THIS IS MATH: Beautiful Geometry

Since this is still April, I will direct you back to the Math Awareness Month Calendar to the window marked The Beautiful Geometry of Crop Circles. You can use a compass and ruler to make beautiful geometric patterns and you can use other media as well. Many of you probably have already done this using a Spirograph.

To find out more about the connection between art and geometry, I will point you to Beautiful Geometry. Eli Maor, who is a mathematician, and Eugen Jost, who is an artist, teamed up to illustrate 51 geometric proofs and assorted mathematical curiosities.

Let’s start with one that most people know about—the Pythagorean theorem or a2 + b2 = c2. No one knows exactly how many proofs there are but Elisha Loomis wrote a book that includes 367 of them. The following illustration is a graphical statement of the theorem that if you draw a square on each of the three sides of a triangle, you will find that the sum of the areas of the two small squares equals the area of the big one.

pythagorean 1

If you look at the colorful figure below by Eugen Jost, you will see something similar, but much more interesting to look at. The figure above is a 30, 60, 90 degree triangle whereas the one below is a 45, 45, 90 degree triangle.

Plate 5 NEW

25 + 25 = 49, Eugen Jost, Beautiful Geometry

Using the Pythagorean formula, we know that 52 + 52  should equal 72. Now this means that

52  + 52 = 72

25 + 25 = 49

I think we all know that is just not true, yet we know that the formula is correct. What is going on here? It seems that the artist is having a bit of fun with us. Mathematics must be precise but art is not bound by the laws of mathematics. See if you can figure out what happened here.

 

Where’s the Math?

We know that there are at least 367 different proofs for the Pythagorean theorem but the most famous of them is Euclid’s proof. Eli Maor will walk you through it below, and, he will not try to trick you.

06-02_maor_fig

 

Important Note: We are going to assume you agree that all triangles with the same base and top vertices that lie on a line parallel to the base have the same area. Euclid proved this in book I of the Elements (Proposition 38).

Before he gets to the heart of the proof, Euclid proves a lemma (a preliminary result): the square built on one side of a right triangle has the same area as the rectangle formed by the hypotenuse and the projection of that side on the hypotenuse. The figure above shows a right triangle ACB with its right angle at C. Consider the square ACHG built on side AC. Project this side on the hypotenuse AB, giving you segment AD. Now construct AF perpendicular to AB and equal to it in length. Euclid’s lemma says that area ACHG = area AFED.

To show this, divide AFED into two halves by the diagonal FD. By I 38, area FAD = area FAC, the two triangles having a common base AF and vertices D and C that lie on a line parallel to AF. Likewise, divide ACHG into two halves by diagonal GC. Again by I 38, area AGB = area AGC, AG serving as a common base and vertices B and C lying on a line parallel to it. But area FAD = 1⁄2 area AFED, and area AGC = 1⁄2 area ACHG. Thus, if we could only show that area FAC = area BAG, we would be done.

It is here that Euclid produces his trump card: triangles FAC and BAG are congruent because they have two pairs of equal sides (AF = AB and AG = AC) and equal angles ∠FAC and ∠BAG (each consisting of a right angle and the common angle ∠BAC). And as congruent triangles, they have the same area.

Now, what is true for one side of the right triangle is also true of the other side: area BMNC = area BDEK. Thus, area ACHG + area BMNC = area AFED + area BDEK = area AFKB: the Pythagorean theorem.

 

 

This post is part of a series, explore additional posts here<< THIS IS MATH: Magic Squares, Circles, and StarsTHIS IS MATH: Wake up and Smell the Functions >>