires trisecting a 120-degree angle. Since this can't be done, obviously trisecting any desired angle is impossible.
Note, by the way, that 2=1+1, 3=21+1, 5=22+1, 17=24+1, 257=28+1, 65537=216+1. The numbers are all primes, and equal to some power of 2 plus one, and the exponents are all powers of 2. 65537 is the largest prime of this type known, although extensive searches for larger ones have been made.
Most people who still send solutions to the three classic problems to mathematics departments don't have a clue how the problems were finally solved. Many seriously think mathematicians just gave up and decreed the problems unsolvable.
The problems actually can't be solved because they require properties that a straightedge and compass simply do not have. You can't draw an ellipse with a straightedge and compass (although you can construct as many points on the ellipse as you like), so why is it a shock that you can't trisect the angle, duplicate the cube, or square the circle? You can't tighten nuts with a saw or cut a board with a wrench, and expecting a straightedge and compass to do something beyond their capabilities is equally futile.
Here's another analogy: if you spend your entire life driving a truck, you might eventually be lulled into thinking you can see the entire country from the highway. It's only if you get out and walk or fly over the landscape that you discover there are a lot of other places as well. Geometry with straightedge and compass creates a similar illusion; eventually we believe the points we can construct are all the points that exist. It was only when mathematicians began studying the properties of numbers that they found out it wasn't so. Just as you have to get off the highway to see that other places exist, to find the limitations of geometry you have to get outside of geometry.
More mathematically literate angle trisectors are sometimes aware of the number-theory approach, but reject it because they think a geometrical problem can only be properly solved geometrically. But if the problem is that the solution requires capabilities beyond those of a straightedge and compass, how in the world can that be discovered from within geometry? Anyway, who says geometrical problems can only be properly solved geometrically? The only thing that would justify that rule is some demonstration that the geometrical solution to a problem and the algebraic solution yielded different results - and then you'd have to prove the geometrical approach was the correct one. But there are no cases where this has ever happened, so there is no justification for rejecting the algebraic solution to the three classic problems. Indeed, it has been shown that if there is an inconsistency anywhere in mathematics, it is possible to prove any proposition whatsoever.
In fact, one of the most famous mathematical proofs of all times, Kurt Godel's Incompleteness Theorem, showed that there is always an "outside" to mathematics. Once a rule system gets complex enough (and Euclidean geometry is plenty complex enough) it is always possible to make true statements that cannot be proven using only the rules of that system. Thus is possible to make true geometrical statements (like "angles cannot be trisected using ruler and compass") that cannot be proven using the rules of geometry alone.
You can construct a 20-sided polygon with central angles of 18 degrees. You can construct a 24-sided polygon with central angles of 15 degrees. Obviously you can superimpose the constructions, and the difference between the two angles is three degrees. That's the smallest integer value we can get using straightedge and compass constructions. Since an arbitrary angle can't be trisected, and we can't find any integer angles except multiples of three degrees,
Therefore you cannot construct an exact one-degree angle with ruler and compass!
I had one correspondent who got rather worked up over this. But what's so special about 360 degrees? It's divisible by a lot of numbers, but so is 240. If we defined a circle as having 240 degrees, each degree would be 1-1/2 of our degrees. And we can construct 1-1/2 degrees exactly.
But so what?
We can't construct a one-degree angle exactly using a ruler and compass, but we can subdivide line segments into any number of intervals. In principle, we could construct a right triangle with one edge 100 million kilometers in length and another edge 1,745,506.492821758576 kilometers long at right angles to it. The hypotenuse and the long edge define an angle of one degree to an accuracy of 10-8 cm, the diameter of an atom. You'll need a very big sheet of paper, a lot of patience, and a very sharp pencil, but in Greek terms it's "possible". So we can construct a one-degree angle to precision so fine it surpasses any possible practical need.
Lots of books cover the mathematics behind the impossibility proofs. A really good recent one is Robin Hartshorne, Geometry: Euclid and Beyond, Springer, 2000. Warning: this is not fireside reading. The book is very clear but it will take serious study to master it.
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Created 10 December 1999, Last Update 31 March 2006
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