A library to do geometric algebra in python.

A multivector is an element of the algebra. It is composed of a sum of scalars, vectors, bivectors, etc., in a way similar to how complex numbers are sums of real and imaginary components. Multivectors and their geometric product are the simplest and most powerful tools for mathematical analysis that I know of.

To create the basis vectors of a geometric algebra with the given signature:

import geometric_algebra as ga

signature = (3, 0)  # 3 dimensions, all vectors with square == +1
print(ga.basis(signature))

The output should be:

[1, e1, e2, e3, e12, e13, e23, e123]

You can add, multiply, etc., those elements to create arbitrary multivectors.

import geometric_algebra as ga

e, e1, e2, e12 = ga.basis((2, 0))

v = 3 + 4*e12
w = 5 + e1 + 3*e2

print('v =', v)                          # 3 + 4*e12
print('w =', w)                          # 5 + e1 + 3*e2
print('3*v =', 3*v)                      # 9 + 12*e12
print('v + w =', v + w)                  # 8 + e1 + 3*e2 + 4*e12
print('v - (1 + w) =', v - (1 + w))      # -3 + -1*e1 + -3*e2 + 4*e12
print('v * w =', v * w)                  # 15 + 15*e1 + 5*e2 + 20*e12
print('w * v =', w * v)                  # 15 + -9*e1 + 13*e2 + 20*e12
print('v / (2*e2) =', v / (2*e2))        # 2.0*e1 + 1.5*e2
print('v**2 =', v**2)                    # -7 + 24*e12
print('exp(e1+e2) =', ga.exp(e1+e2))     # 2.1... + 1.3...*e1 + 1.3...*e2

a = 2*e1 + 3*e2
b = 4*e1 - 0.5*e2
print('dot(a, b) =', ga.dot(a, b))       # 6.5
print('wedge(a, b) =', ga.wedge(a, b))   # -13*e12
print('norm(v) =', (v * v.T)**0.5)       # 5

A geometric algebra is characterized by its signature.

A signature looks like (p, q) or (p, q, r), saying how many basis vectors have a positive square (+1), negative (-1) and zero (0) respectively.

When using the basis() function to create the basis multivectors, you can pass the signature as a tuple. But you can also instead use a dict that says for each basis element its square. For example, astrophysicists normally use for spacetime:

signature = {0: -1, 1: +1, 2: +1, 3: +1}  # t, x, y, z  with e0 = e_t

whereas particle physicists normally use:

signature = {0: +1, 1: -1, 2: -1, 3: -1}

which is the same signature as (1, 3), or (1, 3, 0), in the more common notation.

You can run some tests with:

pytest test.py

• https://bivector.net/

I developed a similar library in clojure, which shares the same internal representation for a multivector: geometric-algebra.