hypercomplex

Changelog

[hypercomplex "0.0.3"]

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.

An example of creating quaternion, 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]])

(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 FIXME

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