Bound on the Number of Geometric Permutations for Congruent Balls

Title: Bound on the Number of Geometric Permutations for Congruent Balls

Research Question: What is the maximum number of distinct geometric permutations possible for a set of congruent balls in a d-dimensional space?

Methodology: The researchers used computational geometry techniques and discrete mathematics to analyze the possible arrangements of congruent balls in a d-dimensional space. They considered the number of distinct geometric permutations that could occur when a line passes through all the balls in a set.

Results: The researchers found that for a set of congruent balls in a 2-dimensional space (the plane), there can be at most 2 distinct geometric permutations. They also proved that for a set of congruent balls in a d-dimensional space (d ≥ 3), there can be at most 4 distinct geometric permutations.

Implications: This research has important implications in various fields such as computer graphics, robotics, and machine learning. The bounds on the number of geometric permutations for congruent balls can be used to optimize algorithms and improve efficiency in these fields. Additionally, this research contributes to the understanding of the behavior of geometric objects in high-dimensional spaces.

Link to Article: https://arxiv.org/abs/0102004v1 Authors: arXiv ID: 0102004v1