Example Morphospaces

Raupian Coiling

Introduction

Raupian coiling is a mathematical model developed by David M. Raup to describe the three-dimensional geometry of mollusk shells. It assumes a cylindrical coordinate space.

Calculation of the Radius from the Coiling Axis

[ r_\theta = r_0 W^{\left(\frac{\theta}{2\pi}\right)} ]

Where:

• ( r_\theta ) = Distance after ( \theta ) revolutions
• ( r_0 ) = Initial distance of point A from the axis
• ( W ) = Rate of whorl expansion
• ( \theta ) = Angle in radians

Calculation of the Angle of the Coiling Axis in Planispiral Forms

[ y_\theta = y_0 W^{\left(\frac{\theta}{2\pi}\right)} ]

Where:

• ( y_\theta ) = y-value of the point after ( \theta ) revolutions in planispiral shells where ( T = 0 )
• ( y_0 ) = Initial y-value of the point
• ( W ) = Rate of whorl expansion
• ( \theta ) = Angle in radians

Calculation of the Angle of the Coiling Axis in Helicoid Forms

[ y_\theta = y_0 W^{\left(\frac{\theta}{2\pi}\right)} + r_c T \left( W^{\left(\frac{\theta}{2\pi}\right)} - 1 \right) ]

Where:

• ( y_\theta ) = y-value of the point after ( \theta ) revolutions in helicoid forms
• ( y_0 ) = Initial y-value of the point
• ( W ) = Rate of whorl expansion
• ( \theta ) = Angle in radians
• ( r_c ) = The r-value of the center of the initial generating curve
• ( T ) = Rate of translation defined as ( \frac{dy}{dr} ), with respect to the center of the generating curve

Figure 1a from Gerber et al 2017

References

1. Gerber, S. The geometry of morphospaces: lessons from the classic Raup shell coiling model. Biological Reviews 92, 1142–1155 (2017).