I would like to discuss the Jordan curve theorem as part of an intrinsic human activity: storytelling.

The Jordan Curve Theorem as a Lusona

Allechar Serrano López
Harvard University

Introduction

The Jordan curve theorem is a result in topology that states that every Jordan curve (a plane simple closed curve) divides the plane into an “inside” region enclosed by the curve and an “outside” region. We can think of a plane simple closed curve as a closed loop that does not intersect with itself. The theorem feels true: it intuitively makes sense and we do not have to spend several minutes trying to convince ourselves it’s true. We have seen this theorem in action in our lives… this is why fences work, right?

The math

In order to state the theorem in a formal manner, we need a definition:

A Jordan curve $C$ is a simple closed curve in $\mathbb{R}^2$. We can construct such a curve as the image of a continuous map $\phi: [0,1] \rightarrow \mathbb{R}^2$ such that:

1. $\phi(0)=\phi(1)$, and
2. the map $\phi$ is injective (also known as: one-to-one) on the interval $[0, 1)$.

Here, condition (1) makes sure that we have a loop, and condition (2) ensures that our loop does not have any self-intersection points.

Then we can state the theorem as follows:

Jordan curve theorem

Let $C$ be a Jordan curve in the plane $\mathbb{R}^2$. Then its complement, $\mathbb{R}^2\setminus C$, consists of two connected components. One of these components is bounded (the interior) and the other one is unbounded (the exterior), and the curve $C$ is the boundary of each component.

So, the Jordan curve separates $\mathbb{R}^2$ into two pieces: an inside region (which has a finite area) and an outside region.

Around the point, around the point…

If we have a point on the plane and draw a closed curve, we can try to figure out if the curve encloses the point (and how many times it goes around it) or if it doesn’t. First, we start by giving the curve an orientation; mathematicians chose a long time ago that counterclockwise is the way to go, so the winding number is positive if the curve encloses the point counterclockwise. The winding number of a closed curve around a given point is an integer that counts the number of times that the curve goes around the point. If the curve does not encircle the point, then the winding number at that point is 0. The winding number of a Jordan curve around a point in its interior is 1 (or -1 if we travel in the other direction!)

While the definition of winding numbers seems straightforward, they come up in some of the advanced undergraduate- and graduate-level courses like differential geometry and complex analysis. In complex analysis, they came to haunt me disguised as line integrals. In fact, Stokes’ theorem and the residue theorem are related to the Jordan curve theorem.

The Jordan curve theorem as a lusona

I was first introduced to the Jordan curve theorem in my algebraic topology class. Like most definitions and theorems, it was introduced in clinical detail: as part of a long list of results with no mention of a context or why would anyone care about it. However, mathematics presents itself in different ways to different groups of people, so I would like to discuss the Jordan curve theorem as part of an intrinsic human activity: storytelling.

The Chokwe people live in Southwestern Africa and they are known for their art, which they also employ in their storytelling. They have a tradition of drawing figures in the sand, these are known as lusona (plural: sona), to illustrate their stories. The sona illustrate fables, games, riddles, proverbs, and stories. Each lusona starts with a series of evenly-spaced dots, in a rectangular array, and the drawing consists of lines weaving in and out around the dots.

The storyteller draws and narrates simultaneously while keeping the audience engaged. The sona and the stories accompanying them played an important role in the passing down of knowledge and traditions from one generation to the next, but many of them were lost due to colonization and slavery. What we know about the sona today comes from documentation kept by missionaries.

The tradition of the mukanda

The mukanda is a rite of passage for boys into adulthood and it begins when the chief of a Chokwe village and his counselors decide that there is a sufficiently large group of children to carry out the rite. The mukanda is a camp enclosed by a fence with huts for the boys; the length of the stay at the camp varies from one year to three years. In the mukanda, they learn rituals, stories, and how to make masks, and they can return home after the prescribed education is complete. Kalelwa, a spirit who is incarnated by a mask of the same name, is who gives the signal for the coming and going from the mukanda, and mothers are not allowed to see their sons while they are going through the rite of passage.

There are several sona referring to the mukanda. A lusona which is a continuous closed curve with no self-intersections includes a story where the line of dots are the children involved in the rite of passage, the two higher dots are the guardians of the camp, and the lower dots represent people who are not involved in the ceremony. The children and the guardians are inside the camp and so they are in the bounded connected component while people not participating in the ceremony are in the unbounded connected component.

The mukanda exemplifies a topological concern of the Chokwe: distinguish between the inside of the mukanda (where the children are) and the outside world, that is, the need to determine two regions with a common boundary (which is exactly the issue that the Jordan curve theorem addresses!).

Further Reading

• Marcia Ascher. Mathematics elsewhere. An exploration of ideas across cultures. Princeton University
  Press, Princeton, NJ, 2002, pp. xii+207. isbn: 0-691-07020-2; 0-691-12022-6.
• Marcia Ascher and Robert Ascher. “Ethnomathematics [Hist. of Sci. 24 (1986), no. 64, part 2, 125–144;
  MR0854948 (88a:01005)]”. In: Ethnomathematics. SUNY Ser. Reform Math. Educ. SUNY Press, Albany,
  NY, 1997, pp. 25–50.
• Paulus Gerdes. Geometry from Africa: mathematical and educational explorations. Classroom Resource
  Materials Series. With a foreword by Arthur B. Powell. Mathematical Association of America, Washington,
  DC, 1999, pp. xvi+210. isbn: 0-88385-715-4.
• Camille Jordan. Cours d’analyse de l’École polytechnique. Tome III. Les Grands Classiques Gauthier-
  Villars. [Gauthier-Villars Great Classics]. Équations différentielles. [Differential equations], Reprint of
  the third (1915) edition. Éditions Jacques Gabay, Sceaux, 1991, pp. iv+631. isbn: 2-87647-018-7.