Pythagorean theorem

The Pythagorean theorem is one of the fundamental theorems of two-dimensional geometry. It states that for all right-angled triangles (cf. Figure 1), the side lengths are in a certain ratio to each other: If you form a square over each of the three sides, the sum of the areas of the two smaller squares (over the cathetae $a$ and $b$) is exactly as large as the area of the large square (over the hypotenuse $c$). Expressed as an equation, the Pythagorean theorem is thus simply $a^2+b^2=c^2$.

Figure 1: A right triangle

Pythagoras himself was born around 570 BC on the Greek island of Samos, the son of a goldsmith. After familiarizing himself with the knowledge of the time, especially Babylonian and Egyptian science, through studies with learned priests and travels, Pythagoras founded his own school with which he wanted to lead his students to “inner purity”. Pythagoras prescribed mathematica to his students and had them focus on arithmetic, geometry, and musicology.

Knowledge of the aspect ratios on a right triangle was known to Babylonian scholars as early as 1800 B.C. and to India by the 6th century B.C. at the latest. The role Pythagoras played in teaching the theorem later named after him and in his mathematical proof is not undisputed (see Figure 2).

Figure 2: Illustration of the Pythagorean theorem

And now … the mathematics of it:

Today, a large number of proofs of the Pythagorean theorem are known. One of them is based on the following consideration (see figures 3a and 3b):

Figure 3a
Figure 3b

The outer square in figure 3a has the side lengths $a+b$ and thus the area $A=(a+b)^2$. But this area is also obtained by adding the areas $A_1=4\cdot\frac{ab}{2}$ of the four right-angled triangles with side lengths $a$ and $b$ and the area $A_2=c^2$ of the twisted square of side length $c$ inscribed in the large square (cf. Figure 3b). So now $$A=(a+b)^2=a^2+2ab+b^2=4A_1+A_2=2ab+c^2,$$

from which follows the desired identity $a^2+b^2=c^2$ by subtracting $2ab$ on both sides.

The exhibit in MATHEMATICS ADVENTURE LAND shows the correctness of the Pythagorean theorem in (another) elementary way (see figures 4a and 4b below):

Figure 4a
Figure 4b

By folding the surface $B$ (to the left) and the surface $C$ (to the right) around the pivots $(I)$ and $(II)$, respectively, up to the stop (cf. Figure 4a), one obtains from a square with the side length $c$ (i.e. the area $c^2$) two squares lying next to each other with the areas $a^2$ and $b^2$ (cf. Figure 4b).


Finally something literary

We conclude with a — mathematically not to be taken quite seriously — literary treatment. The German poet Adalbert von Chamisso (1781–1838) describes the legendary sacrifice that Pythagoras is said to have offered to the gods after he had discovered “his” theorem:

From the Pythagorean Theorem

The truth, it exists in eternity,

When first the stupid world has recognized its light;

The theorem named after Pythagoras

Is valid today, as it was in his time.

A sacrifice has Pythagoras consecrated

To the gods who sent him the ray of light;

It announced, slaughtered and burned,

One hundred oxen showed his gratitude.

The oxen since the day when they scent,

That a new truth is revealed,

Raise an inhuman roar;

Pythagoras fills them with horror;

And powerless to resist the light

They close their eyes and tremble.

(quoted from Project Gutenberg, All the Poems of Adelbert von Chamisso)


Literature

[1] Dewdney, A.K.: Reise in das Innere der Mathematik, Berlin, 2000.

[2] Fraedrich, A.M.: Die Satzgruppen des Pythagoras, Mannheim, 1995.

[3] Maor, E.: The Pythagorean Theorem: A 4,000-year History, Princeton, 2007.

[4] Schupp, H.: Elementargeometrie, Stuttgart, 1977.

[5] Singh, S.: Fermats letzter Satz, München, 2000.

[6] v. Wedemeyer, I.: Pythagoras, Weisheitslehrer des Abendlandes, Ahlerstedt, 1988.