First Clifford Product¶

The Euclidean algebra \(Cl(2, 0)\) is small enough to expose every part of a Clifford calculation. The example below constructs the algebra, embeds its two basis vectors, and evaluates their geometric product. It assumes Python and PyTorch but no prior clifra API knowledge.

Construct an Algebra¶

import torch

from clifra import format_multivector, make_algebra

algebra = make_algebra(2, 0, device="cpu")

The signature says that the algebra has two basis vectors whose squares are positive. It therefore has \(2^2 = 4\) canonical basis lanes: scalar, \(e_1\), \(e_2\), and \(e_{12}\).

Embed Two Vectors¶

embed_vector maps ordinary coordinates into the grade-1 lanes of a canonical multivector tensor.

e1 = algebra.embed_vector(torch.tensor([[1.0, 0.0]]))
e2 = algebra.embed_vector(torch.tensor([[0.0, 1.0]]))

assert e1.shape == (1, algebra.dim)
assert algebra.dim == 4

The final tensor axis always holds Clifford coefficients. Batch and channel axes, when present, come before it.

Multiply Them¶

e1_e2 = algebra.geometric_product(e1, e2)
e1_squared = algebra.geometric_product(e1, e1)

print(format_multivector(algebra, e1_e2))
print(format_multivector(algebra, e1_squared))

The meaningful parts of the output are:

e12
1

Orthogonal vectors produce the oriented plane \(e_{12}\), while the Euclidean basis vector satisfies \(e_1^2 = 1\).

Compare the Exterior Product¶

oriented_area = algebra.wedge(e1, e2)

assert torch.equal(oriented_area, e1_e2)
print(format_multivector(algebra, oriented_area))

For these orthogonal basis vectors the geometric and exterior products agree. For general vectors, the geometric product can also contain a scalar inner-product part.

The example establishes the canonical representation and its product semantics. Layouts and Planned Execution expresses the same class of calculation with compact tensor contracts.