"From Geometry to Algebra and Back Again: 4000 Years of Papers" by Jack Rusher

### Overview In this presentation at Papers We Love, Jack Rusher argues that modern technical mathematics suffers from fragmented, suboptimal notations and suggests refactoring mathematics around geometric algebra (Clifford algebra). Framing mathematical thought through an evolutionary and cognitive lens, Rusher highlights how human spatial planning—shared across hundreds of millions of years of evolution—provides an intuitive foundation that symbolic algebra should directly support rather than obscure. Drawing parallels between mathematical notation and programming languages, he traces historical shifts in numerical systems and demonstrates how geometric algebra unifies disparate geometric concepts across arbitrary dimensions.

### Key Topics - **Evolutionary Origins of Mathematics**: How spatial navigation, physical planning, animal counting, and social deliberation evolved into geometry, number sense, and symbolic logic. - **Evolution of Number Protocols**: The historical emergence and frequent philosophical rejection of new numerical abstractions, including fractions in ancient Babylon, irrational numbers, negative numbers, and complex numbers. - **Notational Shortcomings and Fragmentation**: How standard Vector Calculus notations (e.g., Gibbs and Heaviside) obscure geometric intuition compared to formulations like Grassman's and Clifford's geometric algebra. - **Geometric Algebra and K-Vectors**: How the wedge product and geometric product generalize vectors into higher-dimensional oriented entities (bi-vectors, tri-vectors, or general k-vectors), clarifying concepts such as imaginary numbers and rotations. - **Projective and Metric Extensions**: How parameterized metrics allow modeling diverse geometries, such as Minkowski spacetime, conformal geometric algebra, and projective geometric algebra for graphics and physics.

### Technical Details Rusher examines arithmetic and algebraic systems as functional protocols with generic dispatch. In historical Babylon, scribes computed areas and approximations (such as calculating the square root of two to six decimal places) using place-value base-60 tables of reciprocals and squares. Modern representations evolved past natural numbers and negative numbers to complex numbers, initially cludged by Rafael Bombelli from Girolamo Cardano's algorithms before being understood geometrically as planar rotations.

In standard 3D vector algebra, rotations and cross products do not generalize naturally across dimensions. Geometric algebra resolves this by defining three forms of multiplication: the dot product, the wedge product (producing an oriented bi-vector), and the geometric product (the sum of the dot product and wedge product). Through this formulation, the imaginary unit $i$ is revealed as a unit bi-vector representing an oriented plane of rotation. Extending the algebra with metrics allows unified modeling across dimensions without altering mathematical equations. In a live coding demonstration using Clojure and Emacs, Rusher shows how a physics simulation accounting for gravity, Hooke's law, and damping operates identically across 2D, 3D, and 4D spaces simply by changing the dimensionality parameter.

### Takeaways Rusher advocates for adopting geometric algebra across physics, engineering, and computer graphics, pointing to David Hestenes' research program and 2002 Oersted Medal lecture. Unifying technical mathematics under geometric algebra eliminates redundant tools (such as isolated cross products, Pauli/Dirac matrices, and quarternions), which could reduce physics education by several years while making advanced spatial and physical concepts vastly more accessible and intuitive.