# Math of Seashell Shapes

A simple seashell can be modeled using the following parametric formula:

seashell[u_, v_] := {2*(1 - E^(u/(6*Pi)))*Cos[u]* Cos[v/2]^2, 2*(-1 + E^(u/(6*Pi)))*Cos[v/2]^2* Sin[u], 1 - E^(u/(3*Pi)) - Sin[v] + E^(u/(6*Pi))*Sin[v]}; start = 0; end = 2*Pi; gap = 1; shift = 1; uMin = 0; uMax = 6*Pi; xViewPoint = {1.0992, 3.2001, 0.0255}; gra1 = ParametricPlot3D[Evaluate[seashell[u, v]], {u, uMin, uMax}, {v, end - gap + shift, end + shift}, PlotPoints -> {96, 4}, Mesh -> Full, MeshShading -> {{None},{None}}, PlotRange -> All ] gra2 = ParametricPlot3D[Evaluate[seashell[u, v]], {u, uMin, uMax}, {v, start + shift, end - gap + shift}, PlotPoints -> {96, 20}, Mesh -> Full, PlotStyle -> Directive[White, Opacity[.8], Specularity[White, 20]], PlotRange -> All ] Graphics3D[{ gra1[[1]] , gra2[[1]]}, {PlotRange -> All, Boxed -> False, ViewPoint -> xViewPoint}]

### wrinkled periwinkle

cR=1; "* radius of tube *"; cN=4.6; "* number of turns *"; cH=2.5; "* height *"; cF=80; "* wave frequency *"; cA=0.2; "* wave amplitude *"; cP=1.9; "* power *"; cW = ((#/(2*Pi)*cR) &); seashellF = Function[ {u, v}, { cW[u]*Cos[cN*u]*(1+Cos[v]+Cos[cF*u]*cA) , cW[u]*Sin[cN*u]*(1+Cos[v]+Cos[cF*u]*cA) , cW[u]*Sin[v] + cH*(u/(2*Pi))^cP } ]; ParametricPlot3D[ Evaluate@ seashellF[u, v], {u, 0, 5}, {v, 0, 2 Pi}, PlotPoints -> {160, 10}, PlotRange -> All, ColorFunction -> "TemperatureMap", BoundaryStyle -> Directive[Black, Thin], PlotStyle -> Directive[White, Opacity[0.7], Specularity[10, 20]], Lighting -> "Neutral" ]

### Wentletrap

Clear[cW, seashell, cN, cH, cP, cA, cF] cN = 6.6; ("number of turns"); cH = 4.0; ("height"); cP = 2; (" power "); cA = 0.12;(" Ridge Amplitude "); cF = 15;(" Ridge Frequency "); cW = Function[(#/(2 Pi))^#2 ]; seashell[u_, v_] := {cW[u , cP] (Cos[cN u] - cA Cos[cN cF u]) (1 + Cos[v]), cW[u , cP] (Sin[cN u] + cA Sin[cN cF u]) (1 + Cos[v]), cW[u , cP] Sin[v] + cH (u/(2 Pi))^(1 + cP)}; ParametricPlot3D[ Evaluate@ seashell[u, v] , {u, 0, 6}, {v, 0, 2 Pi}, PlotPoints -> {200,40}, Axes -> True, Boxed -> True, BoundaryStyle -> Directive[Black, Thin], PlotStyle -> Directive[White, Opacity[0.7], Specularity[10, 20]], PlotRange -> All, Lighting -> "Neutral" ]

### Gallery of Shapes

### Tightness of Spiral

### Outline Variations

### Ribs and Folds

### Spikes

### Internal Structure

### Analysis of Seashell Shapes

Seashells are showcasing of spirals. There are great variety of spiral shapes. Suppose we start with a circle winding around a spiral.

• The circle's shape changes periodically like a sine function, creating a corrugated shell somewhat emulate that of

• If instead of a circle we have a triangle, we can simulate that of

• If the rounding shape periodically changes shape to have spikes, then we might emulate shells that have horns such as the

or

• The periodic change might also emulate those shell having ribs such as the

### References and Sources

The Algorithmic Beauty of Sea Shells , by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R Fowler. Buy at amazon

Mike Willams has sent me various formulas, see Math Parametric Equation for Seashell