# Guest Article

By Brudner, Harvey J. | T H E Journal (Technological Horizons In Education), August 1998 | Go to article overview

# Guest Article

Brudner, Harvey J., T H E Journal (Technological Horizons In Education)

A classic mystery locked in a 3,600-year-old Babylonian clay tablet has been solved! How did the Babylonians know the Pythagorean theorem a thousand years before the Greek mathematician and philosopher was born? For those who have forgotten their geometry, the Pythagorean theorem states: "The square of the hypotenuse of a right-angle triangle is equal to the sum of the squares of the two sides."

In October, 1978, I was on an American Airlines flight to Salt Lake City. The stewardess handed me Scientific American. Page 109 of an article contained something that would change my life. It was a Babylonian Clay Cuneiform Tablet dated about 1600 B.C. It is one of the oldest-known documents concerned with number theory. In disguised form, the tablet contains 15 sets of Pythagorean Triples; they are positive, whole numbers. The tenth set yields 4,961, 6,480, 8,161. This means 4,961 x 4,961 + 6,480 x 6,480 = 8,161 x 8,161,

The relationship, [a.sup.2] + [b.sup.2] = [c.sup.2] for whole numbers like (3,4,5), is called a Pythagorean Triple. Even with the formula, it has been very difficult to calculate exact triples. Note that a triple (a,b,c), of natural numbers is called a Pythagorean triple if [a.sup.2] + [b.sup.2] = [c.sup.2]. (3,4,5) is a Pythagorean triple, but (1,1,2) and (1,2,3) are not.

The clay tablet is called Plimpton #322 after its location in the Plimpton Collection of Columbia University. It is in disguised form.

For decades, mathematicians have been trying to determine how this could be. The solution depended on history, geography, and art -- as well as mathematics.

I reasoned that the City of Babylon, capital of Mesopotamia, was laid out as a rectangle, and rectangles can be turned into two triangles by drawing a diagonal line from one corner to another. Squares are equal-sided rectangles, and for the Babylonians who were concerned with area measurements of city blocks and properties, the formula was [b.sup.2] = [c.sup.2] -[a.sup.2] = (c-a) [multiplied by] (c+a), exactly corresponding to the theorem later established by Pythagoras.

From this reasoning, I deduced that the Babylonians were able to produce their perfect triples as 16 = 25-9 by rearranging a basic four-by-four square.

Could a triangle be converted to a rectangle? The simplest right triangle is (3,4,5). I added a Phantom Square; it's imaginary! Notice that 5 and 3 meet at a point and if I rotate the hypotenuse around this point, it suggests small squares below the (t-s) (t+s) rectangle. Adding in the four small squares that aren't there gives rise to the 2 times 2 Phantom Square.

If we had rotated s = 3 around the same point as t meeting s, a rectangle measuring only 6 units high would be created. It is perfectly balanced and symmetrical; it is 2 units wide. On top of it is a real square measuring 2 by 2; below it is a Phantom Square measuring 2 by 2. The real and imagined areas form a rectangle for t. All of these came from r = 4 arm of the original triangle.

Using the Phantom Square, we will now show that all Pythagorean triples can be built from multiples of the (3,4,5) triangle. Consider multiplying the 4 by 4 square by 2. If we apply the Babylonian rectangular rearrangement, we start out with a square: r = 8 = 64 square units. Let's divide it by 2 as we originally did with (3.4.5). We now produce a tall, thin rectangle. It is 32 by 2. To add symmetry, the needed "Phantom Square" is only 2 by 2. However, we see 2s = 30, and 2t = 34. We have produced a new triple from the arm of the triangle measuring 8 across. Since 2s = 30, s = 15, and since 2t = 34, t is 17 units. The produced triple is (15,8,17) and it is a Pythagorean triple.

We repeat the approach starting with the r = 8 square, but this time we divide by 4. We produce a rectangle 4 by 16. We again need a "Phantom Square" for the required balance; it is 4 by 4. Here we see 2s = 12, and 2t = 20; therefore s = 6 and t = 10. …

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