Simplicial (Quadray) coordinate systems use N frame vectors pointing from the origin to the vertices of a regular (N−1) -simplex, providing an overcomplete, permutation-symmetric coordinate description of (N−1) -dimensional Euclidean space whose N frame vectors form an equiangular tight frame (ETF) for ℝN−1. Such coordinates carry a one-dimensional gauge redundancy along the diagonal direction 1 = (1, 1, …, 1), and the physically meaningful geometry lives on the zero-sum hyperplane v: ∑ vi = 0, which is isometric to ℝN−1 under the metric inherited from the ambient Gram matrix. We develop an intrinsic algebraic vector calculus on this hyperplane, comprising three operators — an inner product, a binary cross product, and a rotation; differential operators (∇, div, curl) are not treated here. These three operators are jointly compatible with the gauge action in a precise sense: the inner product is gauge-invariant as a scalar, while the binary cross product and rotation preserve the zero-sum hyperplane and respectively annihilate and fix the gauge direction 1. Once the simplicial Gram data is fixed, their formulas require no ongoing reference to a Cartesian frame: The inner product induced by the simplicial Gram matrix G = N/ (N−1) I − 1/ (N−1) J, whose associated quadratic form collapses on zero-sum vectors to ⟨c, c⟩ = N/ (N−1) ∑i ci2. A skew-symmetric bilinear operator K (u) that, for N = 4, functions as an intrinsic cross product — gauge-annihilating, hyperplane-closing, and satisfying the Lie-algebraic identity K3 = −K when u is a zero-sum unit axis. The exponential map of K (u) into SO (3) for N = 4, yielding the closed-form Rodrigues formula R = I + sin θ · K + (1 − cos θ) K2. For N = 4 we prove K3 = −K and derive the scaling constant 1/√3 from first principles; the resulting rotation matrix fixes the gauge direction (R1 = 1), preserves zero-sum, and supplies a 9-multiplication kernel for applying rotations to zero-sum inputs. We identify N = 4 as the unique case within the simplicial wedge–Hodge framework developed here in which the construction yields a binary cross product and closed-form exponential (higher N proceeds via Hodge duals of wedge products of arity N − 3): the simplicial-coordinate reflection of the exceptional Lie-algebra isomorphism so (3) ≅ (ℝ3, ×). MSC 2020: 15A72, 15A75, 22E60, 53A45, 65D18.
Leonardo Murillo Montero (Wed,) studied this question.