Geometry Constructions: Why?

Factual Questions
1

Geometry constructions were a neat little exercise, but looking back at it all…uh, for what purpose? In these constructions, you are not suppose to measure anything. I think such exercises are from days of old so Euclid could challenge his students. Do they have any practical application?

As an engineer, I have yet to see a co-worker NOT pull out a scale or tape measure. “Oh, I’ll just not measure anything…” No! Not gonna happen!
Even in my academics, these things never surfaced again. I guess it might serve to sharpen the mind’s eye regarding how points are reflected, but I can’t this of much else…

  • Jinx
2

Why not?

3

“Give the man three obols, since he must profit from what he learns.” -Euclid

Those ruler-and-compass constructions go back to the ancient Greeks (Euclid, et al). To them, geometry was an intellectual exercise and, arguably, a way of discovering Absolute Truth—as compared to other civilizations who considered geometry a practical matter useful for building pyramids and measuring farmland.

When you measure, there’s a limit to the precision of your measurements. But when you use a straightedge and compass, your copies are exact, and you can say that one thing is exactly congruent to something else—not necessarily in the drawings you actually put down on paper, but in the mathematical realities those drawings are supposed to respresent.

4

Of course they did if you took any higher mathematics. The Elements is the (somewhat shaky) foundation of the axiomatic method. You’ve obviously forgotten what came before the propositions: the postulates. Most importantly, the first three:

I. To draw a straight line from any point to any point.
II. To produce a finite straight line continuously in a straight line.
III. To describe a circle with any center and radius.

These are the ruler and compass constructions. Imagine the complexity of a system of axioms that would describe the use of a marked ruler, especially for a culture that considered discrete quantities and continuous quantities (numbers and magnitudes) to be two completely different things. You take for granted the natural numbers sitting on the 1cm, 2cm, 3cm… marks within the real line of the ruler, but this wasn’t so clear to the ancient Greeks.

5

I’m an artist, and a lot of my work involves geometry (plain and solid), or even trig. It’s not unusual for me to plan elements of my work with ruler, compass, protractor, etc. - or their computer equivalents. I think the difference is that geometric constructions use both sides of your brain; measuring does not.

6

In addition to their role in illustrating the axiomatic foundations of geometry, constructions also appear in the study of abstract algebra, where the insolubility of the geometric problems of antiquity is proved.

7

Actual diagrams can only take you so far. Suppose you and I both draw a square with sides 100 mm long, and we measure the diagonals of our squares. You get 141 mm, and I get 142 mm. So what would be the length, if we weren’t limited by accuracy of rulers, thickness of lines, errors in drawing, etc.? Well, we can show that it’s an infinite decimal, starting with 141.421356, i.e. the square root of 20 000. We may never be able to reach that accuracy, but isn’t it nice to know tat it can be easily calculated?

8

There is also the problem of the incommensurability of lengths, which came up recently in a thread about irrational numbers. To wit, it had been known quite a while before Euclid’s time that there is no possible length unit that will give integral lengths for both the side of a square and its diagonal. That means that if you want to get truly accurate results in mathematics, you have to work with unmeasured lengths—i.e., line segments in geometric constructions—rather than numerical values. As Giles points out, you need to have a pretty sophisticated theory of real and rational numbers before you can talk about numerical values of irrational numbers with a mathematical rigor that would have satisfied classical Greeks.

9

I think, in a very crude and far-off way, you’re trying to say that higher mathematics finds its roots in classical geometry from Ancient Greece. Perhaps very important in its day, it is nothing more than an academic exercise today. And, while I agree geometry gave us concepts such as two points define a line, that’s hardly a construction, IMHO. - Jinx

10

Well, there are all sorts of mathematical constructs that are still being deverloiped, that use an axiomatic method and use concepts such as “two points define a line”. There are such things as finite geometries, which might have an axiom such as:
“For every two points there exists one and only one line through those points.”
Then you give as an example a “geometry” consisting of only three “points” {a, b, c} and three “lines” {A, B, C}. A is the line {b, c}; B is the line {a, c}; and C is the line {a, b}. There: you have a “geometry”.

Note that it doesn’t have some things that you usually associate with geometries: it doesn’t have “angles” and it doesn’t have “distances”. However, it is interesting just to build on those axioms, and see what consequences follow. It’s also interesting to see what real life ojects correspond to such finite geometries. They might turn up in electric circuits, or in experimental design, for example.

11

No, what I’m saying has nothing to do with geometry. The axioms define what sorts of objects exist and are known and what tools are allowed to combine them into new ones.

