Modeling nature

This activity belongs to the GeoGebra book Attractive projects. 3D Project: model the different types of shells. Note: despite the apparent difficulty, this project is simple because the formulas are reduced to a list of 7 given parameters. The parameterization follows the article Modells for shell mollusks of M.B. Cortie (1989). Surfaces can be parameterized and can also create families of surfaces whose shapes vary by varying the parameters that define them. A spectacular case is the growth pattern defined by self-similarity. The mollusks that create these shells should be enlarged as they grow. In many cases, this entails sealing the previous compartment when the new chamber is ready to reside in it. As all the compartments are added, and the form is preserved, a spiral structure is produced with a self-similar pattern that can be modelled. The syntax of the Surface command is:
  • Surface (x (u, v), y (u, v), z (u, v), u, u1, u2, v, v1, v2)
In this particular case, we have taken note:
  • x(u, v) = r e^(f u) (d + a cos (v)) cos (c u)
  • y(u, v) = r e^(f u) (d + a cos (v)) sin (c u)
  • z(u, v) = r e^(f u) (-1.4e + b sin (v)) + h
where r, f, a, b, c, d, e and h are the parameters of each shell. In this link you can see a more general version, with 15 parameters instead of 7.
Author of the construction of GeoGebra: Rafael Losada