The number of tetrahedra sharing the same metric invariants via symbolic and numerical computations

Abstract

The problem consists in bounding the number of tetrahedra that share the same volume V, circumradius R, and face areas $A_1,A_2,A_3,A_4$. A symbolic computation approach is used to prove that the upper bound of the number of tetrahedra is eight. First, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of the tetrahedron to construct a system of algebraic equations. Then, an investigation of its real roots is done by analyzing the discriminant sequence and extensive numerical computations.

Introduction

A tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces in ${\mathbb{R}}^3$. A tetrahedron is said rigid if it is determined by its volume, the area of its faces, and the radius of its circumscribed sphere. In 1999, Mazur raised the question of whether every tetrahedron is rigid (Mazur, 1999). In 2000, Lisoněk and Israel answered the question negatively by constructing two non-isometric tetrahedra with the same metric invariants (Lisoněk and Israel, 2000). Meanwhile, they also asked whether, for any positive numbers $V,R,A_1,A_2,A_3,A_4$, there are finitely many non-isometric tetrahedra, admitting those quantities as their volume, circumradius, and face areas, respectively. In Yang and Zeng (2005), the authors showed that under the circumstance $A_2=A_3=A_4$, a family of infinitely many non-congruent tetrahedra $T_{(x,y)}$, where $(x,y)$ varies over a component of a cubic curve could be made up such that all the $T_{(x,y)}$ have in common the same volume, circumradius, and face areas. In the same paper, they also conjectured that in case $A_1,A_2,A_3,A_4$ are pairwise distinct, one can construct from the granted parameters at most nine non-congruent tetrahedra. The result presented in the following theorem was first announced in the workshop on Distance Geometry (Yang and Zeng, 2013) with a brief description of the proof. A detailed process of the proof using symbolic computations was presented in Zeng et al. (2019).

Theorem 1

Given six positive numbers $V,R,A_1,A_2,A_3,A_4$, there are at most eight tetrahedra with volume V, circumradius R, and four face areas $A_1,A_2,A_3,A_4$, except when three of $A_1,A_2,A_3,A_4$ are equals.

In Blue and Mcconnell (2012), the concept of pseudo-faces was introduced, besides that, a study of the problem using a different system from that in Yang and Zeng (2005) and Lisoněk and Israel (2000) was done. In 2016, Tsai studied the problem based on the same concept, and meanwhile, through taking different parameters, the author proved that the number of tetrahedra does not exceed eight (Tsai, 2016).

In this paper, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of tetrahedra to construct systems of algebraic equations satisfied by the metric invariants of the tetrahedron, then investigate the number of its real roots. This work is an extended version of the paper published in Zeng et al. (2019), while based on the referees' comments, the detailed proof is given through algebraic expressions instead of the Maple commands, as shown in Zeng et al. (2019). Moreover, profound explanations of the representation of the terms $R_1(1)$ and $R_1(-1)$, and for the coefficients of $R_1$ are given. We also constructed a polynomial $G(x)$ of degree 18 such that the number of real roots of $G(x)=0$ on $]-\infty ,+\infty [$ equals two times of the number of non-congruent tetrahedra determined by parameters $V,R,A_1,\cdots ,A_4$, and investigated the relationship between the number of real roots of $G(x)=0$ and its discriminant sequence, i.e., the sequence formed by the determinants of the principal minors of the Sylvester matrix associated with $G(x)$ and $G^{\prime }(x)$. Furthermore, inspired by the discussion with participants of ISSAC 2019, we constructed sets of rational points on the unit sphere $S^2$ to construct tetrahedra and used an entirely new way to search tetrahedra for which volumes V, circumradius R, and face areas $A_1,\cdots ,A_4$ could provide the number of roots for $R_1(x)=0$.

The paper is organized as follows. In Section 2, the Cayley determinant is used to transform our problem to the algebraic setting and some metric equations of the tetrahedron to reduce it to a parametric polynomials system of two variables. Proof of our main theorem by investigating the number of real roots of our equations is given in Section 3. In Section 4, numerical computations are carried out using the Fray sequences and Monte Carlo method to construct tetrahedra and the sign variations of the Sturm sequence to determine our equation's number of reals roots. Finally, conclusions and a remark for future work based on the numerical computations are presented in Section 5.