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.
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. 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, 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 It will be a huge number. 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 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. 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 and , 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. 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 degrees. For triangles, we can have three, four or five meeting at a vertex six would equal 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 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? Put the point of the compass at and draw a circle of any radius you like. This is the blue circle in the diagram. The circle will cross the two lines at two points: call these and.
Now put the point of the compass at and draw an arc of a circle, as shown in the diagram. Without changing the radius at which the compass is set, move its point to and draw another arc of a circle. These are the red arcs in the diagram. The point where the two red arcs cross, can then be joined to using the straightedge the green line. The angle is exactly half of the angle. It's possible to prove that this method bisects any angle.
You can try this for yourself use similar triangles or see a proof here. What about dividing an angle into three equal parts? Why is that so difficult? There are a few special cases of angles for which it can be done — for example if the angle is equal to radians or 90 degrees see this video.
You can trisect any angle if you allow yourself to use an extra dimension see here. It's also possible to trisect an arbitrary angle if you use a ruler, rather than a plain straightedge, so that you can measure distances. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought.
The problem was algebraically proved impossible by Wantzel Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and radians and , respectively , which can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as Honsberger In addition, trisection of an arbitrary angle can be accomplished using a marked ruler a Neusis construction as illustrated above Courant and Robbins An angle can also be divided into three or any whole number of equal parts using the quadratrix of Hippias or trisectrix.
An approximate trisection is described by Steinhaus Wazewski ; Peterson ; Steinhaus , p. To construct this approximation of an angle having measure , first bisect and then trisect chord left figure above.
The desired approximation is then angle having measure right figure above. To connect with , use the law of sines on triangles and gives. Since we also have , this can be written. Solving for then gives.
This approximation is with of even for angles as large as , as illustrated above and summarized in the following table Petersen , where angles are measured in degrees. Bogomolny, A.
Bold, B. New York: Dover, pp. Conway, J. The Book of Numbers. New York: Springer-Verlag, pp. Courant, R. Oxford, England: Oxford University Press, pp. Coxeter, H. New York: Wiley, p.
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