Version 1
: Received: 22 July 2024 / Approved: 23 July 2024 / Online: 24 July 2024 (15:56:02 CEST)
How to cite:
Balasubramanian, K. Enumeration of N-dimensional Hypercube, Icosahedral, Rubik’s Cube Dice, Colorings and Encryptions Based on Their Symmetries. Preprints2024, 2024071853. https://doi.org/10.20944/preprints202407.1853.v1
Balasubramanian, K. Enumeration of N-dimensional Hypercube, Icosahedral, Rubik’s Cube Dice, Colorings and Encryptions Based on Their Symmetries. Preprints 2024, 2024071853. https://doi.org/10.20944/preprints202407.1853.v1
Balasubramanian, K. Enumeration of N-dimensional Hypercube, Icosahedral, Rubik’s Cube Dice, Colorings and Encryptions Based on Their Symmetries. Preprints2024, 2024071853. https://doi.org/10.20944/preprints202407.1853.v1
APA Style
Balasubramanian, K. (2024). Enumeration of N-dimensional Hypercube, Icosahedral, Rubik’s Cube Dice, Colorings and Encryptions Based on Their Symmetries. Preprints. https://doi.org/10.20944/preprints202407.1853.v1
Chicago/Turabian Style
Balasubramanian, K. 2024 "Enumeration of N-dimensional Hypercube, Icosahedral, Rubik’s Cube Dice, Colorings and Encryptions Based on Their Symmetries" Preprints. https://doi.org/10.20944/preprints202407.1853.v1
Abstract
The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice, and dice of other shapes that exhibit cubic, icosahedral and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as truncated octahedron, dodecahedron and Rubik’s cube shapes. It is shown that all enumerated dice are chiral and we provide the counts of chiral pairs of dice in the n-dimension space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating into the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography and biology are also pointed out. Applications to nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages; “I love U” and “V Elope at 2”.
Keywords
Monte Carlo Enumeration of n-dimensional dice; icosahedral and hypercubic symmetries; buckminsterfullerene; mesoporous and zeolite materials; isochiral polyhedra; cryptography
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
