Chen-Gackstatter Minimal Surface

The Chen-Gackstatter Minimal Surface is a modified Enneper surface with holes in it. The following Mathematica code uses some functions that were adapted from Matthias Weber’sMathematica notebook:
(* runtime: 0.4 second *)
<< Graphics`Shapes`; k = 5; n = (k - 1)/k; rho = 1.0/Sqrt[4^n Gamma[(3 - n)/2] Gamma[1 + n/2]/(Gamma[(3 +n)/2]Gamma[1 - n/2])];
phi[n_, z_] := z^(1 + n)Hypergeometric2F1[(1 + n)/2, n, (3 + n)/2, z^2]/(1 + n); f[z_] := {0.5(phi[n, z]/rho - rho phi[-n, z]), 0.5I(rho phi[-n, z] + phi[n, z]/rho), z};
surface = ParametricPlot3D[Re[f[r Exp[I theta]]], {r, 0, 2}, {theta, 1*^-6, 2Pi}, PlotPoints -> {9, 33}, Compiled -> False, DisplayFunction -> Identity][[1]];
Show[Graphics3D[Table[RotateShape[surface, 0, 0, 2Pi i/k], {i, 0, k - 1}]]]

Scherk-Collins Surface

This surface can be formed by twisting and warping a singly-periodic Scherk’s minimal surface. This idea was originally attributed to Brent Collins. Technically, the surface is no longer considered exactly “minimal” after twisting but it still looks minimal (it is actually very difficult to find the exact shape for most minimal surfaces). Click here to download some POV-Ray code.

Here is some Mathematica code:
(* runtime: 0.3 second *)
<< Graphics`Master`; n = 5; r = 0.75n;
Twist[{x_, y_, z_}, theta_] := {x Cos[theta] - y Sin[theta], x Sin[theta] + y Cos[theta], z};
Warp[{x_, y_, z_}, theta_] := {(x + r) Cos[theta], (x + r) Sin[theta], y};
f[z_] := Module[{t1 = Sqrt[2Cot[z]], t2 = Cot[z] + 1}, Warp[Twist[Re[{0.5xsign(Log[t1 - t2] - Log[t1 + t2])/Sqrt[2], ysign I(ArcTan[1 - t1] - ArcTan[1 + t1])/Sqrt[2], z}], 2Re[z]/n], 2Re[z]/n]];
DisplayTogether[Table[ParametricPlot3D[f[x + I y], {x, 0, n Pi}, {y, 0.001, 0.75}, PlotPoints -> {8n + 1, 5}, Compiled -> False], {xsign, -1, 1, 2}, {ysign, -1, 1, 2}]]

The following Mathematica code can be used to increase the number of edges (or “branches”). This code uses some complicated functions that were adapted from Matthias Weber’s Mathematica notebook:
(* runtime: 1.2 seconds *)
<< Graphics`Shapes`; k = 4; phi = Pi(0.6/k - 0.5)/(1 - k);
f[z_] := Re[NIntegrate[Evaluate[{0.5 (w^(1 - k) - w^(k - 1)), 0.5 I (w^(1 - k) + w^(k - 1)), 1}/(w^(k + 1) + w^(1 - k) - 2w Cos[k phi])], {w, 0, z}]];
alpha = Pi/k; zbeta = Exp[I Pi(phi/alpha - 0.5)];
surface = ParametricPlot3D[Re[f[Exp[I alpha/2]((1 + I zbeta Exp[r + I theta])/(I Exp[r + I theta] -zbeta))^(alpha/Pi)]], {r, 0, 4}, {theta, 0, Pi}, PlotPoints -> 10, Compiled -> False, DisplayFunction -> Identity][[1]];
z0 = f[1][[3]]; surface = {surface, AffineShape[TranslateShape[surface, {0, 0, -2z0}], {1, 1, -1}]};
surface = {surface, AffineShape[surface, {1, -1, 1}]}; surface = Table[RotateShape[surface, 2Pi i/k, 0, 0], {i, 1, k}];
dz = Pi Csc[k phi]/k; Show[Graphics3D[Table[TranslateShape[surface, {0, 0, i dz}], {i, 0, 1}]]]


Links

• “Whirled White Web” – a beautiful snow sculpture by Brent Collins
• Sculpture Generator - C++ program for Scherk-Collins surfaces by Carlo Séquin
• Bathsheba Grossman – metal printed math sculptures
• Torolf Sauermann – math art

