The Geometry of Sea Shells

1. The Geometry of Sea Shells Amanda Daniels 04.21.04

2. Background… • Shells grow at the margins where the inner most part of the adult coil is the baby shell. • Since they have this growth pattern their growth is able to be modeled by a Helico-Spiral.

3. Background continued… • The spiral rotates around a fixed axis which always remains geometrically similar to itself (increasing its dimensions continually). • To create the actual shape of the seashell, a Generating Curve rotates around the spiral increasing its size as it spirals down. • Any dimension of any seashell can be found by one of three parameters.

5. Three Parameters… Assuming that the generating curve is a circle, shells can be expressed as change in 3 parameters: • W (flare) = Expansion rate of the generating curve. • D (verm) = Rate of increase of the distance of the generating curve from the center axis. • T (spire) = Rate at which successive whorls of the spiral creep along the length of the axis.

6. First Parameter… W = Rate of expansion of the generating curve: • W=2 means that in every coil, the opening is twice as large as the previous turn. • Thus larger W’s give larger rate of expansion i.e. in shells such as cockles, W is up in the 1000’s. • Cannot have negative values because that would indicate shrinkage.

7. Second Parameter… D = Rate of increase of the generating curve from the center axis: • D=0.7 means that the distance from the center of the spiral to the inner margin of the tube is 70% of the distance from the center of the spiral to the outer margin of the tube. • A very high D (such as .99), makes a very thin, worm-like tube.

9. Third Parameter… T = Rate of shell expansion along vertical axis: • A small T gives little to no height in a shell. • The Nautilus has a T=0, thus giving a shell with height equal to the diameter of the tube. • No limits on values, negative just indicates the shell is upside down.

11. Three Parameters… • For a shell to be “snug” fitting, it depends on the values of W and D. • For a critical D value to make the shell “snug”, it must be the reciprocal of W. • ex) W=2, D=0.5 W=10, D=0.1 • If the D value is smaller than the critical value then this is presents a case where the shell “takes a bit” out of the previous swirl, thus where the third parameter comes in.

12. Parameters continued… • Parameters remain constant in shells. • The helico-spiral expands either as an Archimedean or a Logarithmic spiral. • Archimedean”sailor on a ship with a rope”, same size, width, etc. • Logarithmicspiral opens as it comes from middle, always the same rate for each spiral.

13. Parameters continued… • Most shells that can be made from the range of parameters are non-livable. • Said to be structurally weak and cannot be used as a source of protection. • The more spaced out shells seem to be more suitable for a worm to live in rather than a snail, and other such “shells” can be seen as horns of bison or antelope.

15. Modeling the Shells… • In order to model shells in 3-D, we must use a more sophisticated series of equations. • The equations relate to W, D, and T but are more complicated in order to create exact replicas of the shells. • These equations can create any shell possible by different values of the parameters. • Each parameter controls a different growth pattern on the shell and can be altered to create an entirely different shell.

16. Modeling the Shells… • The first step to model the shell is to define the helico-curve. Angle of Rotation: Radius: Vertical Displacement:

17. Modeling the Shells… • The first two equations represent the helico-spiral. • The third equation stretches the shell along the z-axis. • The parameter t, which is time, ranges from 0 (the beginning of the shell) to tmax (the opening of the shell).

18. Modeling the Shells… • Since the rate of change of r (and z) with respect to t is proportional to its size at r(t) (and z(t)) at any time then: • , where k is a constant • They usually have the same (or similar) growth rate, • The growth rate alters the silhouette (the slope of the spiral path).

20. where corresponds to Modeling the Shells… • To determine the value of kr and kz, we need to use the value which was taken from a source that used precise measurements on actual shells. • Also need to use 2 equations given to us from the same source

21. Modeling the Shells… • Next, directly related to the curve, you need to introduce the Frenet frame. • The Frenet frame controls the tilt on the shell opening. • It is an orthogonal coordinate system characterized by the unit vectors t, n, and b. • Basically, the Frenet Frame orients the generating curve to the helico-spiral such that the shell opening and the ridges (if any) lie in the plane normal to the helico-spiral. • Otherwise, the shell opening and the ridges will be parallel with the z-axis.

22. Frenet Frame… • T(t) = unit tangent • N(t) = unit normal • B(t) = unit binormal • The three vectors should be conceived as being attached to the point rather than being functions of the parameter t.

25. Modeling the Shells… • Next, the surface of the shell is determined by the Generating Curve. • The Generating Curve increases size as it sweeps down the spiral. • It is controlled by the shape of the opening of the shell, the rate at which the shell opens, and r(t). • Also, you need to orient the Generating Curve with the associated Frenet frame.

26. Relation of W, D, and T… • W  still just the flare or rate of expansion of the generating curve. • D  • T  kz, since it is the rate at which the shell goes along the z-axis and kz is the parameter that controls that in where flare is constant and there is no frenet frame.