The theory of natural numbers starts with

I. There exists an identified element “zero”
II. There exists an injective function which gives the successor of every number
III. Zero is not in the image of the successor function
IV. Given any set S satisfying I, II, and III, there exists a unique function from N to S preserving the zero and the successor function

Axiom II is the analogue of a geometric construction. Until you use the axioms to build more tools, that’s the only tool you’re allowed to use.

Euclidean geometry may be an academic exercise, but it’s by far the easiest instance af an axiomatic system with enough substance to hold the attention of a nonmathematician. The point is not to do the constructions, but to gain facility with the notion of rational, rigorous proof.

12

Geometric constructions also gave rise to complexity theory, which is a big deal nowadays.

13

I think you’re confusing proofs with a construction, perhaps?

Also, with constructions in general and playing the devil’s advocate: How would you know for sure a constructed perpendicular bisector is, indeed, (a) exactly 90 degrees and (b) the bisected halves are congruent? Wouldn’t Euclid have to have measured, or just use the looks-like theory? - Jinx :confused:

14

No, I’m not. The question of how many steps it takes to construct a given point from the points you already have was the earliest notion of algorithmic complexity.

He’d prove it.

15

Since your reply is less of a nonsequitur if you were responding to the subthread you’ve been carrying on with me:

No, I’m not. A construction is a proof. Have you actually read The Elements?

I.1: To construct an equilateral triangle on a given finite straight line.
I.2: To place a straight line equal to a given straight line with one end at a given point.
I.3: To cut off from the greater of two given unequal straight lines a straight line equal to the less.
Your example is I.10: To bisect a given finite straight line.

Read the books, do the proofs, then come complaining.

16

I studied Euclid in college, and our class went through every proof in Euclid’s Elements one by one. While today they may be seen as only cute little execises to challenge middle school students, I think that Thudlow is right that for Euclid and the early mathematicians they had a nearly religious significance. And I think it’s possible to see and understand that significance today.

Geometry is the original mathematics, a word that derives from the Greek mathema, that which is learnable. Meaning that, unlike other arts or skills, when you learn geometry, you understand it completely, because its conclusions are necessary. If you agree with the initial premises of a proof, and follow each of its steps, you must agree with the conclusion. To do otherwise is irrational. So studying geometry teaches you the principles of logic and rigorous modes of thought.

But it’s much more than that. You could study Aristotle’s logic and learn many of the same principles as they are applied to language. But the problem is that formal logic can be misleading or meaningless. We are all familiar with logical fallacies. And even if the logic is correct, it can be empty. To paraphrase a famous syllogism:

All smorks are jorny.
Fred is a smork.
Therefore Fred is jorny.

Logically true, but what does it mean?

Which is to say that when you apply logic to words, you are relying on the meaning of the words, which can be slippery. The logic may be impeccable, but the conclusion may end up being false or meaningless because of problems of definition.

The amazing thing about geometry is that all of the terms and objects it deals with are defined precisely and are real things that we can see and manipulate. Thus logic is applied directly to the real, visible world. It is the beginning of true science: understanding the real world through application of logic and (in a certain sense) experiment. And for this reason, geometry is not merely a formal exercise, but has practical applications at many levels.

The marvel of mathematics in general is that, although it at some level appears to be a only a creation of the mind, a matter of definitions and logical operations, it turns out to have these direct connections to the real world. Although IANAM, I understand that whole fields of theoretical mathematics have been developed with no known connection to the physical world, and subsequent discoveries in physics turn out to be described precisely by this previously theoretical math. (I’m an atheist, but this connection is as close as I get to thinking that there could be something like God. It’s practically mystical.)

It’s a shame that most middle- or high-school students aren’t taught, or don’t get, this insight into the larger meaning of geometry. But perhaps it is in that hope, and not merely because more advanced math is based on it, that geometry is still being taught.

17

It is?

A construction is an algorithm. There’s an associated proof that it’s correct, but the construction itself is not a proof.

18

I’m just starting a bout with On The Sphere And The Cylinder–wish me luck, I’ll need it–and from what I’ve seen so far the constructions were necessary to justify some of the steps of the proofs. Given a circle and two lines of different lengths, for instance, Archimedes needed to rely on the fact that he could divide the circle into segments that would lead in turn to circumscribed and inscribed polygons that differ by less than the two given lines.

19

Consider this more explicit semantics (which, by the way, appeals to intuitionists and classical mathematicians alike and offends only hardcore Platonists):
Prop I.10: Given a finite line segment, one can construct a line perpendicular to the segment which bisects the segment.
Proof: <algorithm to construct>

The construction algorithm is the proof that the construction is possible.

20

It’s a minor point, but I can’t agree. An algorithm is not a proof–they’re two different types of objects. The proof that the algorithm is correct is the proof that the construction is possible.