Punctured Helicoid

Here is a helicoidwith holes in it. The following Mathematica code uses some complicated functions that were adapted from Matthias Weber’sMathematica notebook:
(* runtime: 4 seconds *)
<< Graphics`Shapes`;
tau0 = Exp[1.23409 I]; b0 = 0.629065; theta[z_] := EllipticTheta[1, Pi z, Exp[I Pi tau0]];
r1[z_] := theta[z + 0.5 (b0 - 2) (tau0 + 1)]/theta[z + 0.5 (b0 - 1) (tau0 + 1)]; r2[z_] := theta[z - 0.5 b0 (tau0 + 1)]/theta[z - 0.5 (b0 + 1) (tau0 + 1)];
omega3[z_] := r1[z] r2[z]/(0.386191 - 0.169839 I); G[z_] := (108.37 - 62.8417 I) Exp[I Pi (b0 - 2 z + 2 tau0 + b0 tau0)]r1[z]/r2[z];
f[z0_] := Re[NIntegrate[Evaluate[{-(G[z] omega3[z] - omega3[z]/G[z] )/2, I(G[z] omega3[z] + omega3[z]/G[z] )/2, omega3[z]}], {z, tau0/2, z0}]] + {0.434156, 0, -1};
a0 = -0.409956; r0 = 2.43051; g[z_] := (EllipticF[ArcSin[(a0 + r0 E^z)/(1 - a0 E^z)], 1/r0^2]/(2EllipticF[Pi/2, 1/r0^2]) + 0.5)(1 + tau0)/2;
surface = ParametricPlot3D[f[g[x + I y]], {x, -2.5, 2.5 - 0.8881}, {y, 0.001,0.999Pi}, PlotPoints -> {15, 10}, Compiled -> False, DisplayFunction -> Identity][[1]];
surface = {surface, RotateShape[surface, 0, 0, Pi]};
Show[Graphics3D[{surface, TranslateShape[surface, {0, 0, 2}]}, ViewPoint -> {1, 6, 3}]];

Jorge-Meeks K-Noids

The following Mathematica code uses some functions that were adapted from Matthias Weber’sMathematica notebook:
(* runtime: 0.4 second *)
<< Graphics`Shapes`;
k = 5; phi1[z_] := z^(k - 1) (k/(1 - z^k) - (k - 1) LerchPhi[z^k, 1, 1 - 1/k])/k^2; phi2[z_] := z(1/(1 - z^k) + (k - 1)LerchPhi[z^k, 1, 1/k]/k)/k;
f[z_] := {0.5 (phi2[z] - phi1[z]), 0.5 I (phi1[z] + phi2[z]), 1/(k - k z^k)};
surface = ParametricPlot3D[Re[f[(1 + 2/(I Exp[x + I y] - 1))^(2/k)]], {x,0, Pi/2}, {y, -Pi/2, Pi/2}, PlotPoints -> {8, 16}, Compiled -> False, DisplayFunction -> Identity][[1]];
surface = {surface, AffineShape[surface, {1, -1, 1}]};
Show[Graphics3D[Table[RotateShape[surface, 0, 0, 2Pi i/k], {i, 0, k - 1}]]];

Links

• Mathematica notebook – by Matthias Weber
• Symmetric 4-Noid – at the Virtual Math Museum

Catenoid/Helicoid

This minimal surface is a cross between acatenoid andhelicoid. It would be interesting to see what really happens when a spring is covered with a soap film. Click here to download some POV-Ray code. Here is some Mathematica code:
(* runtime: 0.6 second *)
x := Sin[alpha]Cosh[v]; y := Cos[alpha]Sinh[v];
Do[ParametricPlot3D[{x Cos[u] + y Sin[u], x Sin[u] - y Cos[u], u Cos[alpha] + v Sin[alpha]}, {u, 0, 2Pi}, {v, -2.25, 2.25}, PlotPoints -> {36, 10}], {alpha, -Pi/2, Pi/2, Pi/18}];

Links

• Venice Museum Bridge – bridge proposal, by Eric Worcester
• Marina Bayfront Pedestrian Bridge – double-helix bridge in Singapore
• Soap Film Coil Photo – by the Exploratorium
  

Dodecaplex (120 Cell)

Polychorons are the 4D version of polyhedrons. One way to visualize a polychoron is to apply a 4D to 3D stereographic projection to it. A dodecaplex is a uniform 4D polychoron composed 120 dodecahedral cells. These cells can be divided into 12 rings (Hopf fibrations) of 10 cells each. This picture shows a stereographic projection of 6 rings of the dodecaplex. Each ring is shown in a different color, but only 5 rings are open to direct view because they are wrapped around the 6th ring. I first saw this concept on Matthias Weber’s book page. Click here to download some POV-Ray code.

