Pythagorean quadruple - AbsoluteAstronomy.com

A Pythagorean
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...

quadruple is a tuple

Tuple

In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

of integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s a, b, c and d, such that d > 0 and

, and is often denoted

. Geometrically, a Pythagorean quadruple

defines a cuboid

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

with side lengths |a|, |b|, and |c|, whose space diagonal

Space diagonal

In a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner...

has integer length d. Pythagorean quadruples are thus also called Pythagorean Boxes.

Parametrization of primitive quadruples

The set of all primitive Pythagorean quadruples, i.e., those for which gcd(a,b,c) = 1, where gcd denotes the greatest common divisor

Greatest common divisor

In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

of a, b, and c, is parametrized by,

where m, n, p, q are integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s. Thus, all primitive Pythagorean quadruples are characterized by the Lebesgue Identity

Relationship with quaternions and rational orthogonal matrices

A primitive Pythagorean quadruple

parametrized

Parametrization

Parametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....

corresponds to the first column of the matrix representation

of conjugation

by the Hurwitz quaternion

Hurwitz quaternion

In mathematics, a Hurwitz quaternion is a quaternion whose components are either all integers or all half-integers...

restricted to the subspace of

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

spanned by

, which is given by

where the columns are pairwise orthogonal

Orthogonality

Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

and each have norm d. Furthermore, we have

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

, and, in fact, all 3 × 3 orthogonal matrices with rational

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

coefficients arise in this manner.

Pythagorean quadruples with small norm

(1,2,2,3), (2,3,6,7), (1,4,8,9), (4,4,7,9), (2,6,9,11), (6,6,7,11), (3,4,12,13), (2,5,14,15), (2, 10, 11, 15), (1,12,12,17), (8,9,12,17), (1,6,18,19), (6,6,17,19), (6,10,15,19), (4,5,20,21), (4,8,19,21), (4,13,16,21), (8,11,16,21), (3,6,22,23), (3,14,18,23), (6,13,18,23), (9, 12, 20, 25), (12, 15, 16, 25), (2,7,26,27), (2,10,25,27), (2,14,23,27), (7,14,22,27), (10,10,23,27), (3,16,24,29), (11,12,24,29), (12,16,21,29)

Alternate parametrization

All Pythagorean quadruples (including non-primitives, and with repetition, though a, b and c do not appear in all possible orders) can be generated from two positive integers a and b as follows:

Let p be any factor of