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I am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program.

I have a couple ideas, but I might be missing something or have a gap, so any opinions supported by facts or credible sources is great!

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    $\begingroup$ Aren't there just practical issues? If you've got 35 kids in a public school classroom, you probably don't have 35 computers. And why not just use the right tool for the job, especially if it's a simple, cheap tool? $\endgroup$ – Ben Crowell Jun 13 '17 at 18:39
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    $\begingroup$ Ideas: The choice of permissible drawing tools in a compass and straightedge setup is physically motivated. For computer programs such a choice would appear more arbitrary. Can I use intersection between conics as a primitive operation, even though that involves at least cubic equations? If not, how do I explain that? If I do, how do I make students aware that they are beyond what classical geometry would consider permissible? I guess there may also be an aspect of training motor skills. Which in turn conveys an intuitive idea of issues of numeric precision and the impact of rounding errors. $\endgroup$ – MvG Jun 13 '17 at 21:52
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    $\begingroup$ "why it is necessary ... " not everything which is good is necessary. Few things are really necessary in life. $\endgroup$ – John Coleman Jun 14 '17 at 15:29
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    $\begingroup$ If it was good enough for Euclid, it's good enough for your kids! $\endgroup$ – BruceWayne Jun 14 '17 at 18:49
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    $\begingroup$ @KennethK. - If that's your concern, I'd suggest learning how to read a 'compass'. $\endgroup$ – Mazura Jun 14 '17 at 23:29

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As previous answers have explained, geometric construction with just those two tools has a long and rich history. There is much to learn from playing with ideas this way.

For me, personally, geometric construction is much more pleasurable, and more informative, when done with electronic tools. I love this site (which gamifies geometric construction) and I sometimes enjoy this site. (I get frustrated when I get stuck, because you have to finish each step to go to the next.) Once I've gotten engaged in a problem, I've also used geogebra.

I always felt clumsy with a compass. I would press down too hard, and it would change size. Now I don't have to worry about that.

I have learned so much in the past few years by playing with geometric construction this way.

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One simple answer is that they are such basic tools to make and use. Obviously, there are historical reasons for using a compass and straight edge, but they are also very practical in carpentry.

For example, just last weekend I was helping a friend build a deck, and we had to miter-cut some boards to fit some arbitrary (unmeasured) angle. We needed the miter-cut to bisect the inside obtuse angle so the cut edges would be the same length, and hence fit well together.

Angle Cut

Not being adept with the tools at hand (speed square) for this task, I made marks on the two boards (equidistant from the vertex point) and "made" a beam compass (i.e. held a straight stick) to bisect the angle. It turned out fine, as expected.

Angle Bisection

So, I am thankful that I not only knew the theory of bisecting an angle with a compass and straight edge, but that I had also actually used such devices in various forms. I think exclusive use of a drawing program that allows one to simply "bisect an angle" with a button click would not have given me the requisite understanding to perform this task, but if the program makes you create circles and straight lines individually, that's another story.

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The accuracy limitations of a straight-edge and compass are more obvious than the accuracy limitations of a drawing program. And drawing programs do have accuracy limitations, which are sometimes worse than those of a straight-edge and compass.

  • Floating point numbers are quantized, and not especially consistently.
  • Many drawing programs only pretend to draw arcs and circles. They actually draw regular polygons instead.
  • Many drawing programs only pretend to draw ellipses. Instead they draw "4-circle ellipses" that are actually 4 pieces of regular polygons.
  • Rasterized screenshots and printouts are either quantized enough to be less accurate than a straight-edge, or are fuzzy enough to be less accurate than a straight-edge.
  • Paper sizes can vary by a fraction of a percent from day-to-day. Most drawing software does not have any feature for calibrating print outs.
  • Some drawing programs are not designed to simply shade the area of a shape; they insist on outlining the shape in black as well. Ellipses and circles are especially likely to have this problem.
  • Many drawing programs do not make clear where the mid-line of the outline is with respect to the theoretical shape being drawn.
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In primary school We used to do mental calculations, never touched a calculator back then.. in college, some programming lessons with all the IDE's help deactivated. At university I found myself relaying on papers & pencils because I knew what I was doing & feeling confident about it.

To resume, I think it's because they want to teach kids how not to rely on such programmable helps because they might find themselves in situations with no such tools available that require speed of execution

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Of course there is the deep (~2000 yrs) history of compass/straightedge constructions.

Another more modern alternative is origami constructions:

Robert Lang, "Origami and Geometric Constructions," 1996. PDF download.

In particular, one can trisect an angle via origami folding (under, e.g., Huzita's origami axioms):


Trisection
Figure from Geometric Folding Algorithms: Linkages, Origami, Polyhedra.


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    $\begingroup$ I used to be impressed by origami's ability to do angle trisection until I realized that it relies on the same forbidden technique, "neusis" or sliding, that would also enable it with compass-and-straightedge. You can see the sliding step in the third picture, where the fold is constructed so as to place two points on two existing folds. $\endgroup$ – Ryan Reich Jun 15 '17 at 5:07
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    $\begingroup$ This is not answer, and suitable for MSE. $\endgroup$ – Takahiro Waki Jun 17 '17 at 8:59
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    $\begingroup$ @RyanReich: Axiom 6 of Huzita's origami axioms: Given two points and two lines, one can fold a crease that simultaneously maps one point to one line the other point to the other line. $\endgroup$ – Joseph O'Rourke Jun 17 '17 at 23:25
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    $\begingroup$ Anything can be an axiom; what makes origami notable is that the axioms describe a mechanical activity. My point was that the only way to construct the required crease by actually folding paper is to slide the paper until the points match the lines. If this is allowed, it's not fair to say that origami can construct points that C&S cannot. $\endgroup$ – Ryan Reich Jun 18 '17 at 17:31
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    $\begingroup$ @RyanReich: I added a note to the post re axioms. $\endgroup$ – Joseph O'Rourke Jun 19 '17 at 0:52
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because you learn more when you build something with your hands.

