The Strange World of the Hausdorff Metric Geometry

XII. Segments with a Finite Number of Elements at Each Location

The 2004 REU group (Kris Lund and Patrick Sigmon) found several different interesting families of Hausdorff segments. For example, let A = {a₁, a₂, …, a_k} and B = {b₁, b₂, …, b_m} be points on a Euclidean line with a₁ < b₁ < a₂ < b₂ < …, where either k = m or k = m+1 and the points are uniformly spaced. We call the collection of all such sets (along with A and B) a string segment, and the configuration [A, B] a string configuration (denoted S_k+m). As another example, if we alternately place the points in A and B along the vertices of a 2k gon, then the number of elements C in H(Rⁿ) at each location satisfying ACB is a Lucas number. We call such a segment a polygonal segment and the corresponding configuration [A, B] a polygonal configuration (denoted P_k). In [14] we show that

#(S_k) = F_k-1 and #(P_k) = L_2k,

where F_k is the k^th Fibonacci number and L_k is the k^th Lucas number

Another example of a family of Hausdorff segments is the family of star segments. In this family, the set A consists of (n+1)/2 points and B consists of (n-1)/2 points arranged in a star pattern. In these cases, there are 2ⁿ–1 different elements in H(Rⁿ) at each location between A and B. Examples of these segments can be seen in the next applet. You can choose a type of segment from the Segment Chooser, then enter the appropriate number of points. The element A is shown in red and B in blue. Recall that every element at a specific location between elements A and B in H(Rⁿ) is a subset of the union of (A)_s and (B)_t for suitable values of s and t with s+t = h(A, B). The boundaries of these dilations are shown in red and blue as well. By moving the black point on the slider you can see the maximal point at each location shown in gray.

As we vary the values of s, we can “trace” out the path traveled by the sets (A)_s. In doing so, we obtain a visual representation of a Hausdorff segment between A and B that we call the trace diagram. You can see the traces represented in the applet by toggling the Trace Toggle option.

We have examples of elements A and B in H(Rⁿ) with a variety of different numbers of elements at each location. A natural question to ask is, given a natural number k, can we find elements A and B in H(Rⁿ) so that there are exactly k elements C in H(Rⁿ) at each location satisfying ACB? We have found configurations that allow for any number between 1 and 18 elements at each location. These can also be seen in the applet. In the next section we consider what other integers can appear at the number of elements at each location between sets A and B