Tag
Geometric Algebra
Geometric algebra extends vector algebra with a single associative geometric product, uv = u·v + u∧v, built from a real vector space and an inner product by imposing the contraction v² = g(v,v). The resulting algebra contains scalars, vectors, and higher-grade blades uniformly, and its rotors give a coordinate-free, singularity-free description of rotations that replaces the ad hoc matrix and cross-product machinery of classical vector algebra. Specialising to Cl(3,0) recovers ordinary three-dimensional space, and to Cl(1,3) recovers the spacetime algebra used in relativistic physics.
Blog
July 11, 2026
Quaternions Are the Rotors of Space
Hamilton's quaternions reconstructed on their own terms and then recognised as the even subalgebra of the geometric algebra of space: the imaginaries i, j, k are the basis bivectors, a unit quaternion is a rotor, and the half-angle, the two-to-one cover, and the absence of gimbal lock become plain facts about rotors. A companion to the classical-mechanics article, closing with the native rotation expressions this settles in code.
July 3, 2026
Classical Mechanics from Zero, in Two Languages
Classical mechanics constructed from nothing but an inertial frame, in matrix and linear algebra and in geometric algebra side by side: rotations and rigid-body dynamics without Euler angles, one vector derivative replacing grad, div, and curl, and the Lagrangian and Hamiltonian formalisms in both languages.