The human brain is not a computer. It learns by training, by doing, not by listening or watching or being given a long complicated set of correct steps.

That said, I don't really get the point of geometry and proof exposure. Seems like a diversion from the algebra through calculus track. But that is a different issue.

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There is an idiom in my country.

It depends on how you handle a fool and scissors.

This mean

Everything comes in handy when used right.

and how you handle a scissors has a bit difficult. Even if it's a collapsed scissors, it works by how to use power and method. Therefore it is useful that everyone know how to use some tools. Also it is fundamental work to create image with compass and straightedge on geometry, and that makes us clever, since the geometrical algorithm is some complex. Even now the easy shapes which can be created by those tools are targets for present mathematicians.

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I took drafting and CAD in HS. As a carpenter, I've used my drafting knowledge here and there. The only thing CAD has helped me do is build Doom game levels.

That's my answer to the title, because we never used anything else but a pencil in geometry. As to why you teach a student anything that a tool can do for you, it's to understand the concept; how the ins-and-outs of it all work. That's the real reason, it's not, 'But what if you don't HAVE a calculator?'

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I would like to express here a personal complaint, which will also answer your question: as a child I learnt to work with a compass and straightedge, but there was a great drawback: I learnt that a compass is used to draw circles.

This, obviously, is right, but there also is another use of the compass, which is to estimate a distance between two points: this can be seen in movies about ancient ships, where the captain holds a ruler in one hand, and uses the compass to measure multiples of a certain distance along that ruler.

As I had not learnt to use the compass in that way, I always thought that a circle is something round, and during later education, I first needed to pass via that "round" comprehension before realising that the definition of a circle is a list of points at an equal distance from a central point.

If I had learnt to use the compass in two ways (drawing a circle and measuring distances), I might have realised earlier that a circle is a set of equidistant points directly, without needing to pass via the "round" comprehension, which is for a student something which reduces the understanding of the matter.

Now to answer your question: if you don't use a compass but instead you use a computer program to draw circles and arcs, how will you then pass the most important property of a circle to your students: not the fact that a circle is round but the fact that it consists of equidistant points?

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    $\begingroup$ Not just ancient ships. You should still learn such techniques in navigation today (and apply them - e.g., even with GPS support, courses are still drawn on paper maps, and be it as backup or a kind of "blackbox" that would survive sinking better than any electronics) $\endgroup$ – Hagen von Eitzen Jun 16 '17 at 5:52
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In addition to other good observations in comments and answers, I think (based on my own arc, and on observations of many others) that the physical/tactile issues about "lines and circles" do indeed provide a physically reasonable, but explicitly limited, way to interact with "geometric reality" as well as "axiom-generated/axiom-mediated reality". That is, in particular, I think this is not only an artifact of history, but a fairly natural product of human interaction with the geometric world, as well as humans' interaction with the formality of first-order predicate logic ... sorts of things.

Sufficiently able kids immediately observe that the constraints are a bit artificial, yet interesting. This does echo the 2,000-year-old attitude, too.

But/and of course the limitations of simple/natural things are of meta-interest in themselves...

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The point of compass-and-straightedge constructions is for the students to get experience reasoning about axiomatic systems. The accuracy of the actual drawings is basically irrelevant (as long as it's not so sloppy that it impedes visualization) — the point is that they can prove the accuracy of the idealized constructions.

This method of axiomatic construction was first studied for historical reasons that perhaps aren't as relevant now — having to do with the "synthetic" style of axiomatic geometry used by the Ancient Greeks, as opposed to the "analytic" style more popular in modern mathematics (e.g., Cartesian coordinates). However, geometric constructions can still serve useful educational purposes now, because it's a simple axiomatic system that can be a good way for students to learn to connect geometric intuition with careful, precise reasoning.

For example, once students have learned how to construct bisections of angles, one can ask them to try to trisect an angle, and use this to introduce the concept of an impossibility proof. This could also be used to illustrate how subtle changes to axioms can have a major impact on what's provable/constructible (like how doubling the cube is impossible with compass and straightedge, but becomes possible as soon as you can mark lengths on the straightedge).

These are just a few specific ideas that came to mind — the general idea is to use the constructions as a tool to teach logical and geometric reasoning.

For a case study in how not to teach geometric constructions — which may highlight some pitfalls to avoid — see Schoenfeld 1988, 'When Good Teaching Leads to Bad Results: The Disasters of "Well-Taught" Mathematics Courses'.

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    $\begingroup$ +1 Great article link, thanks for that. $\endgroup$ – Daniel R. Collins Jun 14 '17 at 2:01
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    $\begingroup$ The experience gained by doing this by hand also helps you later, when you do construction in software, to spot any unreasonable results that the software came up with because of bugs or rounding errors. Without the experience of doing it by hand you'd never suspect anything. $\endgroup$ – Sumyrda Jun 14 '17 at 5:00
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    $\begingroup$ The question didn't talk specifically about compass-and-straightedge constructions, just using these as tools; I used compass and straightedge in school geometry to draw pictures, and we didn't do compass-and-straightedge constructions that much. This answer also doesn't mention at all why using the physical tools is considered better than a drawing program. Acutally, there exist many programs for studying straightedge-and-compass constructions. $\endgroup$ – JiK Jun 15 '17 at 13:32

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