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Geometric Algebra for Python

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clifford: Geometric Algebra for Python

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is a numerical Geometric Algebra (a.k.a. Clifford algebra) package for python.
  • Documentation:
  • Source code:
  • Bug reports:

Geometric Algebra (GA) is a universal algebra which among several other independent mathematical systems, subsumes: * Complex numbers * Quaternions * Linear algebra

Scalars, vectors, and higher-grade entities can be mixed freely and consistently in the form of mixed-grade multivectors. Like this,

Visual explanation of blades

Quick Installation

Requires Python version >=3.5

Install using

conda install clifford -c conda-forge
Install using
pip3 install clifford
Detailed instructions


Try out a notebook in binder

Or have a go on your own pc:

from clifford.g3 import *  # import GA for 3D space
from math import e, pi
a = e1 + 2*e2 + 3*e3 # vector 
R = e**(pi/4*e12)    # rotor 
R*a*~R    # rotate the vector  

Syntax Summary

| Syntax | Operation | |:-:|:-:| | | | Symmetric inner product | | << | Left contraction | | ^ | Outer product | | * | Geometric product | | X(i) | Return the section of the multivector X of grade i | | X(ei) | Return the section of the multivector X for which ei is the pseudo scalar | | X[i] | Return the i'th coefficient from the multivector X | X.normal() | Return the normalised multivector so that X*~X is +- 1 |

For installation instructions, api documention, and tutorials, head over to our documentation!

Citing This Library

As linked at the top of this page,

is published to zenodo. DOI 10.5281/zenodo.1453978 refers to all versions of clifford.

To obtain BibTex citation information for a specific release (recommended):

  • Run
    python -m pip show clifford
    to determine which version you are using (or print
    from python)
  • Click on the corresponding version from this list of versions
  • Scroll down to the bottom of the page, and click the "BibTex" link in the "Export" sidebar

If you want to cite all releases, use:

  author       = {Hugo Hadfield and
                  Eric Wieser and
                  Alex Arsenovic and
                  Robert Kern and
                  {The Pygae Team}},
  title        = {pygae/clifford},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.1453978},
  url          = {}

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