Pythagorean Triples--The basics, where they came from and how they are used, some examples and the basic equations

Pythagorean Triples--The Basics

by Richard Brown

Starting around 2500 BC with the Sumerians of the Tigris and Euphrates river basin of the Middle East, humans have expressed an organized interest in right triangles expressed as sides of integer units of length. The most famous example of these is is the Plimpton 322 cuneiform tablet found in the Columbia University collection(See an interesting article by Dr. David Joyce a mathematician at Clark University). I was introduced to this article on the internet by Karen Joseph of the main library of Charleston County in 1997. Since then I have done extensive research on Pythagorean triples both in the literature and on the internet. Dr. Joyce's paper was an inspiration for much further work, some of which is described on these web pages.

Lets look at the first seventeen triples so we can see what they look like:

Exhibit I. The First 17 Primitive Pythagorean Triples
Number	Side A	Side B	Diagonal	Comments
1	3	4	5	The First Triple
2	5	12	13
3	15	8	17
4	7	24	25
5	21	20	29
6	35	16	37
7	9	40	41
8	45	28	53
9	11	60	61
10	63	16	65	The First Duplicate
11	33	48	65	Diagonals
12	55	48	73
13	13	84	85
14	77	36	85
15	39	80	89
16	65	72	97
17	99	20	101

Exhibit Notes:
	Side A values are odd
	Side B values are multiples of 4
	Diagonals are all 4N + 1 values; N positive

The beginnings

Over ninety nine percent of the world's knowledge of Pythagorean triples is embodied in the first triple: 3,4,5. Even today carpenters and surveyors call the "rule of 3 4 5" the right angle rule.

An interesting fact that I have just discovered is that the great pyramids of Ghiza in Egypt used an angle very close to the 3,5 angle of the 3,4,5 triangle for the slope of the Pyramid built around 3000 BC, additonal evidence that the 3,4,5 triangle was known and used at that time.

It is fairly easy to pick a time when knowledge of right triangles came about, walls of structures became straight and square, fields became divided into rectangles, cities and temples were laid out in orderly geometric patterns. These were first found in Egypt and Sumeria and were copied over much of the rest of civiliation.

The surveyor used lengths of rope with knots at unit length. A rope with 12 knots at equal lengths could lay out a corner of a field in Egypt after a flood. The right angle could then be verified by multiples of 3,4,5 to layout the entire field.

The stone mason would use a 3,4,5 measure to assure the walls of a structure were at right angles to each other. Straight walls were assured by a string with a weight. Level rows of stone or bricks were assured by a length of string with a bubble.

The carpenter used the 3,4,5 triangle to make his square and to square up large constructions. The measuring tape and the square are the common tools used by carpenters.

The right angle almost defines civilization, just as the agriculture defines culture. The right angle goes right to the beginning of civilization.

Mathematicians and Astronomers of Antiquity

While the 3,4,5 triangle was in everyday use by a wide range of craftsmen, the priests and scribes of the temple had more complex uses of the triple. The reponsibility of the temple mathematicians and astronomers was to develop calenders and predict future astronomical occurances just as solstices. The intellectual heavy-weights of ancient times were envolved in developing the Pythagorean triple far beyond its everyday use.

While the craftsmen were interested in the triple for its right angle use, the mathematicians and astronomers were looking at the angles on the other sides. The Plimpton 322 tablet mentioned above is an expression of such an ordering of simply constructed triangles.
Before there were numerical expressions of right angles in the Pythagorean triples, there were simple expressions of right triangles:

Exhibit II. The First Observation of Right Angles

| | X | | | | --X------------------------------X--- | | | | X | | Placing two sticks one on top of the other divides the plane into four parts, when these parts are equal, the four angles are equal and become right triangles. If the four parts aren't equal, the two opposite parts are equal but their adjacent sides aren't. So what every good craftsman does is measure equal distances out from the center on each leg, then measure the lengths of the diagonals and adjust until the lengths of two adjacent diagonals are equal.

After doing this many times, someone wondered if there were unit lengths that give a right triangle, empirically:

Exhibit III. Finding the First Pythagorean Triple
Side A	Side B	Diagonal
1	1	1.4	Trying Side/Side
1	2	2.2	Closer
1	3	3.2
2	3	3.6
2	5	5.4
1	1.7	2	Trying a Side/Diagonal
1	2.8	3
1	3.9	4
1	4.9	5
2	4.6	5
3	2.6	4
3	4	5	Eureka!!

Obviously, after the first Pythagorean triple was found, and I am almost sure it was 3,4,5, the next triple was 6,8,10, then 9,12,15,and so on, multiples of the original. Then 5,12,13 was found by trial and error, much like the 3,4,5. The list of triples in Exhibit I could all have been found in such a way without any knowledge of the Pythagorean Theorem. Even in mathematics, the empirical "hit and miss" often precedes theoretical!

Now Pythagorean triples were known at least 1300 years before the expression of the Pythagorean theorem. Most likely a rule preceded the theorem and the rule was discovered in the temples of Sumeria. It is the rule of the integer right angles found in Euclid's Elements, Book X, but known much earlier:

"For every pair of numbers, one odd and one even with no common factor there exists a unique primitive Pythagorean triple."

Exhibit IV lists the same triples as Table I with these special generators and the side angles of the triangles and the radius of the inner circle of the triangles. In general, from the clay tablet of Sumeria and Babylon came rules that became more general mathematical principles.

Exhibit IV. Primitive Pythagorean Triples with Generators, Side Angles,and Incircle Radii
Number	Side A	Side B	Diagonal	Generators(u,v)		Small Angle	Radius
1	3	4	5	1	2	36.8698	1
2	5	12	13	2	3	22.6198	2
3	15	8	17	1	4	28.0725	3
4	7	24	25	3	4	16.2602	3
5	21	20	29	2	5	43.6029	6
6	35	12	37	1	6	18.9247	5
7	9	40	41	4	5	12.6803	4
8	45	28	53	2	7	31.9808	10
9	11	60	61	5	6	10.3888	5
10	63	16	65	1	8	14.2501	7
11	33	56	65	4	7	30.5102	12
12	55	48	73	3	8	41.1121	15
13	77	36	85	2	9	25.0577	14
14	13	84	85	6	7	8.7974	6
15	39	80	89	5	8	25.9892	15
16	65	72	97	4	9	42.0750	20
17	99	20	101	1	10	11.4212	9

From Tables of Numbers to Equations

The right triangle rule was discovered for triangles with integer sides:

side A = v^2-u^2
side B = 2*u*v
diagonal = v^2 + u^2

I think of these rules as the "inner rules" of right triangels and the Pythagorean Theorem as the "outter rule":

sideA^2 +sideB^2 = diagonal^2
Amazingly, and I have never heard it said about Pythagorean triples, is that they are individual expressions of a 4th degree identity, by substitution we have:
(u^4-2u^2v^2+v^4)+(4u^2v^2) = (u^4 + 2u^2v^2 + u^4)

There is no general solution to a fifth order equation, the general solution of Pythagorean triples fits comfortably within the limits of mathematics! For every u and v, one odd and one even with no common factors, there is a unique primitive Pythagorean triple! Primitive Pythagorean triples also have unique side angles. And the sides and diagonals have distincitive expressions. For the next steps from cunieform tablets to modern mathematical number theory go to the next level!

Go to the next Triples level Pythagorean _Triples_II
Or, return to Table_of_Contents
Last Updted 6/14/1998