User Case Study: The Art of Modeling Seashell Morphology

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Mathematical modeling is often related to scientific and engineering applications, with many examples coming from biological phenomena.

Consider a delicately sculpted seashell. We admire its beauty and structural features but we often don’t think for a second that mathematics has anything to do with the way it is crafted.

Modeling the varied shapes of seashells is an appealing application of several mathematical concepts that are introduced in multi-variable calculus. And despite the staggering variety of shell shapes found in nature, the growth of almost all seashells can be described in terms of three exponential functions and a closed curve that describes the shape of the shell’s aperture, or opening.

The challenge in modeling the morphology of these seashells is that the process requires a tool that combines mathematical muscle, outstanding 3D graphics, and a powerful programming language. It’s in this combination of requirements that Maple truly excels.

Dr. Joanna Ellis-Monaghan, a professor in the Mathematics Department at Saint Michael’s College in Colchester, Vermont, teaches various math topics including calculus, algebra, combinatorics, and number theory. She uses Maple extensively throughout the calculus sequence as well as in her other courses. In Calculus III students need to create models that involve mathematical representations, data analysis and curve-fitting and 3D graphics. “It’s in this class that the students really appreciate the power and the beauty of Maple, and as a result, gain a greater appreciation of the subjects being studied,” says Dr. Ellis-Monaghan.

For a project in this particular course students use Maple to:

• Develop the parametric equations that model the surface of the shell.
• Generate 3D models of mollusk shells from growth measurements using Maple’s built-in curve-fitting tools.
• Develop a Maple procedure that takes as inputs the different parameters that define the shape and the color of the shell.
• Generate and visualize the 3D graphics of the seashells.

“Students use Maple on a daily basis, for involved calculations certainly, but in the Calculus III course, primarily for 3D visualization. They began to use it as a creative tool here, and it is not unusual to hear someone in the room hit the enter key, and then just say a quick ‘cool’ as the screen appears,” says Dr. Ellis-Monaghan. “With Maple’s built-in visualization tools, students get a deeper understanding of the math involved, and a greater appreciation for the math that exists in naturally occurring objects – like seashells.”

With colleagues Dr. George Ashline and Dr. Zsuzsanna Kadas from the Mathematics Department and Dr. Declan McCabe from the Biology Department, Dr. Ellis-Monaghan has published a UMAP (Undergraduate Mathematics and Its Applications)/ ILAP (Interdisciplinary Lively Application Project) module describing this seashell classroom project.

To read the full academic paper “Modeling Seashell Morphology” please click here.

Publication Reference: “Modeling Seashell Morphology.” 2009. UMAP Module 801. In UMAP/ILAP Modules 2009: Tools for Teaching, edited by Paul J. Campbell, 101-139. Bedford, MA: COMAP.