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1 The Geometry of Sea Shells Amanda Daniels 04.21.04.

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1 1 The Geometry of Sea Shells Amanda Daniels 04.21.04

2 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 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.

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5 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 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 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.

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9 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.

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11 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 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 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.

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15 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 16 Modeling the Shells… The first step to model the shell is to define the helico-curve. Angle of Rotation: Radius: Vertical Displacement:

17 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 t max (the opening of the shell).

18 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).

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20 20 Modeling the Shells… To determine the value of k r and k z, 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 where corresponds to

21 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 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.

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25 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 26 Relation of W, D, and T… W  still just the flare or rate of expansion of the generating curve. D  T  k z, since it is the rate at which the shell goes along the z-axis and k z is the parameter that controls that in where flare is constant and there is no frenet frame.

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