Once parameters are varied,‭ ‬different forms can be produced.  These forms can either mimic pre-existing or model hypothetical possibilities.  These hypothetical possibilities or morphospaces lead us to the question of why these forms are not present in the fossil record.  Therefore these models can be the foundation of the study of the actual mechanism of evolution and the analysis of organic form.

One method for developing a model of shells is to create a formula to represent its trajectory of aperture movement.  In this study,‭ ‬we estimated the shape of the aperture to be an ellipse.  Coordinates of this ellipse were taken‭ (‬center,‭ ‬horizontal and vertical radial distance.‭)  ‬This data was collected and used in add-on software packages to Mathematica.

CerioShell‭ (‬author:‭ ‬J. Stone‭) ‬is an add-on to Mathematica‭ ‬5.2.  An‭ ‬intermediate software package,‭ ‬Mappoint‭ (‬author:‭ ‬G. Beck‭) ‬bridges the two programs.  Mappoint uses both classical Euclidean and modern differential geometry to develop mathematical functions for graphing.  The utilization of these graphing functions to develop shell surfaces,‭ ‬whorl coiling and patterns of accretion/formation is the focus of CerioShell.  All products run in a Microsoft Windows environment.

PreDraw3D uses arrays based off the coordinates supplied by the user to render two-dimensional shapes‭ (‬i.e. polygon or circle‭) ‬for each aperture.  Using iterations,‭ ‬these shapes are repeated so that a representation of the shell can be made‭ (‬i.e. a series of apertures represented by the‭ ‬2D line drawing‭)‬.  Angles between these coordinates are calculated and then the sin and cosine of these angles are taken to be used to alter the rotational angle of how the aperture will turn through space as it travels the aperture trajectory.

CerioShell uses the above function for example in its GastropodShellRotate function.  This is the function that provides the appropriate angle for the rotation around the columella.  In the beginning of the code,‭ ‬it is possible to see which functions the GSR.M file are calling and which will subsequently be available to this function during its session.

Viewing the accretion of the shell as a series of apertures moving through time is one manner of how to look at growth patterns.  Another approach is to view the shell from the apical side and analyze the curve of the spiral.  Below is a picture that shows this curve on the shell. A juvenile was chosen due to the fact that the origin is clearly visible,‭ ‬during later stages of growth‭; ‬this feature is not easily seen.

For working with spirals,‭ ‬the beginning or origin of the spiral is assigned to be the polar coordinates of‭ ‬0,‭ ‬0.  As the spiral extends and rotates through space,‭ ‬it can be thought of in two forms.  An equiangular or logarithmic spiral is one where the point along the line/curve travels exponentially from the origin.  If there is uniform traveling of the point from the origin,‭ ‬it is considered to be an Archimedean spiral.