Balance in Inequality: Testing the Physical Equilibrium, Precession, and Structural Stability of Magic Gems
Abstract
A magic square exemplifies balance in inequality: its n2 entries {1,…,n2} are distinct, yet every row, column, and main diagonal sums to μn=n(n2+1)/2. Lifting centred grid cells into ℝ3 as radial vectors pij(λ)=(xj,yi,λzij) and taking their convex hull---the Magic Gem 𝒢λ(S)---asks: when does line-sum equality suffice for equilibrium in 3D, and when a nonlinear realization breaks balance, how can an opposing inequality restore it? We test this across all $880$ order-$4$ magic squares in four regimes. For radial spokes from O, natural lengths balance identically, while unit spokes break equilibrium on all $832$ non-associative squares in 3D and on all squares in 2D, where the ring-radius ratio r2/r1=√5 bounds the resultant below by the Pell convergent 9/4≈√5. By length--weight duality, balance is restored via tip weights w*>0 on S2 or power-law spoke scaling (rα* in 2D and (V3)z=0 in 3D). In 3D rotation, equal point masses make every semi-magic square an axisymmetric top with spherical-top transition λiso=(n2+1)−1/2, whereas entry masses mij=sij expose the quadrupole Qxy=∑ zijxjyi, triggering 180∘ Dzhanibekov flips on $824$ squares. Passing to the uniform solid hull 𝒢0(S) shifts the centroid off O on $829$ squares; re-engaging the opposing core 𝒢1(S) restores balance via two-density inclusions, nested wireframes, or the Kn2 strut network, while all $880$ hulls unfold into non-overlapping nets with f0−1 glue tabs. Finally, λ=1/5 stabilizes 74.5% of facets on a table, and alternating 𝒢λ(S) with its mirror twin 𝒢λ(r2S̅)=−𝒢λ(S) locks chiral blocks into a vertical tower.
Related
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BibTeX
@misc{mathewson2026balance,
title = {{Balance in Inequality: Testing the Physical Equilibrium, Precession, and Structural Stability of Magic Gems}},
author = {Kyle Elliott Mathewson},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03991},
eprint = {2611.03991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03991},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Balance in Inequality: Testing the Physical Equilibrium, Precession, and Structural Stability of Magic Gems
AU - Kyle Elliott Mathewson
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03991
UR - https://arxiv.org/abs/2611.03991
AB - A magic square exemplifies balance in inequality: its n^2 entries {1,…,n^2} are distinct, yet every row, column, and main diagonal sums to μ_n=n(n^2+1)/2. Lifting centred grid cells into ℝ^3 as radial vectors p_ij(λ)=(x_j,y_i,λz_ij) and taking their convex hull---the Magic Gem 𝒢_λ(S)---asks: when does line-sum equality suffice for equilibrium in 3D, and when a nonlinear realization breaks balance, how can an opposing inequality restore it? We test this across all $880$ order-$4$ magic squares in four regimes. For radial spokes from O, natural lengths balance identically, while unit spokes break equilibrium on all $832$ non-associative squares in 3D and on all squares in 2D, where the ring-radius ratio r_2/r_1=√5 bounds the resultant below by the Pell convergent 9/4≈√5. By length--weight duality, balance is restored via tip weights w^*>0 on S^2 or power-law spoke scaling (r^α^* in 2D and (V_3)_z=0 in 3D). In 3D rotation, equal point masses make every semi-magic square an axisymmetric top with spherical-top transition λ_iso=(n^2+1)^−1/2, whereas entry masses m_ij=s_ij expose the quadrupole Q_xy=∑ z_ijx_jy_i, triggering 180^∘ Dzhanibekov flips on $824$ squares. Passing to the uniform solid hull 𝒢_0(S) shifts the centroid off O on $829$ squares; re-engaging the opposing core 𝒢_1(S) restores balance via two-density inclusions, nested wireframes, or the K_n^2 strut network, while all $880$ hulls unfold into non-overlapping nets with f_0−1 glue tabs. Finally, λ=1/5 stabilizes 74.5% of facets on a table, and alternating 𝒢_λ(S) with its mirror twin 𝒢_λ(r^2S̅)=−𝒢_λ(S) locks chiral blocks into a vertical tower.
ER -