Hypercomplex

A Python library for working with quaternions, octonions, sedenions, and beyond following the Cayley-Dickson construction of hypercomplex numbers.

The complex numbers may be viewed as an extension of the everyday real numbers. A complex number has two real-number coefficients, one multiplied by 1, the other multiplied by i.

In a similar way, a quaternion, which has 4 components, can be constructed by combining two complex numbers. Likewise, two quaternions can construct an octonion (8 components), and two octonions can construct a sedenion (16 components).

The method for this construction is known as the Cayley-Dickson construction and the resulting classes of numbers are types of hypercomplex numbers. There is no limit to the number of times you can repeat the Cayley-Dickson construction to create new types of hypercomplex numbers, doubling the number of components each time.

This Python 3 package allows the creation of number classes at any repetition level of Cayley-Dickson constructions, and has built-ins for the lower, named levels such as quaternion, octonion, and sedenion.

Installation

pip install hypercomplex

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This package was built in Python 3.9.6 and has been tested to be compatible with python 3.6 through 3.10.

Basic Usage

from hypercomplex import Complex, Quaternion, Octonion, Voudon, cayley_dickson_construction

c = Complex(0, 7)
print(c)        # -> (0 7)
print(c == 7j)  # -> True

q = Quaternion(1.1, 2.2, 3.3, 4.4)
print(2 * q)  # -> (2.2 4.4 6.6 8.8)

print(Quaternion.e_matrix())  # -> e0  e1  e2  e3
                              #    e1 -e0  e3 -e2
                              #    e2 -e3 -e0  e1
                              #    e3  e2 -e1 -e0

o = Octonion(0, 0, 0, 0, 8, 8, 9, 9)
print(o + q)  # -> (1.1 2.2 3.3 4.4 8 8 9 9)

v = Voudon()
print(v == 0)  # -> True
print(len(v))  # -> 256

BeyondVoudon = cayley_dickson_construction(Voudon)
print(len(BeyondVoudon()))  # -> 512

For more snippets see the Thorough Usage Examples section below.

Package Contents

Three functions form the core of the package:

• reals(base) - Given a base type (float by default), generates a class that represents numbers with 1 hypercomplex dimension, i.e. real numbers. This class can then be extended into complex numbers and beyond with cayley_dickson_construction.

  Any usual math operations on instances of the class returned by reals behave as instances of base would but their type remains the reals class. By default they are printed with the g format-spec and surrounded by parentheses, e.g. (1), to remain consistent with the format of higher dimension hypercomplex numbers.

  Python's decimal.Decimal might be another likely choice for base.

  # reals example:
  from hypercomplex import reals
  from decimal import Decimal
  
  D = reals(Decimal)
  print(D(10) / 4)   # -> (2.5)
  print(D(3) * D(9)) # -> (27)

• cayley_dickson_construction(basis) (alias cd_construction) generates a new class of hypercomplex numbers with twice the dimension of the given basis, which must be another hypercomplex number class or class returned from reals. The new class of numbers is defined recursively on the basis according the Cayley-Dickson construction. Normal math operations may be done upon its instances and with instances of other numeric types.

  # cayley_dickson_construction example:
  from hypercomplex import *
  RealNum = reals()
  ComplexNum = cayley_dickson_construction(RealNum)
  QuaternionNum = cayley_dickson_construction(ComplexNum)
  
  q = QuaternionNum(1, 2, 3, 4)
  print(q)         # -> (1 2 3 4)
  print(1 / q)     # -> (0.0333333 -0.0666667 -0.1 -0.133333)
  print(q + 1+2j)  # -> (2 4 3 4)

• cayley_dickson_algebra(level, base) (alias cd_algebra) is a helper function that repeatedly applies cayley_dickson_construction to the given base type (float by default) level number of times. That is, cayley_dickson_algebra returns the class for the Cayley-Dickson algebra of hypercomplex numbers with 2**level dimensions.

