Residual Corner Associate Simplices in 3D Tetrahedra and n-Dimensional Space: Multilinear Volume Laws, Central Inversion Congruence, and a Complete Resolution of the András–Kiss Higher-Dimensional Open Problem
Year: 2026
Type: preprint
Abstract: We extend residual (corner) associate configurations from triangles to tetrahedra and n-simplices. An anchor-ray decomposition gives a square-free multilinear volume law for the associate simplex of any weight vector across any inscribed simplex. In R^3 we prove a volume formula for the tetrahedron of residual centroids, 64 Vol(GA GB GC GD) = 2 Vol(ABCD) + Σ Vol(Tk) − 2 Vol(W1W2W3W4), and an alternating-sum formula in R^n. For cevian simplices, the self-associate simplex is the point reflection of the cevian simplex in every dimension. In orthocentric simplices the associate simplex is orthologic to the reference simplex, with reciprocal center the n-dimensional isogonal conjugate. All results are verified with SymPy.
Author: Nguyen, Hoang Long
Institution: Foxlink (China)
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225348554