Mullikin's Nautilus

Click here to see a QuickTime
movie of Mullikin's Nautilus.

 

 
Objective: To demonstrate limits geometrically, first on the real line and then in the plane.  We begin with an example on the real line that motivates the extension to the plane that is based on an example provided by Anna Mullikin in her one published paper.

Level: Calculus, advanced calculus, or real analysis.

Prerequisites: Basic understanding of convergence and limit points.

Platform: PC or MAC with QuickTime; routines in both Excel and Matlab are included.

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, ).  Clearly the closure of  is = [0, ].  Students accustomed to an algebraic approach should be reminded that the sequence

 

of numbers in  converges to .  They might also be reminded that the formal reasoning for confirming this assertion proceeds as follows: whenever you take an open interval U containing , 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.

Credits:  This demo was submitted by 

David R. Hill and David E. Zitarelli
Mathematics Department
Temple University

and is included in Demos with Positive Impact with their permission.


Created 6/5/2007 DRH         Last updated 7/9/2007 DRH

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