Master of Science - Research
School of Mathematics and Applied Statistics
Habib, Hamza Barakat, On the equitable distribution of points on the circle, Master of Science - Research thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2014. https://ro.uow.edu.au/theses/4102
In this thesis, we discuss the problem of breaking a stick representing resources, so that, as recipients arrive one at a time they get approximately even parts. To do this, we model the placement of points evenly distributed on the circle.
The thesis is in two parts, firstly a review of the various models including the Stick Breaking model of Erdos, hopping around the circle based on the Steinhaus three gap theorem (with Golden (Fibonacci) hops being optimal), a Binary Splitting model and Random distributions.
The second part consists of comparing and contrasting these models using various measures, such as, largest and smallest gap and overall discrepancies.
The second part consists of comparing and contrasting these models using various measures, such as, largest and smallest gap and overall discrepancies. Some of the results we have obtained are; the demonstration that of all uniform distributions, the hop model is the most even, and the random model the least even. We also have found simpler methods for describing the various distributions of points and discrepancy in Stick Breaking, as well as, the Golden hop. A comparison is also made with the distributions in the Benford first digit problem which has led to a novel approach to calculate the first digit distributions other than Benford.