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The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geometry classes even today.

What I'm getting at is this: Are the rules of construction just arbitrarily imposed restrictions, like a form of poetry, or is there a meaningful reason for prohibiting, say, the use of a protractor?

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    There is a lot taught in high school with little reason for it; they should stop teaching trigonometry and instead do some intro statistics (at the level of, say, **How to Lie With Statistics**); geometric constructions with straightedge and compass are likely a remnant of the time when instruction in geometry was viewed more as a way to "exercise the mind" of the student than because it is significant or useful in any way.2011-02-18
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    http://www.maa.org/devlin/LockhartsLament.pdf2011-02-18
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    @Qiaochu: Yes, I know. But the purpose of instruction in high school includes preparing the individual for the world, not just introducing them to the wonders of knowledge and science. A good intro to statistics, or a good discrete math/intro to logic course does not have to be a drudgery of rules, or pure results-oriented, but it would give the students something which they really do need in the world. We don't just teach children to read so that they can enjoy the wonders of Shakespeare, but because to function in society they must read; today, they should understand basic statistics!2011-02-18
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    Trig is extremely important for physics, engineering and, last but not least, mathematics. And even if your children don't care for any of these, they probably care for video games, and, oh wonder, it's of much use there too. I think it's a really bad example of a piece of mathematics that can be dropped from the curriculum. However, I completely agree with the OP that compass-and-ruler constructions are almost useless as of today and can be dropped. I have done elementary geometry for some years (there's indeed a lot to be done there), and compass-and-ruler constructions ...2011-02-18
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    ... have always stricken me as ugly, useless and uninstructive. For example, the radical axis of two circles is very easy to construct if the circles intersect at two points (just connect these two points), rather easy to construct if they touch (it's their common tangent) and pretty hard to construct if they don't meet (in this case, it can be obtained as the perpendicular bisector of the segment which joints the inverse of the center of the first circle wrt the second circle with the inverse of the center of the second circle wrt the first circle - a fun thing to prove, but out ...2011-02-18
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    ... of reach for school). What have non-intersecting circles done that justifies this complication? Algebraically, the radical axis is a rational object, and there should be no reason to distinguish cases according to whether circles intersect or not. By solving rational problems in irrational ways (i. e. taking the two points of intersection of two circles - this comes down to solving a quadratic equation), compass-and-ruler constructions are solving easy problems in artificially complicated ways. Nice as a curiosity to spend some time thinking about, but why is this considered an ...2011-02-18
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    ... integral part of education?2011-02-18
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    @darij, thanks for your comments. I am not, as your first comment suggests, however, saying that constructions are useless and should be dropped. When I ask "What's the point?", I'm asking out of genuine curiosity and not trying to make an editorial statement.2011-02-18
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    This to me is like asking what is the point of music?2011-02-19
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    I believe euclid's answer to a similar question was: "give this man an obolus, as he must make gain of everything he learns."2011-02-19
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    @darij: I'm not saying trig is not important for later studies; I just don't know why we teach it in high school, and not through JITT ("Just In Time Teaching") in those later studies. I'm no suggesting dropping trig from **all** curricula, but why do we teach middle and high school students to solve triangles? Students coming out of **high school** need to be literate (able to read and comprehend what they read, able to write coherently) and numerate; again, we don't teach them to read because they should know Shakespeare or to show them how cool poetry is.2011-02-19
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    @darij: *I* learned all the trig I know either in JITT in calculus class (which introduced the functions from the ground up, as the parametrization of the unit circle), or by having to teach pre-calculus (I had never "solved a triangle" before then, nor was this any particular obstacle to my education). By all means, trig is important for physics, engineering, video-gaming, etc. It should be taught to *them*. Measure theory is extremely important for statistics and probability, we don't demand that it be taught to everyone.2011-02-19
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    @Arturo: that wasn't intended to be a response to your comment, but a response to the OP. I totally agree with your comment. But Lockhart (if I am remembering correctly) points out, among other things, that high school geometry exists for historical reasons and that it is being parroted without a real understanding of its point, or something like that. Perhaps I should have pointed this out.2011-02-19
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    @Qiaochu: Sorry if I seemed to be jumping at you. A subject of frequent frustation for me, as you can probably guess from my rant below.2011-02-19
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    From Stukeley's memoir on Newton: "he never observed [Newton] to laugh but once. 'twas upon occasion of asking a friend to whom he had lent Euclid to read, what progress he had made in that author, & how he liked him? he answerd, by desiring to know what use & benefit in life that study would be to him? upon which Sir Isaac was very merry."2011-05-19

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