The circled surfaces generated by the circles centered on a conical helix, located on the plane passing by the axis of the cone, and with radii proportional to the distance of their center to the vertex of the cone, have a shape very similar to that of various snails or sea-shells.
 

Cartesian equation: , when the equation of the cone on which the conical spiral is traced is  (k = cotangent of the half-angle at the vertex of this cone). For , each coil is tangent to the next; opposite m = –1/10 and k = 3.

 
 

The case m = –1/20 and k = 200 gives a shape very similar to that of the turritella!

A real turritella

See the same sea-shells thanks to Povray rendering.
 
 

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© Robert FERRÉOL  2017