The Strange World of the
Hausdorff Metric Geometry
XI. Segments with a Finite Number of Elements at Each
Location
The 2004 REU group found several different interesting families of Hausdorff segments. For example, let A = {a1, a2, , an} and B = {b1, b2, , bm} be points on a Euclidean line with a1 < b1 < a2 < b2 < , where either n = m or n = m+1 and the points are uniformly spaced. In this case, for each positive number s, 0 < s < h(A.B), the number of elements C in H(Rn) satisfying ACB and h(A, C) = s (in other words, the number of elements at each location between A and B) is a Fibonacci number. We call the collection of all such sets (along with A and B) a string segment. 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. These we call polygonal segments. See [13] for details. 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 2n1 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 extensions 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