hypercomplex

Changelog

[hypercomplex "0.0.4"]

A hypercomplex number library written in Clojure.

Includes Cayley-Dickson construction for generating and working with hypercomplex algebras.

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

Usage

• hypercomplex.core for operators on or between hypercomplex numbers: times, plus, minus, neg, c (conjugate), scale, norm, inv, and mag.
• hypercomplex.cayley-dickson-construction for constructing hypercomplex numbers: complex, quaternion, octonion, sedenion, pathion, and n-hypercomplex for constructing higher-order algebras of arbitrary order, provided it is a power of 2.
• All constructions can take an :impl, which can be either :plain, or :apache, which represent either pure Clojure or org.apache.commons.math3.complex.Complex implementations to be used under the hood for constructing the hypercomplex numbers. Hypercomplex numbers with different underlying implementations are not interoperable (TODO).

An example of creating complex numbers, quaternions, performing multiplication, and checking equality as a result of associativity:

(:require [hypercomplex.core :refer :all]
          [hypercomplex.cayley-dickson-construction :refer
            [complex quaternion octonion sedenion pathion n-hypercomplex]])

;create a complex number using pure Clojure impl:
(complex {:a 1.1 :b 2.2 :impl :plain})
; => #hypercomplex.core.Complex2{:a 1.1, :b 2.2, :order 2}

;create a complex number using Apache Commons Complex2 impl:
(complex {:a 1.1 :b 2.2 :impl :apache})
; => #hypercomplex.core.Complex2Apache{:a 1.1, :b 2.2, :order 2, :obj #object[org.apache.commons.math3.complex.Complex 0x3249a1ce (1.1, 2.2)]}

;similar for other hypercomplex numbers:
(quaternion {:a 1 :b 2 :c 3 :d 4 :impl :plain})
; => #hypercomplex.core.Construction{:a #hypercomplex.core.Complex2{:a 1, :b 2, :order 2}, :b #hypercomplex.core.Complex2{:a 3, :b 4, :order 2}, :order 4}

;example passing test case:
(is
  (= (times
       (quaternion {:a 1 :b 2 :c 3 :d 4})
       (times (quaternion {:a 8 :b 7 :c 6 :d 5})
              (quaternion {:a 9 :b 10 :c 11 :d 12})))
     (times
       (times
         (quaternion {:a 1 :b 2 :c 3 :d 4})
         (quaternion {:a 8 :b 7 :c 6 :d 5}))
       (quaternion {:a 9 :b 10 :c 11 :d 12}))))

An example of several other operators and constructions:

(is 
  (= (quaternion {:a 1.5 :b 1.5 :c 1.5 :d 1.5})
     (scale 
       (quaternion {:a 1 :b 1 :c 1 :d 1})
       1.5)))
       
(is (= 32
       (norm (pathion {:a 1 :b 1 :c 1 :d 1 :e 1 :f 1 :g 1 :h 1
                       :i 1 :j 1 :k 1 :l 1 :m 1 :n 1 :o 1 :p 1
                       :q 1 :r 1 :s 1 :t 1 :u 1 :v 1 :w 1 :x 1
                       :y 1 :z 1 :aa 1 :bb 1 :cc 1 :dd 1 :ee 1 :ff 1}))))       

An example of creating much higher order algebras using pure Clojure implementation, :plain:

(n-hypercomplex [0 1 2 3 4 5 6 7] :plain)
=> 
  #hypercomplex.core.Construction
  {:a #hypercomplex.core.Construction
      {:a #hypercomplex.core.Complex2
          {:a     0
           :b     1                                                                     
           :order 2}
       :b #hypercomplex.core.Complex2
          {:a     2
           :b     3
           :order 2}
       :order 4}
   :b #hypercomplex.core.Construction
      {:a #hypercomplex.core.Complex2
          {:a     4
           :b     5
           :order 2}
       :b #hypercomplex.core.Complex2
          {:a     6
           :b     7
           :order 2}
       :order 4}
   :order 8}

And a very high order hypercomplex number:

(n-hypercomplex (range 4096) :plain)
=>
  #hypercomplex.core.Construction
  {:a #hypercomplex.core.Construction
      {:a #hypercomplex.core.Construction
          {:a #hypercomplex.core.Construction
              {:a #hypercomplex.core.Construction
                  {:a #hypercomplex.core.Construction
                      {:a #hypercomplex.core.Construction
                          {:a #hypercomplex.core.Construction
                              {:a #hypercomplex.core.Construction
                                  {:a #hypercomplex.core.Construction
                                      {:a #hypercomplex.core.Construction
                                          {:a #hypercomplex.core.Complex2
                                               {:a 0 
                                                :b 1 
                                                :order 2}
                                           :b #hypercomplex.core.Complex2
                                               {:a 2 
                                                :b 3 
                                                :order 2}
                                           :order 4}
                                                ...
                                                :order 2}
                                           :order 4} 
                                       :order 8} 
                                   :order 16} 
                               :order 32} 
                           :order 64} 
                       :order 128} 
                   :order 256} 
               :order 512} 
           :order 1024} 
       :order 2048} 
   :order 4096}

Tests

lein test

License

Credits to:

• https://github.com/hamiltron/py-cayleydickson (MIT License)
• https://nakkaya.com/2009/09/29/fractals-in-clojure-mandelbrot-fractal/
• https://github.com/clojure-numerics/image-matrix/blob/master/src/main/clojure/mikera/image_matrix/colours.clj (EPL License)

Copyright © 2018 Octavian Geagla

Distributed under the Eclipse Public License either version 1.0 or (at your option) any later version.