Some time ago I looked at questions around trisecting an edge by compass and straightedge, i beg your pardon entailed discussing the rules for such constructions. Us left open an additional common question: Why room such constructions important, and why execute we usage those particular tools? This probably isn’t defined as regularly as it need to be.

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Why does it matter? Axioms

I’ll begin with this concern from 1998:

The prominence of Geometry ConstructionsI to be doing a report on constructions in geometry. Ns would favor to recognize why constructions are important. I realize that they an obstacle us come use different tools yet there must be more to it then that. So i was wonder if you could give me more of a reason why constructions room so important?Since plenty of things we ask children to execute are mainly to get them offered to particular ways to use their hand or bodies, that is understandable the Kel would suppose that us teach build just because compasses and straightedges are worth knowing just how to use. Yet that isn’t yes, really it. Ultimately, it’s since our minds are worth knowing how to use!

I answered:

Hi, Kel. That"s a an excellent question. We have tendency to teach it the end of tradition, and also forget to think around why it"s worth doing!Certainly learning just how to usage the devices is useful. Few of the techniques are helpful in building and construction (of buildings, furniture, and also so on), though in truth sometimes over there are less complicated techniques building contractors use that we forget come teach. But I think the main reason for finding out constructions is their close link to axiomatic logic. If girlfriend haven"t heard the term, I"m talking about the entirety idea that proofs and also careful thinking that we regularly use geometry to teach.I’ve provided compass constructions once I aided renovate a church building; yet then the “compass” was a length of string. It to be the idea behind it the really mattered.

Euclid, the Greek mathematician who wrote the geometry text supplied for centuries, stated many of his theorems in regards to constructions. His axioms are carefully related come the tools he provided for construction. simply as axioms and also postulates let us prove everything with a minimum the assumptions, a compass and also straightedge let us construct whatever precisely v a minimum that tools. There room no approximations, no guesses. Therefore the an abilities you require to figure out exactly how to construct, say, a square without a protractor, are very closely related come the thinking skills you have to prove theorems around squares.A building is, in ~ root, a theorem: If you monitor this succession of steps, the result will necessarily be the object you claim to be creating, such as the bisector of one angle, or a triangle that meets particular requirements. So discovering to architecture a building and construction is practice in “constructing” geometrical proofs. Exercise in building is not mostly practice in using your hands, but your mind.

I close up door my quick answer by referring to the an initial proof in Euclid’s Elements, Proposition I.1:

Proposition 1

It is forced to build an it is intended triangle ~ above the directly line AB.

Describe the circle BCD with center A and radius AB. Again describe the circle ACE with center B and radius BA. Join the straight lines CA and CB from the point C at i m sorry the circles cut one another to the points A and B.

Now, because the point A is the center of the circle CDB, therefore AC equals AB. Again, because the point B is the center of the circle CAE, therefore BC equals BA.

But AC was proved equal to AB, therefore each of the directly lines AC and BC equals AB.

And points which same the exact same thing additionally equal one another, therefore AC also equals BC.

Therefore the three directly lines AC, AB, and BC equal one another.

Therefore the triangle ABC is equilateral, and it has been constructed on the offered finite straight line AB.

This to organize is in reality a construction. Keep in mind that the procedures involve making circles (with a compass) and also making currently (with a straightedge); and also at the end he place “Q.E.F.”, short for “Quod erat faciendum”, Latin because that “Which to be to be done”. (Euclid, of course, actually offered Greek, “ὅπερ ἔδει ποιῆσαι”, “hoper edei poiēsai”.)

Why no rulers and protractors? Axioms

In 2002, we obtained a similar question from a teacher, that referred to as for a little much more detail on exactly how the axioms (Euclid’s Postulates) relate to the compass and straightedge:

Why Straightedge and also Compass Only?My Geometry students want to recognize why constructions can only be done using a straightedge and also a compass. They want to know why lock can"t simply measure a heat segment to copy the or usage a protractor to construct an angle. What"s the difference? We have actually searched our book and some internet sites include constructions, but to no avail.I referred ago to the ahead answer, then elaborated.

There space two methods that I have the right to see to explain the limit rules for constructions, which come to us indigenous the old Greeks:1. Castle are simply the rule of a game mathematicians play. There are many other methods to do constructions, but the compass and also straightedge were preferred as one collection of tools that make a construction challenging, by limiting what girlfriend are permitted to do, simply as sports restrict what you have the right to do (e.g. Touching yet not tackling, or tackling however no atom weapons) in stimulate to save a video game interesting. Various other tools might have been preferred instead; for example, geometric constructions have the right to be done making use of origami.Euclid can have began with any tools that wanted; yet a significant goal to be to border what could be done, as kind of a video game to check out how small we have the right to use, to do just how much.

For much more on axioms or postulates, view my collection in July 2018, beginning with Why go Geometry begin With Unproved Assumptions?

