The Pythagorean Theorem

Introduction

            a² + b² = c²  where a and b are the sides of a right triangle and c is the hypotenuse is the answer most students will give when asked to define the Pythagorean Theorem. Ask them to prove it and most can’t unless they are fairly advanced in mathematics. By and large, most anyone who has taken high school algebra has encountered this theorem without every really understanding it.

            In words, the Pythagorean Theorem states that the square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides._[1]

            The Egyptians, Babylonians and Chinese knew of the right triangle long before Pythagoras came along, and the Egyptians had devised a right triangle of rope with twelve evenly spaced knots for measurements. They knew that “3, 4 and 5 make a 90 degree angle and used the triangle of rope as an instrument of measurement:

            This paper will explore several aspects of the Pythagorean Theorem, from its history to how it found in art, how it relates to music and how it occurs in everyday life, whether people know it or not.

1. The History Of the Pythagorean Theorem

            It is not conclusively known that Pythagoras can be solely credited with his famous theorem, but it is known that he was the first to prove why a right triangle behaves the way it does.

            Born in Samos (now in Turkey) in about 582 B.C., Pythagoras came from a humble family. Various scenarios of his early life put him in Egypt, Babylonia, and Italy, where he founded his school that survived for a few hundred years.

            Pythagoras was not interested in solving mathematical problems; he focused on pure mathematics and geometric relationships._[3]

            Pythagoras and his Brotherhood (the Pythagoreans) traveled fairly extensively, and it is known that they visited and studied in Egypt. The Pythagoreans were a mystical group, and Pythagoras himself identified numbers with mysticism.

            It is possible that Pythagoras saw the knotted triangle in Egypt and went about assigning a numerical value to the geometry of the triangle. Nothing of his was written down, though, and while the theorem is attributed to him it could well have been one of his brethren or students that invented the theorem; since all in the Pythagorean school was community property, and since his name was the name of the school, we have come to know the famous theorem as it is today.

            How do we know that they Pythagorean Theorem was known before Pythagoras? So far, nothing has been shown in terms of an equation in relation to a right triangle before Pythagoras, but the right triangle was certainly known for its special properties, as can be seen in the figure below, which depicts the pattern of Oriental tiles in use at least one thousand years before Pythagoras:

            Looking at this pattern, it is easily determined that the proof of the theorem lies within the pattern itself; yet its proof is credited to Pythagoras.

            The Pythagorean Theorem is pure mathematics. It involves no irrational numbers but is found just about everywhere we look. The Pythagoreans apparently abhorred irrational numbers, and since Pythagoras’ motto was “all is number,” irrational numbers had no place within the school._[5]

            Since algebra had not yet been invented, the mathematicians in Pythagoras’ time used numbers only in terms of counting and geometry. Yet the Pythagorean Theorem is seen in the most profound occurrences of nature, architecture and music. One problem exists with the formula though…it only relates to integers, and therefore cannot fully translate itself into geometry as it is known today, although in plane geometry the theorem is undeniably valuable. Or can it?

1. Famous Proofs of the Pythagorean Theorem

            The Bride’s Chair

            This proof is a product of Euclid and while neither simple nor obvious, it remains the most famous of proofs for the Pythagorean Theorem_[6]

            Pythagoras’ proof was much more simple, involving dissection. His own proof states that if the areas of both squares are equal, and the sides of each square are denoted by a + b. The area of the square on the left is given by (a+b) ² + a ²  + b²   and the area of the square on the right is given by (a+b)²  + c² . Setting the areas equal to each other and subtracting equals yields the proof of the theorem.[7]

            Similarly, Basakara’s first proof used dissection in the 2^nd Century to prove the Pythagorean Theorem. This proof shows that the area of the square is the same no matter what final shape it takes:_[8] 

            One final proof shows the practicality of the Theorem; it is not directly historical, yet it is still extremely useful and displays the versatility of the Pythagorean Theorem:

III. Evolution Of the Pythagorean Theorem

            Given that the Pythagoreans shunned anything other than whole numbers, an entire universe of geometric applications in terms of describing harmony and unity was lost to them. While the Pythagorean Theorem was indispensable for building and measuring perfect squares and rectangles it could not stand alone in terms of the patterns of nature without revision and the introduction of irrational numbers. By applying the Pythagorean theorem to irrational numbers, a beautiful array of patterns emerges, as was initially discovered by Euclid. Other mathematicians have even managed to take the Theorem into three dimensions. Incredibly, the Pythagorean theorem as it was originally expressed, in whole numbers only, was limited. But using it as a platform for other geometries sparked an entire cosmology in mathematics and philosophy, demonstrating that the Theorem is fairly universal in its ability to be the springboard to other dimensions of mathematical expressions and mechanics.