  # cayley_dickson_algebra example:
  from hypercomplex import *
  OctonionNum = cayley_dickson_algebra(3)
  
  o = OctonionNum(8, 7, 6, 5, 4, 3, 2, 1)
  print(o)              # -> (8 7 6 5 4 3 2 1)
  print(2 * o)          # -> (16 14 12 10 8 6 4 2)
  print(o.conjugate())  # -> (8 -7 -6 -5 -4 -3 -2 -1)

For convenience, nine internal number types are already defined, built off of each other:

# built-in types example:
from hypercomplex import *
print(Real(4))               # -> (4)
print(C(3-7j))               # -> (3 -7)
print(CD4(.1, -2.2, 3.3e3))  # -> (0.1 -2.2 3300 0)
print(CD[3](1, 0, 2, 0, 3))  # -> (1 0 2 0 3 0 0 0)

The names and letter-abbreviations were taken from this image (mirror) found in Micheal Carter's paper Visualization of the Cayley-Dickson Hypercomplex Numbers Up to the Chingons (64D), but they also may be known according to their Latin naming conventions.

Thorough Usage Examples

This list follows examples.py exactly and documents nearly all the things you can do with the hypercomplex numbers created by this package.

Every example assumes the appropriate imports are already done, e.g. from hypercomplex import *.

1. Initialization can be done in various ways, including using Python's built in complex numbers. Unspecified coefficients become 0.

   print(R(-1.5))                        # -> (-1.5)
   print(C(2, 3))                        # -> (2 3)
   print(C(2 + 3j))                      # -> (2 3)
   print(Q(4, 5, 6, 7))                  # -> (4 5 6 7)
   print(Q(4 + 5j, C(6, 7), pair=True))  # -> (4 5 6 7)
   print(P())                            # -> (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

2. Numbers can be added and subtracted. The result will be the type with more dimensions.

   print(Q(0, 1, 2, 2) + C(9, -1))                   # -> (9 0 2 2)
   print(100.1 - O(0, 0, 0, 0, 1.1, 2.2, 3.3, 4.4))  # -> (100.1 0 0 0 -1.1 -2.2 -3.3 -4.4)

3. Numbers can be multiplied. The result will be the type with more dimensions.

   print(10 * S(1, 2, 3))                    # -> (10 20 30 0 0 0 0 0 0 0 0 0 0 0 0 0)
   print(Q(1.5, 2.0) * O(0, -1))             # -> (2 -1.5 0 0 0 0 0 0)
   
   # notice quaternions are non-commutative
   print(Q(1, 2, 3, 4) * Q(1, 0, 0, 1))      # -> (-3 5 1 5)
   print(Q(1, 0, 0, 1) * Q(1, 2, 3, 4))      # -> (-3 -1 5 5)

4. Numbers can be divided and inverse gives the multiplicative inverse.

   print(100 / C(0, 2))                      # -> (0 -50)
   print(C(2, 2) / Q(1, 2, 3, 4))            # -> (0.2 -0.0666667 0.0666667 -0.466667)
   print(C(2, 2) * Q(1, 2, 3, 4).inverse())  # -> (0.2 -0.0666667 0.0666667 -0.466667)
   print(R(2).inverse(), 1 / R(2))           # -> (0.5) (0.5)

5. Numbers can be raised to integer powers, a shortcut for repeated multiplication or division.

   q = Q(0, 3, 4, 0)
   print(q**5)               # -> (0 1875 2500 0)
   print(q * q * q * q * q)  # -> (0 1875 2500 0)
   print(q**-1)              # -> (0 -0.12 -0.16 0)
   print(1 / q)              # -> (0 -0.12 -0.16 0)
   print(q**0)               # -> (1 0 0 0)

6. conjugate gives the conjugate of the number.

   print(R(9).conjugate())           # -> (9)
   print(C(9, 8).conjugate())        # -> (9 -8)
   print(Q(9, 8, 7, 6).conjugate())  # -> (9 -8 -7 -6)

7. norm gives the absolute value as the base type (float by default). There is also norm_squared.

   print(O(3, 4).norm(), type(O(3, 4).norm()))  # -> 5.0 <class 'float'>
   print(abs(O(3, 4)))                          # -> 5.0
   print(O(3, 4).norm_squared())                # -> 25.0

8. Numbers are considered equal if their coefficients all match. Non-existent coefficients are 0.

   print(R(999) == V(999))         # -> True
   print(C(1, 2) == Q(1, 2))       # -> True
   print(C(1, 2) == Q(1, 2, 0.1))  # -> False

