Two-sided bounds for dihedral angle sums of path and 4-ball tetrahedra

TL;DR

This work derives two-sided, tight bounds on the dihedral-angle sum $\Sigma_T$ for two tetrahedron classes: path tetrahedra and $4$-ball tetrahedra. For path tetrahedra, a constructive parametric family yields $2\pi < \Sigma_T < 2.5\pi$, with the entire interval realized; for $4$-ball tetrahedra, the bound is $6\arccos\left(1/3\right) \le \Sigma_T < 3\pi$, also shown to be tight and attainable across the interval. The analysis for $4$-ball tetrahedra leverages the mid-sphere framework, tangent-lengths $l_i$, the Cayley–Menger determinant $D_3$, and a spherical-geometry relation $\Sigma_T+\Gamma_T=6\pi$, along with constructive examples and decompositions (e.g., into 24 path tetrahedra) to establish both bounds and their realizability. The results advance understanding of dihedral-angle sums in constrained tetrahedra and have implications for geometric modeling and finite-element methods where nonobtuse or well-behaved angle sums are desirable.

Abstract

A tetrahedron is called a path tetrahedron, if it has three mutually orthogonal edges that do not intersect at a single point. A tetrahedron is called a 4-ball tetrahedron, if there exists a sphere tangent to all its edges. We derive two-sided tight bounds for dihedral angle sums of such tetrahedra. In particular, we prove that this sum lies in the interval (2π, 2.5π) for path tetrahedra and in [6 arccos 1/3, 3π) for 4-ball tetrahedra. Also some of their useful properties are presented.

Figures (7)

• Figure 1: Left: Path tetrahedron $OABC$ with orthogonal edges $a$, $b$, and $c$ forming a path (in the sense of graph theory). Its four faces are right triangles and its volume is $abc/6$. Right: Using midlines of its faces and the inner diagonal connecting the midpoints $M_1$ and $M_2$ of the edges $OB$ and $AC$, it can be partitioned face-to-face into $8$ path subtetrahedra, see BOOK.
• Figure 2: A sketch of a $4$-ball tetrahedron $A_1A_2A_3A_4$. Here $l_i$ are lengths of tangents from its vertices to the corresponding mid-sphere. The six touching points are marked by bullets.
• Figure 3: Left: Inscribed circles to unfolded quadrilaterals constructed from triangular faces. Right: If there exists an inscribed circle to a convex unfolded quadrilateral, then $a+d = b+e$ (cf. Remark \ref{['rem:no-analog']}). Two smaller circles correspond to the mid-sphere.
• Figure 4: Two typical cases when all triangle inequalities for all four faces hold, but the Cayley-Menger determinant is negative. Left: for $a=b=c=d=e=1$ the remaining thick red edge $f=7/4$ is too long to construct a tetrahedron (see Example 4). Right: for given edges $a=b=1.1$ and $c=d=e=2$ the remaining thick red edge $f=1.1$ is too short to construct a ($4$-ball) tetrahedron, see Remark \ref{['rem:no-analog']}.
• Figure 5: For the given three (blue) mutually externally touching circles there exists a unique fourth (red) circle externally touching all three circles (the 10th Apollonios problem).

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Theorem 2
• Definition 4
• Remark 1
• Remark 2
• Theorem 3
• Theorem 4