            Here is one example of the use of irrational numbers plugged into the Pythagorean Theorem, creating a crude spiral: Applying the Pythagorean Theorem to an isosceles triangle with sides 1,1, we can begin to add new triangles, each with a hypotenuse that increases by using the first hypotenuse as the leg of the new triangle:[10]

Here is the second triangle added:

Continuing, we see the building blocks for the Golden Spiral:

            By viewing this diagram, it is easy to imagine how the Pythagorean theorem has been used for building spiral staircases, applying the Theorem in three dimensions. But it is also important to recognize how the Pythagorean theorem is hidden within innumerable other theorems, and by deduction it can be found even when it is not obvious. This was made possible by the innovative thinking of geometers after Pythagoras’ time, which acknowledged irrational numbers and seemed to see beyond the strict boundaries of Pythagorean thinking.

            Euclid greatly advanced geometry, including the Pythagorean theorem, resulting in the Golden Spiral (or Golden Ratio or Golden Mean), which uses the Pythagorean theorem as an integral part. The Golden Ratio was explosive in the exploration of natural occurrences, harmonics, architecture and physics. There is such beauty in the Golden Ration and so much was done with it in ancient times that it lends a particular appreciation to the Pythagorean Theorem; take the Golden Ratio apart and there is the right triangle, ready to expand again into something greater.

            Right triangles were known long before Pythagoras’ time, as has already been mentioned. But it was Euclidian thinking that took the Pythagorean theorem to a whole new level by applying the Theorem to irrational numbers and circular functions. Instead of stopping at the 3:4:5 right triangles, square roots were introduced that were not whole numbers.

            Below is a diagram of the beginnings of the Golden Section, beginning with the Golden Rectangle and the geometric construction of the Golden Section, both based on the Pythagorean theorem:

Bibliography 

Bulaevsky, Jacobo. “The History of Pythagoras and his Theorem.” Archytech Website. 2003, 23 February 2006, http://www.arcytech.org/java/pythagoras/history.html

Head, Angie. “The Pythagorean Theorem.” Angela Head’s EMT 668 & EMT 669 Page. University Of Georgia  Department of Mathematics Education. 1997, 25 February 2006, http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html

Lindauer, Heather. “The Pythagorean Theorem.” University Of Kentucky: Math Sciences Web Page. 2 December, 1998, 28 February 2006, http://www.ms.uky.edu/~lee/ma502/pythag/pythag.htm

Martin Gardner. “The Pythagorean Theorem.” Sixth Book of Mathematical Games from Scientific American. Charles Scribner’s Sons: New York, 1971. Chapter 16.

“Pythagoras.” Math Mania, Oracle ThinkQuest Library (n.d.), 27 February 2006, http://library.thinkquest.org/J0110961/pyth.htm

Regula, Timothy. “The Pythagorean Theorem.” The Perseus Project Website. Tuft’s University, 1995, 28 February 2006, http://www.perseus.tufts.edu/GreekScience/Students/Tim/Pythag’sTheorem.html

University Of Minnesota, “A Brief History of the Pythagorean Theorem,” Geometry Center website 1994 – 1998, 26 February 2006, http://www.geom.uiuc.edu/~demo5337/Group3/hist.html

Weisstein, Eric W. “Pythagorean Theorem.” From MathWorld–A Wolfram Web Resource. 1999 – 2006, 27 February 2006,  http://mathworld.wolfram.com/PythagoreanTheorem.html

            This figure (popularly known as “The Bride’s Chair) represents the visual aspect of the Pythagorean Theorem, which originated with Euclid (Heather Lindauer, 1998).

[1] Martin Gardner, 1971. Ch. 16, p. 152

[2] Jacobo Bulaevsky, 2003, http://www.arcytech.org/java/pythagoras/history.html

[3] “Pythagoras,” Math Mania, Oracle ThinkQuest Library (n.d.), http://library.thinkquest.org/J0110961/pyth.htm

[4] University Of Minnesota, “A Brief History of the Pythagorean Theorem,” Geometry Center website 1994 – 1998, http://www.geom.uiuc.edu/~demo5337/Group3/hist.html

[5] U of Minnesota Geometry Center.

[6] Eric W. Weisstein. “Pythagorean Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanTheorem.html

[7] Angie Head, 1997, University of Georgia Dept. Of Mathematics.

[8] Ibid.

[9] Timothy Reluga. “The Pythagorean Theorem.” The Perseus Project. Tufts University, 1995. http://www.perseus.tufts.edu/GreekScience/Students/Tim/Pythag’sTheorem.html

[10] Cathleen V. Sanders, 1997, 2005. “Right Triangles.” Connecting Geometry. Ch. 6, http://www.punahou.edu/acad/sanders/connectinggeometry/ch_06Pythag.html

[11] Museum Of Harmony and Golden Section, http://www.goldenmuseum.com/0202Geometry_engl.html

[12] ibid.