Beautiful Seashells: Patterns, Shapes, Mathematics and Non-linearity: Their Pedagogical and Scientific Applications.

Presenter: Karen VanderVen
Discussant: Fred Abraham

These two seashell collectors will relate the shapes and patterns of seashells to some basic mathematical concepts and to non-linear dynamical systems theory. Illustrated with real shells and photographs, the presentation will include mathematical concepts as the Fibonacci numbers, the Golden Mean, and exponential spiral formation with their application to shell formations and patterns, and discuss how dynamical systems processes such as bifurcation determine various complex pattern formations in seashells. We have used these concepts to motivate elementary and high school students to ride their curiosity into science, math, and computers, and into the sheer enjoyment of learning. Some pedagogical examples will be shown from projects in elementary and high schools in Vermont and the Philippines (Abraham, 1997).

In addition to the basic spiral features of shells, their patterns of colorations are more challenging to portray. Following the use of neural net theory to this pattern formation (Ermentrout, Campbell, & Oster, 1986), recent evidence suggests a sensory-neural dynamical system that may further explain the development of these patterns (Boettiger, Ermentrout, & Oster, 2009). These models may provide understanding the basis of the bifurcations in these patterns, like those seen in Oliva porphyria.

Abraham, F.D., & friends (1997-8). Synergistic dynamics of art, mathematics, & science In nature and culture: An interdisciplinary International Project, Silliman University High School, 1997-1998, and Stowe Middle School, VT, USA.
http://www.blueberry-brain.org/syndyn/suhs/index.html and
http://www.blueberry-brain.org/syndyn/index.htm

Ermentrout, B., Campbell, J., & Oster, G. (1986). A model for shell patterns based on neural activity. Veliger, 28(4), 369-388.

Boettiger, A., Ermentrout, B., & Oster, G. (2009). The neural origins of shell structure and pattern in aquatic mollusks. Proc Natl Acad Sci USA, 106(16):6837-42.

image from Richard Ott (supreme Jungian, Molluskovite, and Plastic Surgeon), unfortunately unable to attend the conference