9. coefficients gives a tuple of the components of the number in their base type (float by default). The properties real and imag are shortcuts for the first two components. Indexing can also be used (but is inefficient).

   print(R(100).coefficients())   # -> (100.0,)
   q = Q(2, 3, 4, 5)
   print(q.coefficients())        # -> (2.0, 3.0, 4.0, 5.0)
   print(q.real, q.imag)          # -> 2.0 3.0
   print(q[0], q[1], q[2], q[3])  # -> 2.0 3.0 4.0 5.0

10. e(index) of a number class gives the unit hypercomplex number where the index coefficient is 1 and all others are 0.

    print(C.e(0))  # -> (1 0)
    print(C.e(1))  # -> (0 1)
    print(O.e(3))  # -> (0 0 0 1 0 0 0 0)

11. e_matrix of a number class gives the multiplication table of e(i)*e(j). Set string=False to get a 2D list instead of a string. Set raw=True to get the raw hypercomplex numbers.

    print(O.e_matrix())                        # -> e1  e2  e3  e4  e5  e6  e7
                                               #   -e0  e3 -e2  e5 -e4 -e7  e6
                                               #   -e3 -e0  e1  e6  e7 -e4 -e5
                                               #    e2 -e1 -e0  e7 -e6  e5 -e4
                                               #   -e5 -e6 -e7 -e0  e1  e2  e3
                                               #    e4 -e7  e6 -e1 -e0 -e3  e2
                                               #    e7  e4 -e5 -e2  e3 -e0 -e1
                                               #   -e6  e5  e4 -e3 -e2  e1 -e0
                                               #
    print(C.e_matrix(string=False, raw=True))  # -> [[(1 0), (0 1)], [(0 1), (-1 0)]]

12. A number is considered truthy if it has has non-zero coefficients. Conversion to int, float and complex are only valid when the coefficients beyond the dimension of those types are all 0.

    print(bool(Q()))                    # -> False
    print(bool(Q(0, 0, 0.01, 0)))       # -> True
    
    print(complex(Q(5, 5)))             # -> (5+5j)
    print(int(V(9.9)))                  # -> 9
    # print(float(C(1, 2))) <- invalid

13. Any usual format spec for the base type can be given in an f-string.

    o = O(0.001, 1, -2, 3.3333, 4e5)
    print(f"{o:.2f}")                 # -> (0.00 1.00 -2.00 3.33 400000.00 0.00 0.00 0.00)
    print(f"{R(23.9):04.0f}")         # -> (0024)

14. The len of a number is its hypercomplex dimension, i.e. the number of components or coefficients it has.

    print(len(R()))      # -> 1
    print(len(C(7, 7)))  # -> 2
    print(len(U()))      # -> 128

15. Using in behaves the same as if the number were a tuple of its coefficients.

    print(3 in Q(1, 2, 3, 4))  # -> True
    print(5 in Q(1, 2, 3, 4))  # -> False

16. copy can be used to duplicate a number (but should generally never be needed as all operations create a new number).

    x = O(9, 8, 7)
    y = x.copy()
    print(x == y)   # -> True
    print(x is y)   # -> False

17. base on a number class will return the base type the entire numbers are built upon.

    print(R.base())                      # -> <class 'float'>
    print(V.base())                      # -> <class 'float'>
    A = cayley_dickson_algebra(20, int)
    print(A.base())                      # -> <class 'int'>

18. Hypercomplex numbers are weird, so be careful! Here two non-zero sedenions multiply to give zero because sedenions and beyond have zero devisors.

    s1 = S.e(5) + S.e(10)
    s2 = S.e(6) + S.e(9)
    print(s1)                                    # -> (0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0)
    print(s2)                                    # -> (0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0)
    print(s1 * s2)                               # -> (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
    print((1 / s1) * (1 / s2))                   # -> (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
    # print(1/(s1 * s2)) <- zero division error

About

I wrote this package for the novelty of it and as a math and programming exercise. The operations it can perform on hypercomplex numbers are not particularly efficient due to the recursive nature of the Cayley-Dickson construction.

I am not a mathematician, only a math hobbyist, and apologize if there are issues with the implementations or descriptions I have provided.