tfga 0.1.10

Clifford and Geometric Algebra with TensorFlow

TFGA - TensorFlow Geometric Algebra

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Python package for Geometric / Clifford Algebra with TensorFlow 2.

This project is a work in progress. Its API may change and the examples aren't polished yet.

Pull requests and suggestions either by opening an issue or by sending me an email are welcome.

Installation

Install using pip: pip install tfga

Requirements:

• Python 3
• tensorflow 2
• numpy

Basic usage

There are two ways to use this library. In both ways we first create a GeometricAlgebra instance given a metric. Then we can either work on tf.Tensor instances directly where the last axis is assumed to correspond to the algebra's blades.

import tensorflow as tf
from tfga import GeometricAlgebra

# Create an algebra with 3 basis vectors given their metric.
# Contains geometric algebra operations.
ga = GeometricAlgebra(metric=[1, 1, 1])

# Create geometric algebra tf.Tensor for vector blades (ie. e_0 + e_1 + e_2).
# Represented as tf.Tensor with shape [8] (one value for each blade of the algebra).
# tf.Tensor: [0, 1, 1, 1, 0, 0, 0, 0]
ordinary_vector = ga.from_tensor_with_kind(tf.ones(3), kind="vector")

# 5 + 5 e_01 + 5 e_02 + 5 e_12
quaternion = ga.from_tensor_with_kind(tf.fill(dims=4, value=5), kind="even")

# 5 + 1 e_0 + 1 e_1 + 1 e_2 + 5 e_01 + 5 e_02 + 5 e_12
multivector = ordinary_vector + quaternion

# Inner product e_0 | (e_0 + e_1 + e_2) = 1
# ga.print is like print, but has extra formatting for geometric algebra tf.Tensor instances.
ga.print(ga.inner_prod(ga.e0, ordinary_vector))

# Exterior product e_0 ^ e_1 = e_01.
ga.print(ga.ext_prod(ga.e0, ga.e1))

# Grade reversal ~(5 + 5 e_01 + 5 e_02 + 5 e_12)
# = 5 + 5 e_10 + 5 e_20 + 5 e_21
# = 5 - 5 e_01 - 5 e_02 - 5 e_12
ga.print(ga.reversion(quaternion))

# tf.Tensor 5
ga.print(quaternion[0])

# tf.Tensor of shape [1]: -5 (ie. reversed sign of e_01 component)
ga.print(ga.select_blades(quaternion, "10"))

# tf.Tensor of shape [8] with only e_01 component equal to 5
ga.print(ga.keep_blades(quaternion, "10"))

Alternatively we can convert the geometric algebra tf.Tensor instance to MultiVector instances which wrap the operations and provide operator overrides for convenience. This can be done by using the __call__ operator of the GeometricAlgebra instance.

# Create geometric algebra tf.Tensor instances
a = ga.e123
b = ga.e1

# Wrap them as `MultiVector` instances
mv_a = ga(a)
mv_b = ga(b)

# Reversion ((~mv_a).tensor equivalent to ga.reversion(a))
print(~mv_a)

# Geometric / inner / exterior product
print(mv_a * mv_b)
print(mv_a | mv_b)
print(mv_a ^ mv_b)

Notebooks

Generic examples

Quantum Electrodynamics using Geometric Algebra

Projective Geometric Algebra