Pythagorean triples

The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It can be written as an equation,

a2 + b2 = c2,

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

The theorem, which appears in Euclid Book I Proposition 47, has been called by Jacob Bronowski the most important theorem in the whole of mathematics. It enables us to define distance s between two arbitrary points in the plane as

s2 = (x2 – x1)2 + (y2 – y1)2

and leads on to the much more general metric of Riemannian geometry in curved space:

ds2 = gmn dxmdxn

Perhaps proved first by Pythagoras, it was known more than 1000 years earlier to the Babylonians and also to the Indians and Chinese.

Pythagorean triples

The most familiar application of the result is the 3:4:5-rule, known for millennia to builders and carpenters. If a rope with knots spaced one metre apart is used to form a triangle with sides 3, 4 and 5 metres, the sides of length 3 and 4 meet at a right angle. This is the simplest example of a Pythagorean triple, 32 + 42 = 52.

Left: Right-angled triangle with sides of length 3, 4 and 5, made from 12 matches. Right: General Pythagorean triangle.

Left: Right-angled triangle with sides of length 3, 4 and 5, made from 12 matches. Right: General Pythagorean triangle.

More generally, a Pythagorean triple is any set of any three whole numbers (a, b, c) that satisfy

a2 + b2 = c2

giving three integral sides of a right triangle. If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any integer k. If a, b and c have no common factors, the triple is called primitive. There are an infinite number of primitive triples.

Plimpton 322

A clay tablet originating from Mesopotamia around 1800 BC shows that the Babylonians were familiar with Pythagorean triples. Known as Plimpton 322, it contains columns of numbers in cuneiform script.

These have been interpreted in terms of right-angles triangles: two of the columns list the largest and smallest elements of a Pythagorean triple or, in other words, a and c, the lengths of the hypotenuse and shortest leg of a right-angled triangle.

The Babylonian clay tablet known as Plimpton 322.

The Babylonian clay tablet known as Plimpton 322.

Another column lists the square of the ratio of these values, effectively the squared cosecant of the angle opposite the shortest side. The order of the rows corresponds to increasing values of the angles, suggesting that the tablet may be an early trigonometric table, although this is controversial.

The tablet has 15 rows of numbers, but it is broken, and what additional entries might have been found on the original is unknown. John Conway and Richard Guy give a reconstruction with 34 rows, with sides corresponding to all regular numbers, that is, divisors of 60.

Generating Triples

Diophantus of Alexandria showed that, for any whole numbers m and n, the three numbers

a = 2mn, b = n2 – m2, c = n2 + m2

form a Pythagorean triple. It is trivial to prove that a2 + b2 = c2. Moreover, if mn is odd (m and n are of opposite parity) and m and n are coprime (no common factors) then the triple (a, b, c) is primitive, that is, a, b and c have no common factors.

We can tabulate the first few triples generated in this way:

m

n

a = 2mn

b = n2 – m2

c = n2 + m2

1

2

4

3

5

1

3

6

8

10

1

4

8

15

17

1

5

10

24

26

2

3

12

5

13

2

4

16

12

20

2

5

20

21

29

3

4

24

7

25

3

5

30

16

34

4

5

40

9

41

10

20

400

300

500

Nobody knows how the Babylonians calculated the values on the tablet, but they must have had a method something like the Diophantine algorithm.

 Trigonometry

Today, we can easily derive all the Pythagorean triples with a simple geometric construction. We map point Q on the unit circle by linear projection to point P on the y-axis. The equations of the circle and line are

x2 + y2 = 1      and      y = t x + t

and it is a simple matter to solve for the coordinates of (x,y):

x = (1–t2) / (1+t2)      and      y = 2t / (1 + t2) .

Mapping of rational points on unit circle to rational points on the real line.

Mapping between rational points P on the real line and rational points Q on the unit circle.

Clearly, if t is rational, i.e., the slope of the line is rational, then x and y are also rational. We have a mapping between rational points on the line and rational points on the circle. It is trivial to confirm that x2 + y2 = 1.

Conway and Guy call x / y = (1–t2) / 2t a Pythagorean ratio. Defining

a = (1+t2) x = (1–t2), b = (1+t2) y = 2t / (1 + t2) and c = (1+t2)

we get a Pythagorean triple. Writing t = m/n this is just a = n2 – m2, b = 2mn, c = n2 + m2.

We note that the expressions for x and y can be interpreted in trigonometric terms. Writing x = cos θ and y = sin θ we find that t = tan ½θ and get the half-angle formulae:

x = cos θ = ( 1tan2 ½θ ) / ( 1 + tan2½θ )

and

y = sin θ = 2 tan ½θ / ( 1 + tan2½θ ) .

These formulae are invaluable for proving trigonometric identities and evaluating integrals.

Sources:

Conway, John H and Richard K Guy, 1996: The Book of Numbers. Springer-Verlag, New York, ISBN 0-387-97993-X

Maor, Eli, 2007: The Pythagorean Theorem: a 4000-year History. Princeton Univ. Press. ISBN 9-780-691-14823-6

Plimpton 322: The Ancient Roots of Modern Mathematics (Half-hour video documentary). Kirby, Laurence (2011).

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