Links

• Jenn 3D – polytope program, by Fritz Obermeyer
• The HyperSphere, from an Artistic point of View – explanation by Rebecca Frankel
• 120 Cell Soap Bubbles – by John Sullivan
• Regular Polytopes – Mathematica notebook by Russell Towle
• 4D Star Polytope Animations – data by Russell Towle
• POV-Ray include files – by Russell Towle
• Magic 120 Cell – OpenGL program by Roice Nelson, et al.

Stereographic Projection of a Dodecahedron

Here is a stereographic projection of a dodecahedron. This is the 3D counterpart to the 4D dodecaplex. Here is some Mathematica code:
(* runtime: 0.4 second *)
z1 = (Sqrt[5] - 3)/Sqrt[30.0 - 6 Sqrt[5]]; z2 = Sqrt[(1 + 2/Sqrt[5])/3.0]; r1 = Sqrt[2(1 + 1/Sqrt[5])/3.0]; r2 = Sqrt[2(1 - 1/Sqrt[5])/3.0];
vertices = Join[Table[{r2 Cos[theta], r2 Sin[theta], z2}, {theta, 0, 2Pi - 0.4Pi, 0.4Pi}], Table[z1 = -z1; {r1 Cos[theta], r1 Sin[theta], z1}, {theta, 0, 1.8Pi, 0.2Pi}], Table[{r2 Cos[theta], r2 Sin[theta], -z2}, {theta, 0.2Pi, 1.8Pi, 0.4Pi}]];
edges = {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 1}, {1, 6}, {2, 8}, {3, 10}, {4, 12}, {5, 14}, {6, 7}, {7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 6}, {7, 16}, {9, 17}, {11, 18}, {13, 19}, {15, 20}, {16,17}, {17, 18}, {18, 19}, {19, 20}, {20, 16}};
Show[Graphics3D[Map[Line[vertices[[#]]] &, edges]]]
Norm[x_] := x.x; Normalize[x_] := x/Sqrt[x.x]; Rx[theta_] := {{1, 0, 0}, {0, Cos[theta], -Sin[theta]}, {0,Sin[theta], Cos[theta]}};
ProjectPoint[{x_, y_, z_}] := 2{x, y}/(1 - z);
ProjectSegment[{v1_, v2_}] := Module[{p1 = ProjectPoint[v1], p2 = ProjectPoint[v2]}, {nx, ny, nz} = Normalize[Cross[v1, v2]]; If[nz != 0, p0 = -2{nx, ny}/nz; r = 2/Abs[nz]; theta = Sign[nz]Re[ArcCos[(p1 - p0).(p2 - p0)/Sqrt[Norm[p1 - p0]Norm[p2 - p0]]]], theta = 0]; If[Abs[theta] > 0.001, theta1 = ArcTan[p1[[1]] - p0[[1]], p1[[2]] - p0[[2]]]; theta2 = theta1 + theta; If[theta1 > theta2, t = theta1; theta1 = theta2; theta2 = t]; Circle[p0, r, {theta1, theta2}], Line[{p1, p2}]]];
Do[Show[Graphics[Map[ProjectSegment[Map[Rx[phi].# &, vertices[[#]]]] &, edges], PlotRange -> 6{{-1, 1}, {-1, 1}}, AspectRatio -> Automatic]], {phi, 0, 2Pi, Pi/18}];

Double Spiral

One way to create a Double Spiralis by applying a light projection from the top of a Riemann sphere (loxodrome) onto a plane.

This type of projection is called a stereographic projection.

Click here to download a Mathematica notebook. Here is some Mathematica code:
(* runtime: 3 seconds *)
<< Graphics`Shapes`; a = 0.25; Rx[phi_] := {{1, 0, 0}, {0, Cos[phi], -Sin[phi]}, {0, Sin[phi], Cos[phi]}};
Do[loxodrome = Table[Rx[phi].{Sin[t], -a t, -Cos[t]}/Sqrt[1 + (a t)^2], {t, -100, 100, 0.1}]; projection = Map[Module[{r = 2/(1 - #[[3]])}, {r #[[1]],r #[[2]], -1}] &, loxodrome]; Show[Graphics3D[{EdgeForm[], Sphere[0.99, 37, 19], Polygon[{{4, 4, -1}, {-4, 4, -1}, {-4, -4, -1}, {4, -4, -1}}],Line[loxodrome], Line[projection]},PlotRange -> {{-4, 4}, {-4, 4}, {-1, 1}}]], {phi, 0, Pi -Pi/12, Pi/12}]

Another kind of double spiral can be made by applying a special homography to a single logarithmic spiral:
(* runtime: 0.05 second *)
Show[Graphics[Table[Line[Table[z = Exp[r + (2 r + theta)I]; z = (1 + z)/(1 - z); {Re[z], Im[z]}, {r, -10, 10, 0.1}]], {theta, -Pi, Pi, Pi/3}], PlotRange -> {{-2, 2}, {-2, 2}}, AspectRatio -> Automatic]]

