Converging and Diverging Ball Arches

Two Ball Arches over Mathemalchemy

When you first saw Mathemalchemy, what struck you the most? Let’s guess it’s the two arches showing balls (spheres) of different sizes. Although the spheres in both arches become arbitrarily small, the spheres in one arch extend indefinitely, crashing into the ocean and plummeting into its depths. The spheres in the other arch approach a single point in space—that arch does not grow without bound.

These arches serve the artistic purpose of adding color to the vertical elements of the piece. In early versions of the project, they were important as drawing the eye to the highest point of the east side of the work. The arches continue to serve as a physical connection between the page honoring five women mathematicians and the surrounding ocean, but also communicate a fundamental principle in mathematics.

Sequences and series

Figure 1: Partial sums sequence

Converging principle with partial sums sequence
This example shows that .9 repeating equals 1.

The mathematical principle behind these arches is that of sequences and series. If we think of a sequence of numbers, we  typically figure an infinite list of numbers. We can create a second sequence, called the sequence of partial sums, by creating a running total of the sum of the numbers in the first sequence. The sum of a sequence (its series) might be finite (convergent) or not (divergent).

Converging Ball Arch

First, let’s focus on the short arch, which we call the converging arch.  Consider the list of diameters of the spheres in decreasing order, beginning with the largest sphere. This list comprises the sequence in which we are interested.  We constructed this arch by choosing a corresponding series that we knew to converge.  That is, we knew that the sum of the terms was finite.  We scaled the terms so that the arch would have our desired length. 

Converging sequence on Desmos
Converging sequence in orange and its sequence of partial sums in black.
Converging ball arch
Converging arch figuring in Dominique Ehrmann’s maquette.

In other words, we worked backwards: we made the first sphere have diameter equal to the first term in the sequence, the second sphere have diameter equal to the second term in the sequence, and so on. Quite quickly, after only 23 spheres, the diameters of the spheres are less than one inch!  Hence the length of the arch is growing very slowly.  Because there are theoretically an infinite number of spheres in this arch, we were only able to include a portion of the arch. Nonetheless, the viewer can get a good sense of its full length from what is shown.

Figure 2: Illustration of the convergent principle

Convergent principle in animation

The example in figure 1 shows that even an infinite sum of numbers can add up to be finite. For a visual illustration of the same principle (using a different sum), see figure 2.

Notice that in figure 3, we see that each color comprises 1/4 of the figure at the first stage of coloring. We can then intuit that at the next stage of coloring each color comprises 1/4 of the remaining 1/4 or 1/16 of the total figure, and so forth. 

Figure 3: First stage of the convergence

Each color comprises 1/4 of the figure at the first stage of coloring

This leads us to the series:

C = n=1
1 / 4n

We can see that the sum of this series is 1/3, as the three colored regions are congruent and cover the entire figure.

What kinds of visualizations have people made for convergent series? Can you come up with new visualizations?

Diverging Ball Arch

Turn now to the long arch, which we call the diverging arch. The sphere diameters are still growing arbitrarily small, but the diameter size is not shrinking as fast as for the converging arch. We scaled this series so the first term would be the same as for the converging arch, thereby they would share the same first sphere. 

Diverging ball arch
Diverging arch figuring in Dominique Ehrmann’s maquette.
Diverging sequence on Desmos
Diverging sequence in orange and its sequence of partial sums in black.

Like the converging arch, this arch theoretically contains an infinite number of spheres, so we can only include a finite number.  Yet the length of this arch is infinite, or so we should imagine it to be.  (An infinitely long arch would not fit inside of our world.)

One of the beautiful aspects of mathematics is how the imagination naturally comes into play as we move into the theoretical realm.

Converging geometric series principle in video

You understand french? This Youtube video about converging geometric series is very well done and illustrates this concept.

Zeno’s Path in Tess the Tortoise‘s scene is also based on the Converging theory

Published by Carolyn Yackel

Traditionally trained in mathematics with a PhD in commutative algebra from the University of Michigan, Mercer University professor Carolyn Yackel has dedicated two decades to developing the field of mathematical fiber arts, including co-editing three books on the topic. Her other professional interests include recreational mathematics and undergraduate mathematics education. She actively works to engage others in exciting mathematics. Her own approaches to making mathematics visible through art involve a variety of mathematical ideas, techniques and media, including temari balls, knitting, crocheting, and more recently digital art, laser cutting, and shibori dyeing.

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