Novel required properties of, and efficient algorithms to seek, perfect cuboids
Aubrey de Grey, Philip Gibbs, Louie Helm
TL;DR
The paper advances the search for perfect cuboids by introducing two new algorithms rooted in the geometry of perfect plinths and perfect picture frames, connecting PC existence to bi-perfect cuboids (BPCs) and their four subtypes. It develops a robust elliptic-curve framework, deriving master equations that encode NPCs and 3-BPCs through EF1, EF2, EF3, EF4, EI1, EI2 curves, and demonstrates that many face and internal-rectangle aspect ratios are prohibited when these curves have rank zero. Computationally, the authors produce extensive enumerations: 288 F-BPCs and 122 I-BPCs up to large bounds with the first algorithm, and 1319 total F-BPCs/I-BPCs via the second algorithm, while finding no PCs to date; these results strengthen the conjecture that a PC may not exist. The work also provides two parametric families of edge cuboids and identifies intriguing relationships among F-BPCs, Leech pentacycles, and cousin transformations, offering new directions for theoretical and computational exploration. Overall, the study sharpens nonexistence evidence for perfect cuboids and lays a scalable, elliptic-curve–based path toward resolving the PC question.
Abstract
We present a novel approach to the age-old question of whether perfect cuboids exist. Our approach consists of two new computer search algorithms, arising from the analysis of "perfect plinths" reported by one of us recently, that are much more efficient than any prior algorithm of which we are aware. Using this approach, we also (i) identify some new two-parameter families of edge cuboids, and (ii) show that large proportions of aspect ratios of faces or internal rectangles cannot be present in a perfect cuboid, a property that was previously reported only for faces with the aspect ratio 4:3.