Tag
Clifford Algebra
A Clifford algebra Cl(V,g) is the associative algebra generated by a vector space V and a quadratic form g, subject to the single relation v² = g(v,v) for every vector v. It is constructed as a quotient of the tensor algebra on V, has dimension 2ⁿ for an n-dimensional V, and contains the exterior algebra as its underlying vector space. Clifford algebras furnish the double covers Spin(n) of the rotation groups SO(n) and the spinor representations used throughout quantum mechanics and general relativity.
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.