Twice Punctured Euclidean and Hyperbolic Manifolds, Revisited as a Hypothetical Explanation for Quantum Dots

Abstract

Upon reading a newspaper article from December 2023, I realized that my 1984 conference paper might provide a possible explanation relevant to the Nobel Prize in Chemistry awarded to Alexey Yekimov, Luis E. Brus, and Moungi G. Bawendi. After corresponding with my colleague, Academician István Hargittai, he extended his congratulations. I subsequently presented the topic at several conferences in 2024, including those held in Budapest, Sopron, Belgrade, and Senj.
Naturally, the author could not have anticipated the relevance and significance of the topic at that time. This was an incidental consequence of my earlier erroneous paper [2], intended to construct an infinite series of non-orientable compact hyperbolic manifolds as a polyhedral tiling series in the Bolyai-Lobachevsky hyperbolic space H3. Fortunately, I observed and improved the mistake soon. Specifically, those constructions were not manifolds because the two fixed-point orbits represent punctures where point reflections (central inversions) occur in the symmetry group of the tricky polyhedral tilings.
But these singular points, interpreted as 'quantum dots' (e.g., copper and chlorine ions), respectively, in a silicon-based glass fluid that subsequently solidifies produce optical effects (due to electron transitions) whose colors might depend on the sizes of crystal particles.
This suggests that the unexpected result was more significant than the original intention that could be reached easily later!
The main point of this paper is that – in addition to an Euclidean crystal group 61. Pbca – I constructed infinitely many hyperbolic space groups (in Bolyai-Lobachevsky space) possibly providing such a material.