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Is there a way of trisecting an angle using a compass, straight edge and pencil?

  1. Aug 9, 2004 #1
    I know that there is a way to bisect an angle using that equipment, but is there a way to trisect it. I have heard it is possible to do it by constructing a regular polygon with sides of a multiple of 3 around the angle.
    Does any1 have anything?
     
  2. jcsd
  3. Aug 9, 2004 #2
    Nope sorry.

    Its an old problem, and its been known that its impossible for quite some time now. It was one of the main puzzles the greeks came across, the others being squaring the circle and doubling the cube, all of which are impossible. You can trisect certain angles, but it remains impossible to trisect any arbitrary angle.
     
    Last edited: Aug 9, 2004
  4. Aug 9, 2004 #3
    Ahh well.
    There's always hope.
    Has it been proven that it is impossible?
     
  5. Aug 9, 2004 #4

    NateTG

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    Trisection has no general solutions with a finite number of steps, but many particular angles can be trisected, and there are many examples where additional information can allow angle trisection as well.
     
    Last edited: Aug 9, 2004
  6. Aug 9, 2004 #5
    The trisection of an angle was performed by Archimedes. Heinrich Dorrie, "100 Great Problems of Elementary Mathematics," lists this on page 173. As he explains it is done with a paper strip construction. The fact that it can not be done with a compass and straightedge is a very artificial restriction that axiomatic individuals have greatly overblown. People do not use straightedges, they use rulers even in class, which have markings on them.

    Were the Greeks interested in finding solutions or proving things could not be done? Obviously they wanted solutions in their day. There is an internet reference here: http://www.cut-the-knot.org/pythagoras/archi.shtml, where it is called an "illict" solution, which somehow supposes it is better to not solve problems than solve them.
     
    Last edited: Aug 9, 2004
  7. Aug 9, 2004 #6
    Trisection of an angle with a straight edge and compass is impossible. It can be proved using abstract algebra. I know it has to do with fields, my professor showed me the proof before. I will post it, if anyone has not when I get home and look through my old text.
     
  8. Aug 9, 2004 #7

    enigma

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    Not if you're a drafter and know what you're doing. Using a ruler or scale to draw with ruins the ticks very quickly.

    Luckily, board drafting rapidly becoming an obsolete art.
     
  9. Aug 9, 2004 #8
    yeah it's proved in most books on Galois theory. I like Hadlock's "Field Throey & its Classical Problems".

    & it's compass and straightedge, not ruler. A ruler has a scale on it, a straightedge has no markings.
     
  10. Aug 9, 2004 #9

    HallsofIvy

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    The Greeks were very good at geometry but didn't have a good numeration system which restricted their algebra. The main way they handled numbers was as constructed lengths. The whole question of "construction with compass and straightedge" was a question of "what numbers can be constructed". Of course, a ruler, with ticks already on it was was begging the question- you couldn't assume you already had numbers before you constructed them.

    The answer, by the way, due to Galois theory, is that the "constructible numbers" are precisely those real numbers that are "algebraic of order a power of 2". (A number is "algebraic of order n" if it a solution to a polynomial equation with integer coefficients of degree n but no such equation of lower degree.)
    One can show that if one can trisect, for example, a 60 degree angle, then one can construct a segment whose length satisfies a cubic equation but no lower degree equation- it is algebraic of order 3, not a power of 2. Thus such a construction is impossible.
    One can also show that if, given a circle and taking the radius to be 1, one could construct a square having the same area, then one could construct a length (a side of the square) which has length of √(π). Since π is transcendental (not algebraic of any order), such a length is not "constructible" and "squaring the circle" is impossible with compasses and straightedge.
    Finally (to complete the classic trilogy), given a cube of side 1, if one were able to construct a cube of exactly twice the volume, one would be constucting a length cube root of 2. Of course, that is algebraic of order 3, not a power of 2. That is, it is impossible to "duplicate the cube" using 3 dimensional versions of straightedge and compasses (that, given 3 points construct a plane and, given 2 points, construct the sphere with center at one point and passing through the second).
     
  11. Aug 9, 2004 #10

    selfAdjoint

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    Every construction in Euclid's Elements can be don with compass and straightedge - in fact it was discovered in the 18th century that it suffices to have one fixed circle and a straightedge. The Greeks did both axiomatics and computation. But the trisection of an angle with any old tool doesn't give you a regular method. Why not use a protractor if that's what your after.
     
  12. Aug 10, 2004 #11
    I wish to add, that the conditions for Euclidian construction, were figured out by Gauss, who at the age of 19 constructed the 17 sided polygon. Along with this, Gauss found the conditions for constructions in terms of Fermat primes and powers of 2. This was done before there was a Galois or a Galois theory.
     
  13. Aug 12, 2004 #12

    mathwonk

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    Ihnot makes a valid if esoteric point, that the theory of constructibility only uses the theory of dimension of field extensions, and not the interplay between field automorphisms and group automorphisms perfected by galois.

    Of course Gauss apparently understand quite a bit about quadratic field extensions and particular groups, such as groups of units mod n, before Galois, but I am not expert enough to suggest that a rudimentary form of Galois theory preceded Galois.

    The reason for the fact quoted by Halls of Ivy, that the only numbers, or lengths, constructible by straight edge and compass, are those satyisfying equations whose degree is a power of two, is almost intuitively obvious.

    I.e. the constructions involve (or can be reduced to) intersecting two lines or one line and one circle. So you are solving simultaneously two linear or one linear and one quadratic equation, so the degree of the resulting equation at each stage is one or two. Repeating the constructions, gives field extensions of a power of two, once you know that the dimensions ("degree") of repeated field extensions multiply.
     
    Last edited: Aug 12, 2004
  14. Aug 12, 2004 #13
    Quoting from 100 Great Problems of Elementary Mathematics by Heinrich Dorrie, he credits Gauss with publishing the regular heptadecagon in Disquisitiones Arithmeticae in 1801, thought I hear Gauss discovered this proof at 19.

    Anyway, Gauss remarks,"The division of the circle into 3 and 5 equal parts was already known in Euclid's time, but it is amazing that nothing new was added to these discoveries in the next two thousand years."

    Perhaps then we can assume that Gauss was principally interested in advancing the understanding of Euclidean construction, and never intended to take his result any further.
     
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