cayley-dickson

implements Cayley-Dickson construction to produce a sequence of algebras over a field

Installation | Usage : Real > Complex > Quaternion > Octonion > Sedenion | License

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Installation

With npm do

npm install cayley-dickson

Usage

Every code snippet below it is intended to be contained in a single file.

Define real operators, see also algebra-ring. Note that you could use any operators definition, for example using a big numbers lib.

const real = {
  zero: 0,
  one: 1,
  equality: (a, b) => (a === b),
  contains: (a) => (typeof a === 'number' && isFinite(a)),
  addition: (a, b) => (a + b),
  negation: (a) => -a,
  multiplication: (a, b) => (a * b),
  inversion: (a) => (1 / a)
}

Import cayley-dickson.

const iterateCayleyDickson = require('cayley-dickson')

Now you can use Cayley-Dickson constructions to build algebras. Every iteration doubles the dimension. Let’s take a trip through Cayley-Dickson algebras.

Real

start from here

Well, iteration 0 gives the common Real numbers. The result is just the return value of the algebra-ring function, nothing really exciting.

// Real numbers.
const R = iterateCayleyDickson(real, 0)

R.equality(2, 2) // true
R.disequality(1, 2) // true
R.contains(Math.PI) // true
R.notContains(Infinity) // true
R.addition(1, 2) // 3
R.subtraction(1, 2) // -1
R.negation(2) // -2
R.multiplication(-3, 2) // -6
R.division(10, 2) // 5
R.inversion(2) // 0.5

Complex

a beautiful plane

First iteration gives Complex numbers, they are a field like the Real numbers.

// Complex numbers.
const C = iterateCayleyDickson(real, 1)

C.equality([1, 2], [1, 2]) // true
C.disequality([1, 2], [0, 1]) // true
C.contains([Math.PI, 2]) // true
C.notContains(1) // true
C.addition([1, 2], [-1, 2], [2, 2]) // [2, 6]
C.subtraction([1, 1], [2, 3]) // [-1, -2]
C.negation([1, 2]) // [-1, -2]
C.multiplication([1, 2], [1, -2]) // [5, 0]
C.division([5, 0], [1, 2]) // [1, -2]
C.inversion([0, 2]) // [0, -0.5]
C.conjugation([1, 2]) // [1, -2]

Quaternion

here you loose commutativity

Second iteration gives Quaternion numbers, usually denoted as ℍ in honour of sir Hamilton. They are used in computer graphics cause rotations are far easier to manipulate in this land.

Let’s check the famous formula for Quaternion multiplication ijk = i² = j² = k² = -1

ijk-1

// Quaternion numbers.
const H = iterateCayleyDickson(real, 2)

const minusOne = new H([-1, 0, 0, 0])
j
const i = new H([0, 1, 0, 0])
const j = new H([0, 0, 1, 0])
const k = new H([0, 0, 0, 1])

H.equality(H.multiplication(i, i), minusOne) // true
H.equality(H.multiplication(j, j), minusOne) // true
H.equality(H.multiplication(k, k), minusOne) // true

// ijk - 1 = 0
H.subtraction(H.multiplication(i, j, k), minusOne) // [0, 0, 0, 0]

Octonion

here you loose associativity

Third iteration gives Octonion numbers. A byte could be seen as an octonion of bits, which should define a new kind of bit operator.

// Octonion numbers.
const O = iterateCayleyDickson(real, 3)

const minusOne = [-1, 0, 0, 0, 0, 0, 0, 0]

const i1 = [0, 1, 0, 0, 0, 0, 0, 0]

O.equality(O.multiplication(i1, i1), minusOne) // true

O.conjugation([1, 2, 3, 4, 5, 6, 7, 8]) // [1, -2, -3, -4, -5, -6, -7, -8]

Sedenion

hic sunt leones

Fourth iteration gives Sedenion numbers, that are out of my scope sincerely. They are not a division ring, there are elements that divide zero 😱.

// Sedenion numbers.
const S = iterateCayleyDickson(real, 4)

License

MIT