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 29, 2026
A High School Student's Introduction to Physics
Physics built from arithmetic, assuming nothing beyond addition and subtraction. The number systems and the equals sign, then units, then the affine space of positions, then the geometric product forced by the demand that a vector square to its own length. The calculus is constructed on the way, with the limit stated in full and the derivative defined by Caratheodory's slope function. In the geometric algebra of the plane: the kinematic formulas derived as a corollary of integrating a constant, centripetal acceleration by differentiating a rotor twice, the harmonic oscillator's cosine proved rather than announced, angular momentum as a conserved bivector with Kepler's equal areas as its consequence, and work and energy as the fundamental theorem along a path.
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.