On Exact Dihedral Angle Sum Ranges for Orthocentric and Isogonic Tetrahedra
Abstract
For a nondegenerate Euclidean tetrahedron T, let Σ(T) denote the sum of its six interior dihedral angles. We establish the exact range [6 arccos(1/3), 3π) for the orthocentric and isogonic classes. The orthocentric part gives a short alternative proof of the sharp lower bound; the main new result is the corresponding isogonic theorem. Starting from classical insphere-contact geometry, we derive a reciprocal half-angle identity and a sharp defect inequality controlled by the scale-invariant dispersion of the reciprocal tangent-length parameters. This quantitative refinement shows that any nonzero dispersion forces a positive dihedral-angle-sum gap, and that near equality forces the normalized reciprocal parameters to be close to equality. Finally, the common kite subfamily realizes every value in the interval and proves sharpness of both endpoints.
Related
- The Exact Dihedral Angle Sum Range for Circumcentral-Cevian Jaw Tetrahedra
- Conformal tetrahedra and global rigidity of sphere packings in constant curvature background geometry
- Rational quadrilaterals and tetrahedra -- the involutions of Kummer, Heegner and Regge
- Necessary and Sufficient Criteria for the Realizability of Kaleidocycles Composed of Six Congruent Isosceles Tetrahedra
- Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra
BibTeX
@misc{korotov2026exact,
title = {{On Exact Dihedral Angle Sum Ranges for Orthocentric and Isogonic Tetrahedra}},
author = {Sergey Korotov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26529},
eprint = {2610.26529},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26529},
}
RIS
TY - UNPB
TI - On Exact Dihedral Angle Sum Ranges for Orthocentric and Isogonic Tetrahedra
AU - Sergey Korotov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26529
UR - https://arxiv.org/abs/2610.26529
AB - For a nondegenerate Euclidean tetrahedron T, let Σ(T) denote the sum of its six interior dihedral angles. We establish the exact range [6 arccos(1/3), 3π) for the orthocentric and isogonic classes. The orthocentric part gives a short alternative proof of the sharp lower bound; the main new result is the corresponding isogonic theorem. Starting from classical insphere-contact geometry, we derive a reciprocal half-angle identity and a sharp defect inequality controlled by the scale-invariant dispersion of the reciprocal tangent-length parameters. This quantitative refinement shows that any nonzero dispersion forces a positive dihedral-angle-sum gap, and that near equality forces the normalized reciprocal parameters to be close to equality. Finally, the common kite subfamily realizes every value in the interval and proves sharpness of both endpoints.
ER -