[Paper Review] Parametric Solutions for a Nearly-Perfect Cuboid

Mamuka Meskhishvili|arXiv (Cornell University)|Feb 9, 2015

Coding theory and cryptography6 references7 citations

TL;DR

This paper presents three rational one-parameter parametrizations for nearly-perfect cuboids (NPCs), where only one face diagonal is irrational. By leveraging rational solutions to Diophantine conditions involving squares, the author derives explicit polynomial formulas for side lengths and diagonals, offering a constructive basis for computational searches for perfect cuboids.

ABSTRACT

We consider nearly-perfect cuboids (NPC), where the only irrational is one of the face diagonals. Obtained are three rational parametrizations for NPC with one parameter.

Motivation & Objective

• To construct explicit rational parametrizations for nearly-perfect cuboids (NPCs), where only one face diagonal is irrational.
• To provide a systematic method for generating NPC solutions using one rational parameter, advancing prior incomplete or two-parameter parametrizations.
• To establish a foundation for computer-assisted searches by deriving conditions under which the face diagonal becomes rational.
• To explore the equivalence of the perfect cuboid problem to the existence of rational solutions satisfying three square conditions on rational parameters.

Proposed method

• Derives a rational parametrization of the identity $(1 - T^2)(1 - (4T^3 - 3T)^2) = [(1 - T^2)(1 - 4T^2)]^2$ to generate solutions for two of the three square conditions in the NPC problem.
• Uses the substitution $T = \frac{t^2 + 3}{4t}$ to generate rational parameters $\xi$ and $\zeta$, satisfying $\xi\zeta = \square$ and $(1 - \xi^2)(1 - \zeta^2) = \square$, where $\square$ denotes a rational square.
• Applies three different generating systems from prior work to derive three distinct parametrizations (I, II, III), each corresponding to a different configuration of the irrational diagonal.
• Transforms the parametrized rational parameters $\alpha$, $\beta$, $\gamma$ into explicit polynomial expressions for the side lengths $a$, $b$, $c$, and all face and space diagonals.
• Derives three distinct one-parameter families of NPC solutions, each with rational expressions for all sides and diagonals except one face diagonal.
• Establishes that if any of the three parametrized expressions for $d_{ab}^2$ becomes a rational square for some nontrivial rational $t$, a perfect cuboid exists.

Experimental results

Research questions

• RQ1 Can rational one-parameter families be constructed for nearly-perfect cuboids where only one face diagonal is irrational?
• RQ2 What algebraic identities and parametrizations can satisfy the necessary square conditions for the existence of such cuboids?
• RQ3 How can the Diophantine system for the perfect cuboid be reduced to rational parametrizations that facilitate computational search?
• RQ4 Do the derived parametrizations cover all possible NPC solutions, or are there missing rational solutions to the square conditions?

Key findings

• Three rational one-parameter parametrizations (I, II, III) are constructed for nearly-perfect cuboids with one irrational face diagonal, using a single rational parameter $t \neq 0, \pm1, \pm3$.
• The first parametrization yields $a = 16t^2(t^4 - 9)$, $b = (t^4 - 10t^2 + 9)(t^4 + 2t^2 + 9)$, and $d_{ab}^2 = [a]^2 + [b]^2$ is rational if $t$ satisfies a specific square condition.
• The second parametrization gives $a = 16t^2(t^4 - 9)(t^4 - 2t^2 + 9)$, with $d_{ab}^2$ rational under the same condition, derived from a different generating system.
• The third parametrization produces $a = (t^4 - 1)(t^4 - 81)$, and again $d_{ab}^2$ is rational if the corresponding expression becomes a rational square.
• The paper establishes that a perfect cuboid exists if and only if one of the three parametrized expressions for $d_{ab}^2$ is a rational square for some nontrivial rational $t$.
• The parametrizations provide a complete algebraic framework for a computer search: if any such $t$ is found, a perfect cuboid is confirmed.