First Clifford Product

In the Euclidean algebra \(Cl(2,0)\), the basis vectors satisfy \(e_1^2=e_2^2=1\) and \(e_1e_2=-e_2e_1\). Represent them with a compact grade-1 layout:

import torch
from clifra import format_multivector, make_algebra

algebra = make_algebra(2, 0)
vectors = algebra.layout((1,))
e1 = torch.tensor([1.0, 0.0])
e2 = torch.tensor([0.0, 1.0])

The final axis stores coefficients of \(e_1,e_2\). The layout supplied to the operation explicitly assigns the tensor's geometric meaning.

products = algebra.layout((0, 2))
e1_e2 = algebra.geometric_product(
    e1, e2, left=vectors, right=vectors, output=products,
)
e1_squared = algebra.geometric_product(
    e1, e1, left=vectors, right=vectors, output=products,
)
assert torch.equal(e1_e2, torch.tensor([0.0, 1.0]))
assert torch.equal(e1_squared, torch.tensor([1.0, 0.0]))
print(format_multivector(algebra, e1_e2, layout=products))  # e12

The result lanes are scalar and \(e_{12}\). For general vectors, \(ab=\langle ab\rangle_0+a\wedge b\).

The scalar part is the Euclidean inner product in this signature. The bivector part measures oriented area: reversing the order reverses its sign. This yields a mixed-grade multivector, distinct from elementwise multiplication of the arrays.

a = torch.tensor([2.0, 1.0])
b = torch.tensor([1.0, 3.0])

ab = algebra.geometric_product(a, b, left=vectors, right=vectors, output=products)
ba = algebra.geometric_product(b, a, left=vectors, right=vectors, output=products)

assert torch.equal(ab, torch.tensor([5.0, 5.0]))
assert torch.equal(ba, torch.tensor([5.0, -5.0]))

Here \(\langle ab\rangle_0=2\cdot1+1\cdot3=5\) and the \(e_{12}\) coefficient is \(2\cdot3-1\cdot1=5\). In a mixed signature, the scalar part changes with the basis squares. The exterior coefficient does not.

bivectors = algebra.layout((2,))
area = algebra.wedge(e1, e2, left=vectors, right=vectors, output=bivectors)
assert torch.equal(area, torch.tensor([1.0]))

The exterior product retains only the oriented-area coefficient. It has a different output layout from the geometric product, even when both represent \(e_{12}\).

Nothing wraps these tensors or takes ownership of their gradients. For example, the derivative of the scalar part of \(ab\) with respect to \(a\) is \(b\) in this Euclidean signature:

a = a.clone().requires_grad_()
ab = algebra.geometric_product(a, b, left=vectors, right=vectors, output=products)
ab[0].backward()
assert torch.equal(a.grad, b)

The layout defines which coefficient space the final axis represents; PyTorch owns the values and the differentiation graph. Continue with Layouts, Storage, and Tensor Composition to make this distinction explicit for larger algebras and batches.