Here is some Mathematica code that uses the inverse method:
(* runtime: 17 seconds *)
Show[Graphics[RasterArray[Table[r1 = (x - 1)^2 + y^2; r2 = (x + 1)^2 + y^2; Hue[(Sign[y]ArcCos[(x^2 + y^2 - 1)/Sqrt[r1 r2]] -Log[r1/r2])/(2Pi)], {x, -2, 2, 4/274}, {y, -2, 2, 4/274}]], AspectRatio -> 1]]
and here is some POV-Ray code:
// runtime: 2 seconds
camera{orthographic location <0,0,-2> look_at 0 angle 90}
#declare r1=function(x,y) {(x-1)*(x-1)+y*y}; #declare r2=function(x,y) {(x+1)*(x+1)+y*y};
#declare f=function{(y/abs(y)*acos((x*x+y*y-1)/sqrt(r1(x,y)*r2(x,y)))-ln(r1(x,y)/r2(x,y)))/(2*pi)};
plane{z,0 pigment{function{f(x,y,0)}} finish{ambient 1}}

Links

• Old version
• other double spiral formulas – Cornu spiral (clothoid), tanh spiral
• Whirlpools – famous double spiral tessellation by M.C. Escher
• Loxodrome animation – by Frank Jones

Clifford Torus

The Hopf map is a special transformation invented by Heinz Hopf that maps to each point on the ordinary 3D sphere from a unique circle of points on the 4D sphere. Taken together, these circles form a fiber bundle called a Hopf Fibration. If you apply a 4D to 3D stereographic projection to the Hopf Fibration, you get a beautiful 3D torus called a Clifford Torus composed of interlinked Villarceau circles.

By applying 4D rotations to the Hopf Fibration, you can transform the Clifford Torus into a Dupin cyclide or you can turn it inside-out.

Click here to download some POV-Ray code. The right picture shows a complex cubic polynomial.

Here is some Mathematica code:
(* runtime: 7 seconds *)
HopfInverse[theta_, phi_, psi_] := {Cos[phi/2] Cos[psi], Cos[phi/2]Sin[psi], Cos[theta + psi]Sin[phi/2], Sin[theta + psi]Sin[phi/2]};
Ryw[theta_] := {{1, 0, 0, 0}, {0, Cos[theta], 0, Sin[theta]}, {0, 0, 1, 0}, {0, -Sin[theta], 0, Cos[theta]}};
StereographicProjection[{x_, y_, z_, w_}] := {x, y, z}/(1 - w);
Table[Show[Graphics3D[Table[{Hue[(4 phi/Pi - 1)/3], Table[Line[Table[StereographicProjection[Ryw[alpha].HopfInverse[theta, phi, psi]], {psi, 0.0, 2Pi, Pi/18}]], {theta, 0.0, 2 Pi,Pi/9}]}, {phi, Pi/4, 3Pi/4, Pi/4}], PlotRange -> 3{{-1, 1}, {-1, 1}, {-1, 1}}]], {alpha, 0, Pi, Pi/18}];

Links

• Dimensions – interesting video animated by Jos Leys, see their fibration explantion and video
• The Flat Torus in 3-Sphere – by Thomas Banchoff, explanation1, explanation2, animations
• Stereographic Projection – by Davide Cervone
• Visualization of the Hopf Bundle – Mathematica notebook by Srdjan Vukmirovic

Moiré Pattern

A Moiré pattern is the interference of two similar overlapping patterns. Here is the Moiré pattern on a twisted IKEA wastepaper basket. The mesh on the wastepaper basket was ray-traced from 100,000 tiny cylinders.

Here is some Mathematica code to plot Moiré contours around radiating lines:
(* runtime: 1.7 seconds *)
f[dx_] := Sin[200ArcTan[x - dx, y]];
DensityPlot[f[0.1] - f[-0.1], {x, -1, 1}, {y, -1, 1}, PlotRange -> {0, 1}, PlotPoints -> 275, Mesh -> False, Frame -> False]

Here is some Mathematica code to plot a Moiré pattern from rapidly varying contours of a function:
(* runtime: 0.8 second *)
f[z_] := z^3; DensityPlot[Sin[20Pi Abs[f[x + I y]]], {x, -2.5, 2.5}, {y, -2.5, 2.5}, PlotPoints -> 275, Mesh -> False, Frame -> False]

Link:

• Mandelbrot Set Moire Pattern – by Bernard Helmstetter