The Strange World of the Hausdorff Metric Geometry

 

XII. What is Special about the Number 19?

 

As mentioned in the last section, we can find sets A and B so that the number of elements at each location between A and B can be any Fibonacci or Lucas number, or any integer between 1 and 18. Now we investigate the number 19.

 

As we have seen, the PFAEL conditions determine when there can be a finite number of elements at each location between sets A and B. If A and B satisfy the PFAEL conditions, then we call X, the union of A and B, a configuration. We denote by #(X) the number of elements at each location between the sets A and B that comprise the configuration X. If both A and B are finite sets, we call X a finite configuration. Otherwise, X is an infinite configuration. Through an exhaustive argument, we have shown that there is no finite configuration XF for which #(XF) = 19. The question remains, however, if there is an infinite configuration X for which #(X) = 19? The answer turns out to be no. The Finite Conversation Algorithm and the Configuration Construction Theorem (described in the next section) show us that given any infinite configuration X so that #(X) is finite, we can construct a corresponding finite configuration XF with #(XF) = #(X). Therefore, infinite configurations provide no additional possibilities beyond the ones provided by finite configurations. We conclude the surprising result that there are no configurations X for which #(X) = 19.

 

Subspace Topology and Elements Satisfying ACB.

Let (X,T) be a topological space, and let C be a subset of X.  The subspace topology of C, denoted TC, is the set {O ∩ C : O is in T}. If U is in TC, we say that U is a relatively open set.

 

The subspace topology TC on a set C is a topology on C. We will eventually be able to connect the subspace topology of C with the elements at each location between A and B, but we need to make several more definitions in order to have the necessary tools to make this connection.

 

Let A and C be elements in H(Rn) and let a be an element of A. We define the adjacency set of a relative to C as the set

 

[a]C = {c in C : dE(a, c) = h(A,B)}.

 

We will let [A]C denote the set {[a]C : a in A}.

 

Figure 5

 

For example, in the configuration shown in Figure 5, we can see that [a1]C = {c1, c2} and [b0]C = C1.

 

Let A and B satisfy the PFAEL conditions and C be the largest element in H(Rn) satisfying ACB at a given location.

 

·        Let qA : C → [A]C be defined by qA(c) = [a]C, where c is in [a]C.

·        Let qB : C → [B]C be defined by qB(c) = [b]C, where c is in [b]C.

 

For example, consider the configuration X in R2 defined by A = {(0,0)} and B = {(x,y) : x2 + y2 = 1}.  The only element between A and B that is 1/2 units from A is the set C = {(x,y) : x2 + y2 = 1/4}. Then [(0,0)]C will be the set of all points on C, since (0,0) is adjacent to every one of these points.  Also, for any b in B, we can see that [b]C will be the single point on C that intersects a line drawn between b and the origin.  For any point c in C, we have qA(c) = [(0,0)]C.  The adjacency set given by qB(c) will be different for every c.

 

A straightforward computation shows that qA and qB are both well-defined and onto.  Now we can begin to define the function that will take an element of the subspace topology of C to an element between A and B.

 

Define Ω as the set of all U in UC such that

·        for all [a]C in qA(A), there exists c in [a]C such that c is not in U

·        for all [b]C in qA(B), there exists c in [b]C such that b is not in U

Now let Z be the set of all C* such that C* satisfies AC*B with h(A,C*) = h(A,C).  Note that every C* must be a subset of C [2].

Finally, define f : Ω  → Z by f(U) = C - U.

 

As an example, let A = {a1, a2}, B = {b1, b2}, and C = {c1, c2, c3} as shown in Figure 6. We can see that A and B satisfy the PFAEL conditions.

 

Figure 6

 

For this configuration, we have Ω = {Ø, {c2}}.  It would be impossible to have c1 in any U in Ω because qA(c1) = [a1]C = {c1}, which would mean that there would not exist another c in [a1]C such that c is not in U.  A similar argument shows that c3 cannot be in any U in Ω.  In this example, f(Ø) = C and f({c2}) = {c1, c­3}, both of which lie at the location shown between A and B.

 

In the next section we introduce the Finite Conversion Algorithm and the Configuration Construction Theorem.