Euclidean Arrows: Algebra, Geometry and Calculus

Physics speaks many mathematical languages, and most of them are not Euclidean: the Minkowski space-time of relativity, the algebra of Dirac's electron, and the many-dimensional worlds of string theory all allow squared lengths that are negative. We ask whether one Euclidean language, the geometry of ordinary lengths and angles, could serve instead. We call a number system Euclidean when its numbers are the points of ordinary space, and dividing one number by another measures their lengths and the angle between them, undoes multiplication, and never leads outside the system. We prove that exactly three such systems exist: the complex numbers, the quaternions, and the octonions, of two, four, and eight dimensions, which we call Euclidean Arrows. A single formula, Euler's relation, lets one arrow describe propagation along a straight line, rotation, and oscillation. In the arrows, Euclid's five postulates become theorems, and the famous fifth, on parallel lines, follows from how lengths are measured rather than from lines that never meet. The calculus of arrows is built on division as well. It starts from finite steps, with no limits, so every signal, even a noisy or fractal one, has a derivative and an integral, and the two undo each other exactly. The derivative exists in every direction, and its average over all directions is the gradient, with a minus sign that division brings and the traditional gradient lacks, which corrects the signs that troubled Maxwell's quaternion electromagnetism. The traditional gradient instead tests whether the derivative depends on the direction, like the Cauchy-Riemann condition. Fourier and Laplace transforms turn derivatives into products with the frequency. An effective zero and infinity, set by the resolution and range of measurement, make every derivative and integral exist. Resting on a single physical assumption, that a measurement returns a length, the arrows offer a complete Euclidean mathematics, ready to describe time and matter.