But it’s not simply a game; it’s the game:

2. They are the basis of one axiomatic system, through the goal of ensuring that geometry is constructed on a hard foundation. Euclid want to begin with as few assumptions as possible, for this reason that every one of his conclusions would be particular if friend just accepted those few things. So he detailed five postulates (in addition to some other assumptions even an ext basic); I"ve taken this from the reference provided in my answer above: Postulate 1. to attract a directly line from any allude to any type of point. Postulate 2. to develop a finite right line repeatedly in a directly line. Postulate 3. to explain a circle with any type of center and radius. Postulate 4. That all ideal angles same one another. Postulate 5. That, if a directly line falling on two right lines renders the internal angles top top the exact same side less than two best angles, the two straight lines, if produced indefinitely, fulfill on that side top top which room the angles much less than the two best angles.He starts v the visibility of lines and also circles, climate adds only two extr facts. (His device is not quite complete, and extr axioms space now recognized to be necessary.)

The very first two postulates say that you have the right to use a straightedge: line it up through two offered points, and also draw the line between them, or heat it up through an existing segment, and also draw the line past it. That"s the an initial tool girlfriend are allowed to use, and also those room the only methods you are permitted to use it.As is often noted, you are not permitted to do various other things, like measure or copy a length by making point out on the straightedge. This is no just since Euclid wanted to save his devices clean! It’s because he want to minimization his assumptions, proving as lot as possible starting through as tiny as possible.

The 3rd postulate says you can use a compass to draw a circle, given the center and radius (or a suggest on the circle). The is the only way you are permitted to usage the compass; friend can"t, because that example, attract a one tangent to a line by adjusting that is radius till it _looks_ tangent, without understanding a details point the circle needs to pass through.The last 2 postulates relate to angles, and also are less associated with the construction process itself than v what girlfriend see once you are done.Again, the limitations are to minimization the assumptions, not since his compass was defective. I actually understated the border in this case. (More on that later.)

So really the two devices Euclid compelled for a building just stand for the assumptions he to be willing to make: if these two devices work, climate you can construct every little thing he talks about. For example, you can use these tools, in the prescribed manner, to construct a tangent to a provided circle v a given point; but it bring away some thought to find how to execute so (without just drawing a line the _looks_ tangent), and it takes numerous theorems to display that it yes, really works.Here we are back to the challenge! and the goal is not just to do something the looks right, however to have the ability to prove something.

Of food you can just measure up a heat or an angle, if her goal is just to do a drawing - and also usually that will certainly be much more accurate 보다 a complex compass construction! however when you use only the tools enabled in this game, you room actually play within one axiomatic system, getting a feel for just how proofs work. Friend are simultaneously playing a challenging game, and doing one of the couple of things in life the can provide you pure certainty: if these lines and circles were specifically what castle pretend to be (with no thickness, etc.), climate the point I construct would be exactly what I case it is. And also it"s that sense of certainty that the Greeks to be looking for.

Why walk the compass collapse? Axioms!

I didn’t mention over a distinct restriction ~ above the compass, which turns out come be completely theoretical. We gained a question around that in 2003:

Collapsible CompassI need to understand what a collapsible compass is and also what it is offered for. All I recognize is that as soon as you choose it increase from the paper, you shed your place.Again, i answered the question, keeping it brief:

The collapsible compass is not something the is "used"; rather, it to represent the fact that Euclid want to make as couple of assumptions (postulates, or axioms) at the basic of his proofs as possible. So quite than assume that it was feasible to move a heat around, maintaining the same size (as you might do through a real, fixed compass), or equivalently the you can attract a circle through a given center and also length, he assumed just that friend can attract a circle v a provided center and also through a given point. Climate he go on come prove the if you might do that, you can then construct a circle through a given radius, or relocate a line to a offered place: Collapsible Compass http://mathforum.org/library/drmath/view/52601.htmlThe recommendation is come a short answer that web links to the proposition ns am around to discuss.

Where ns quoted Euclid’s postulates above, it might look as if you can just set the compass to any type of radius you want, contradictory to what I’ve stated here: “ to define a circle with any kind of center and radius.” but the indigenous “radius” come Euclid go not describe a number, as we think of that today, but to a details segment! This is made explicit in the commentary to the aspects on Joyce’s site that I’ve described before, Postulate 3:

Circles were characterized in Def.I.15 and also Def.I.16 as aircraft figures with the residential or commercial property that there is a particular point, called the center of the circle, such the all straight lines native the center to the boundary are equal. The is, all the radii space equal.

The given data are (1) a suggest A to be the center of the circle, (2) another point B to be on the circumference of the circle, and (3) a plane in i beg your pardon the 2 points lie. …

Note the this postulate does not allow for the compass to be moved. The usual means that a compass is provided is that is is opened up to a provided width, climate the pivot is placed on the illustration surface, climate a one is drawn as the compass is rotated approximately the pivot. Yet this postulate go not permit for delivering distances. It is together if the compass collapses as shortly as it’s removed from the plane.

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Proposition I.3, however, provides a building and construction for moving distances. Therefore, the exact same constructions that can be made through a constant compass can additionally be made with Euclid’s collapsing compass.