The Jonah Pyramid: A Numerically Validated Saddle-Like Structure in Tetrahedral Shape Dynamics

This paper introduces a three-dimensional analogue of the "Jonah Triangle," discovered in previous work on planar triangle dynamics. We define a generalized pedal map FtF_tFt acting on the space of tetrahedra, driven by a point Pt=(1−t)I+tN′P_t = (1-t)I + tN'Pt=(1−t)I+tN′ moving along the line connecting the incenter III and a simplicial generalization of the Nagel point N′N'N′.We prove that the family of regular pyramids (tetrahedra with an equilateral base and apex on the symmetry axis) is invariant under this map. Within this reduced one-dimensional dynamic system, we identify a unique non-regular fixed point—the Jonah Pyramid—which exists for all parameters t>2/3t > 2/3t>2/3.Key findings include:Geometric Characterization: The side lengths of the Jonah Pyramid are determined by the formula ℓ2/b2=3t2−3t+1(3t−2)2\ell^2/b^2 = \frac{3t^2 - 3t + 1}{(3t - 2)^2}ℓ2/b2=(3t−2)23t2−3t+1, where ℓ\ellℓ is the lateral edge and bbb is the base edge. This formula was derived via rational interpolation and verified numerically to machine precision (<10−16< 10^{-16}<10−16 error) across multiple test values.Spectral Analysis: Unlike the strictly hyperbolic saddle nature of the 2D Jonah Triangle, the linearization of FtF_tFt at the Jonah Pyramid exhibits partial degeneracy. While it retains one unstable direction (λu>1\lambda_u > 1λu>1) and two stable directions (λs<1\lambda_s < 1λs<1), it possesses two eigenvalues equal to −1-1−1. These correspond to involutive symmetries related to disphenoidal deformations of the base.Honesty on Status: The existence of the fixed point and its spectral properties are supported by rigorous numerical verification using NumPy. However, a complete symbolic algebraic proof of the edge-length formula remains open and is presented here as a strong conjecture backed by exact arithmetic checks.This work extends the theory of geometric saddles from dimension 2 to dimension 3, revealing how higher-dimensional symmetries introduce spectral complexity (eigenvalues on the unit circle) absent in the planar case. It serves as a foundational step toward understanding the hierarchy of Schloss objects in arbitrary dimensions. Lo