# Math Parametric Equation for Seashell

The following are some math formulas for seashell, by Mike Willams. (slightly modified)

From: mike@econym.demon.co.uk Subject: Paramnetric equations of seashells Date: January 19, 2005 12:35:17 AM PST To: xah@xahlee.org

I hear that you're interested in parametric equations that approximate spiral seashells. Here are a few that I've made.

They're all based on this parameterisation of the torus

(* a torus *) R = 1; R2 = 2; ParametricPlot3D[ { R* Cos[u]*(R2+Cos[v]), R*Sin[u]*(R2+Cos[v]), R*Sin[v] }, {u, 0, 6}, {v, 0, 10} ]

I then replaced R with R*u/(2*pi) so that the radius of the tube increases linearly as u increases, and replaced sin(u) with sin(N*u) so that the tube spirals round the origin N times. Then I added an offset factor H*(u/2*pi)^P so that the turns of the spiral are offset in the z direction by a distance that varies from 0 to H.

This gives a periwinkle seashell:

### periwinkle seashell

R =1; (* radius of tube *) cN = 4.6; (* number of turns *) H =2; (* height *) P =2; (* power *) W = Function[ {u}, u/(2*Pi)*R ]; (* W = (#/(2*Pi)*R &) *) ParametricPlot3D[ { W[u]*Cos[cN*u]*(1+Cos[v]), W[u]*Sin[cN*u]*(1+Cos[v]), W[u]*Sin[v] + H*(u/(2*Pi))^P }, {u, 0, 6}, {v, 0, 6}, PlotPoints -> 60, PlotRange -> All, ColorFunction -> "TemperatureMap" ]

### top shell

And a top

R=1; (* radius of tube *) cN=7.6; (* number of turns *) H=2.5; (* height *) P=1.3; (* power *) W = Function[ {u}, u/(2*Pi)*R ]; ParametricPlot3D[ {W[u]*Cos[cN*u]*(1+Cos[v]), W[u]*Sin[cN*u]*(1+Cos[v]), W[u]*Sin[v] + H*(u/(2*Pi))^P }, {u,0,7}, {v,0,6}, PlotPoints -> {160, 10}, PlotRange -> All, ColorFunction -> "TemperatureMap" ]

### cone shell

A cone shell is something like this:

A cone shell is something like this: R=1; // radius of tube N=4.6; // number of turns H=0.5; // height P=2; // power W = u/(2*pi)*R Fx = W(u)*cos(N*u)*(1+cos(v)) Fy = W(u)*sin(N*u)*(1+cos(v)) Fz = W(u)*sin(v)*1.25 + H*(u/(2*pi))^P +W(u)*cos(v)*1.25

### wrinkled periwinkle

R=1; (* radius of tube *) cN=4.6; (* number of turns *) H=2.5; (* height *) F=80; (* wave frequency *) A=0.2; (* wave amplitude *) P=1.9; (* power *) W = ((#/(2*Pi)*R) &); ParametricPlot3D[ { W[u]*Cos[cN*u]*(1+Cos[v]+Cos[F*u]*A) , W[u]*Sin[cN*u]*(1+Cos[v]+Cos[F*u]*A) , W[u]*Sin[v] + H*(u/(2*Pi))^P }, {u, 0, 5}, {v, 0, 2 Pi}, PlotPoints -> {160, 10}, PlotRange -> All, ColorFunction -> "TemperatureMap" ]

-- Mike Williams Gentleman of Leisure

From: private@econym.demon.co.uk Subject: Parametric equations of seashells Date: January 20, 2005 9:41:46 PM PST To: xah@xahlee.org

You wrote

hi Mike,

Great work!

of your results, i find the wrinkled version of greatest interest. Now it would be great if one can add to the equation periodic spikes, as to make one of those horny ones,

I reckon that this general approach would only be able to place spikes that point in the +-x and +-y directions. This might make something that at first glance resembles a Venus Comb, but the rows of spikes should really point in directions separated by about 120 degrees rather than 90.

For spikes that point in other directions I think you'd need an entirely different approach.

I think the ideal “top” one should have a flat surface, very angular triangular opening, and with a cone shaped hole at the bottom. See * Seashell Gallery: Top Shell

We only have one species of “top” in my country (Britain), the “Toothed Topshell” (Monodonta lineata) and it is quite rounded. A flat sided topshell is more like this:

R=1; // radius of tube N=7.6; // number of turns H=2.5; // height P=1.3; // power T=0.8; // Triangleness of cross section A=-0.3; // Angle of tilt of cross section (radians) W = u/(2*pi)*R Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fz = W(u)*(sin(v+A)+cos(2*v+A)*T/4) + H*(u/(2*pi))^P

Which also gives the possibility of creating something like a turret shell (oddly, my book of British fauna doesn't include these, but there's loads of them on the local beach). I can't quite seem to get the coils to touch each other correctly at the tip.

R=1; // radius of tube N=9.6; // number of turns H=5.0; // height P=1.5; // power P1=1.1; // another power T=0.8; // Triangleness of cross section A=0.1; // Angle of tilt of cross section (radians) S=1.5; // Stretch W = (u/(2*pi)*R)^P1 Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fz = S*W(u)*(sin(v+A)+cos(2*v+A)*T/4) + S*H*(u/(2*pi))^P

Whilst trying to get the turret working, I happened to create this by mistake.

R=1; // radius of tube N=7.6; // number of turns H=2.5; // height P=1.3; // power T=0.8; // Triangleness of cross section A=-0.3; // Angle of tilt of cross section (radians) S=3.0; // Stretch W = u/(2*pi)*R Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fz = S*W(u)*(sin(v+A)+cos(2*v+A)*T/4) + S*H*(u/(2*pi))^P

I should have mentioned (but I guess you've worked it out for yourself) that all these surfaces are designed to be generated with u and v both being in the range 0 to 2*pi.

PS may i forward your results to the curves-surfaces mailing list?

Yes.

Is there an actual mailing list? All I could find at the URL you gave me was one of those horrible Google Groups.

-- Mike Williams Gentleman of Leisure

spindle shell

R=1; // radius of tube N=5.6; // number of turns H=4.5; // height P=1.4; // power L=4; // Controls spike length K=9; // Controls spike sharpness W = (u/(2*pi)*R)^0.9 Fx = W(u)*cos(N*u)*(1+cos(v)) Fy = W(u)*sin(N*u)*(1+cos(v)) Fz = W(u)*(sin(v)+L*(sin(v/2))^K) + H*(u/(2*pi))^P

top

R=1; // radius of tube N=7.6; // number of turns H=2.5; // height P=1.3; // power T=0.8; // Triangleness of cross section A=-0.3; // Angle of tilt of cross section (radians) W = u/(2*pi)*R Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4) Fz = W(u)*(sin(v+A)+cos(2*v+A)*T/4) + H*(u/(2*pi))^P

back to Mathematics of Seashell Shapes