Instructor's
Notes:
Anna Mullikin (1893-1975) was a virtually
unknown doctoral student of R. L. Moore until recently, when her life and work
were brought to light in an article by Thomas L. Bartlow and David E.
Zitarelli [1]. Her Ph.D. dissertation, which was her only published work,
includes an example that can be used to illustrate limits of geometric figures
in the plane [2].
We begin by illustrating
what is now called Mullikin’s nautilus. Our ultimate goal is to find the set
of limit points of the nautilus. Since that involves determining limits in
the plane geometrically, we backtrack to the real line to demonstrate how
students (in a calculus or a real analysis course) can be introduced to
geometric limits of figures as opposed to the usual algebraic approach using
functions. We provide an example in which the limit of a set can be
calculated in dimension one. Only then do we move to the plane and Mullikin’s
nautilus.
Mullikin’s nautilus
consists of an infinite collection of arcs Mj, for j
=1, 2, ... , each of which is composed of four line segments. In her paper
she provided a sketch of the first three arcs, M1, M2,
and M3, which we display in Figure 1 using different colors to
distinguish the arcs. In each case, the arc begins at a point on the x-axis,
then connects to points in the first, second, and third quadrants, and finally
ends at a point in the fourth quadrant whose abscissa is x = 1.
Figure 1. |
We provide two ways to
visualize Mullikin’s nautilus. The first generates it one arc at a time using
either Excel or Matlab. To use the Excel file, click on Figure 2 to execute
or download this file.
Figure 2. |
Alternately, a MATLAB
m-file allows for zooming in for more detail; to download this m-file click
here. An even more dynamic visualization can be seen with a QuickTime
movie that can be viewed by clicking
here.
Classroom use
Beginning calculus
students encounter the concept of a limit in both differentiation and
integration. In differentiation, the approach to limits is usually algebraic,
centered around the limit of a function as the variable approaches a fixed
number. In definite integration the student encounters areas as limits of
partial sums, a concept met again in series, where sums (when they exist) are
defined as limits of sequences of partial sums.
Our approach is more
geometric than algebraic. We begin with an example illustrating limit points
geometrically on the real line that is different from what is generally
presented in an introductory calculus course. Next, we extend the limit
concept to the plane, again in a geometric manner.
Geometric limits on
the real line
For n
= 1, 2, 3, …, define the closed-open interval
.
For
instance, J1 = [0, ½).
Clearly the closure of is
=
[0, ½]. Students accustomed to an algebraic approach should be reminded that
the sequence
of numbers in
converges
to ½. They might also be reminded that the formal reasoning for confirming
this assertion proceeds as follows: whenever you take an open interval U
containing ½, no matter how small, there is some number in this sequence
beyond which all subsequent numbers in the sequence lie in U.
The goal is to calculate
all limits points of the set J that is the union of these intervals:
J =
. To motivate this in a particular case, form
the set S = J1
J2
J3 that
is the union of three closed-open intervals and find its closure
S.
Click here
to see both S and
S geometrically. With this in mind, it is easy to see that the closure J
of
J = is
the unit interval [0, 1].
Geometric limits in
the plane
We now move from limits
on the line to limits in the plane, with open circles taking the place of open
intervals. Recall that a sequence of points in the plane converges to a point
P if for each open circle
C(P, ε) with
center P and radius epsilon, there is some
point in this sequence beyond which all subsequent points in the sequence lie
in C. For example, the point is a
limit of the sequence
because for each open
circle C with center there
is some point in the sequence for which all subsequent points in the sequence
lie in C.
Mullikin’s nautilus
Now we introduce
Mullikin’s nautilus formally and use it to illustrate how to compute a set of
limit points geometrically. For each positive integer n, define an arc Mn to be composed
of four line segments drawn from the x-intercept
to
,
thence to
and 
, and finally to
.
The abscissa of the final point is
always x = 1.
For example, M3 is composed of the line segments
connecting
to
to
to
to
.
As noted in the introduction, the topologist Anna Mullikin sketched the first
three arcs in the only research article she published in her lifetime. Her
insightful idea was to form the union
,
which is called “Mullikin’s nautilus” and is denoted by M in her honor.
We proceed to find the
closure
M by
calculating the limit points of M. Miss Mullikin stated that that each
arc
Mj
contains a
limit point of every infinite subset of the collection
{Mn, n ≠ j}.
We consider a special case of the statement by showing that
is
a point on
M3 that
is a limit of the entire collection {Mn,
n ≠ 3}. Click
here to see a QuickTime
movie demonstrating the convergence geometrically. In similar fashion, all
points of the form
for
n = 0, 1, 2, … are limit points of Mullikin’s nautilus
,
so they all lie in
M.
What about an arbitrary
point , where x lies in
the unit interval ? Notice that for each
open circle C with center P, the sequence of points
converges to P and
each lies on the fourth
line segment of the arc
Mn.
Consequently is
a limit point of M. Therefore
M consists
of Mullikin’s nautilus M and this interval of points. Click
here to
see a QuickTime movie depicting
M geometrically.
References:
1.
Thomas L. Bartlow and David E.
Zitarelli, Who Was Miss Mullikin? [to appear]
2.
Anna M. Mullikin, Certain
theorems relating to plane connected point sets, Trans. AMS 24,
(1922) 144-162.
