** **

The Strange World of the
Hausdorff Metric Geometry

** **

## XIII. 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.

Through an
exhaustive argument in which we checked all finite configurations *X * that could possibly satisfy #(*X*) = 19, in the 2004 GVSU REU, Kris Lund, Patrick Sigmon, and I showed that
there is no finite configuration *X*_{F}
for which #(*X*_{F}) = 19 (see
[3]). 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 *X*_{F}
with #(*X*_{F}) = #(*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. To discuss the next theorems we
need to review subspace topologies.

**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 *T*_{C}, is the set {*O* ∩ *C *: *O *is in *T*}. If *U* is in *T*_{C}, we
say that *U* is a *relatively open* set.

The
subspace topology *T*_{C} 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*(**R**^{n})
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* : *d*_{E}(*a*, *c*) = *h*(*A*,*B*)}.

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

Figure 7

For
example, in the configuration shown in Figure 7, we can see that [*a*_{1}]_{C} = {*c*_{1},
*c*_{2}} and [*b*_{0}]_{C} = *C*_{1}.

Let [*A, B*] be a configuration and *C* be the largest element in *H*(**R**^{n}) satisfying *ACB* at a given location.

·
Let *q*_{A}
: *C* → [*A*]_{C} be defined
by *q*_{A}(*c*) = [*a*]_{C}, where *c* is in [*a*]_{C}.

·
Let *q*_{B}
: *C* → [*B*]_{C} be defined
by *q*_{B}(*c*) = [*b*]_{C}, where *c* is in [*b*]_{C}.

For
example, consider the configuration *X*
in **R**^{2} defined by *A* = {(0,0)} and *B* = {(*x*,*y*) : *x*^{2}* *+ *y*^{2}
= 1}. The only element between *A* and *B* that is 1/2 units from *A *is
the set C = {(*x*,*y*) : *x*^{2}* *+ *y*^{2}
= 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 *q*_{A}(*c*) = [(0,0)]_{C}. The adjacency
set given by *q*_{B}(*c*) will be different for every *c*.

A
straightforward computation shows that *q*_{A}
and *q*_{B} 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 *U*_{C} such that

·
for all [*a*]_{C} in *q*_{A}(*A*), there
exists *c *in [*a*]_{C} such that *c *is not in *U*

·
for all [*b*]_{C} in *q*_{A}(*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* = {*a*_{1}, *a*_{2}}, *B* = {*b*_{1}, *b*_{2}}, and *C* = {*c*_{1}, *c*_{2}, *c*_{3}}
as shown in Figure 8. We can see that [*A*,
*B*] is a configuration.

Figure 8

For
this configuration, we have Ω = {Ø, {*c*_{2}}}. It would be impossible to have *c*_{1} in any *U* in Ω because *q*_{A}(*c*_{1})
= [*a*_{1}]_{C} = {*c*_{1}},
which would mean that there would not exist another *c* in [*a*_{1}]_{C} such that *c* is not in *U*. A similar argument shows
that *c*_{3} cannot be in any *U* in Ω. In this example, *f*(Ø) = *C* and *f*({*c*_{2}})
= {*c*_{1}, *c*_{3}}, 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.