The Hausdorff Metric Geometry REU 2005 Report ∗

The Hausdorff metric h gives us a method of measuring the distance between non-empty compact subsets of n-dimensional Euclidean space. Unlike the Euclidean concept of betweenness, there is not necessarily a unique set at each location between two other sets in the Hausdorff metric geometry. A configuration defines two sets (infinite or finite) for which it is possible to have a finite number of elements at each location between sets. We will first consider infinite and finite sets for which it is possible to have a finite number of sets at each location between two sets, noting that there is no configuration with exactly 19 elements at each location between two sets. We will conclude by connecting the number of sets at each location between two sets, making up specific finite configurations, and Fibonacci numbers.