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Math Artwork
On this page you will find some tessellations, surfaces, and other math stuff along with some basic Mathematica code.

Loxodrome Sconce - POV-Ray version: 10/7/08; 3ds Max, MaxScript model: 3/15/11
This sconce light demonstrates how to create a double spiral by applying a light projection from one side of a loxodrome to a wall on the other side. In mathematics, we call this type of projection a stereographic projection. I finished the small 4 inch prototype on 10/10/11, but I did not present my large 8 inch model until recently, at the 3D Printer World Expo 2014. It received many positive responses from viewers, and numerous requests to purchase one. Unfortunately, the model is not available for purchase at this time, but I am hoping to change that soon.

The loxodrome is a nautical term referring to the path that a ship follows on the Earth if it maintains a constant heading. For example, if a ship maintains a heading of 45° North-East, then it will spiral around the North pole as it approaches the pole, but it will never reach the North pole because it is not pointing due North. Similarly, the ship will spiral around the South pole in the opposite direction if it follows the path in reverse.


Dodecaplex (120 Cell) - Mathematica 4.2, POV-Ray 3.6.1, 10/21/08
Polychorons are the 4D version of polyhedrons (in general, for arbitrary dimensions, these are called polytopes). A dodecaplex is a uniform polychoron composed 120 dodecahedral cells. These cells can be divided into 12 rings (Hopf fibrations) of 10 cells each. One way to visualize a polychoron is to apply a 4D to 3D stereographic projection to it. These projections resemble soap bubbles because all intersecting surfaces meet each other at angles of exactly 120°. The left picture shows a stereographic projection of the complete dodecaplex. The right picture shows 6 of the inner rings (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. You can purchase this as a poster here.

Links
Flatland - movie based on Edwin Abbott's novel
Jenn 3D - polytope program that uses the Todd-Coxeter algorithm, by Fritz Obermeyer
The HyperSphere, from an Artistic point of View - explanation by Rebecca Frankel
120 Cell Soap Bubbles - by John Sullivan
Magic 120 Cell - YouTube video demonstrating how to solve this complicated variation of the Rubik's Cube, by Noel Chalmers
Regular Polytopes - Mathematica notebook by Russell Towle
POV-Ray include files - by Russell Towle
Magic 120 Cell - OpenGL program by Roice Nelson, et al.

Geodesic Sphere Knots - Mathematica 4.2, POV-Ray 3.6.1, 2/15/11
There are many ways to cover surfaces with a network of knots. You can purchase these 2 inch or 1 inch geodesic sphere knots here. They are not recommended for children's toys, but they are durable and bouncy (my 2 year old son approved of them). You can insert small LEDs and hang them as pendant lamps, or use them as Christmas tree ornaments, or hang them from your rear view mirror.

Links
Geo Weave, Geodesic Weave, Worm Weave, Takraw Ball Toy (Dick Esterle's Nobbly Wobbly) - beautiful renderings by Taff Goch, see also this analysis
International Guild of Knot Tyers - we recently saw this group at the Tall Ships Festival, be sure to see the animals and Turk's Head globes
Celtic knotwork tutorial - by Christian Mercat
Bending a Soccer Ball - animations by Michael Trott
KnotsBag - software by Hypatiasoft
Knots3D - free Celtic knot program by Steve Abbott, see also KnotsBag by Geraud Bousquet
The KnotPlot Site - interesting knots by Robert Scharein, some of my favorites are his Brunnian Links, Ashley Knot #2445 and #2462, bbframe-bain6b, and 13jun03d2

3D Printing Links
Homemade SLA printer - by Junior Veloso
3D Printed Candy - trefoils, dodecahedrons, moebius strips, created with the CandyFab 6000
D-Shape - large scale 3D printing for buildings

“Dinosaur Tessellation” - AutoCAD 2000, AutoLisp, Adobe Photoshop 5.0, 12/9/02
A tessellation is a regular tiling of figures without any gaps or overlapping. If you have access to AutoCAD, you can use this AutoLisp routine to help you design your own tessellations. You can purchase pattern as a tie or as a printed fabric.

Tessellation Links
Penrose Tiling Java Applet - by Craig Kaplan
Maurits Cornelis Esher (M.C. Escher) - famous artist who used tessellations in his artwork
Tessellation Java Applet - create your own tessellations
I Love Dragons - a nice looking reproduction of this tessellation by a student at Lizzie M. Burges Alternative School

Hopf Fibration - Mathematica 4.2, POV-Ray 3.6.1, 8/12/08
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. As it turns out, there is a single point of rotation in 2D, there are 3 axes of rotation in 3D (x, y, z), and there are 6 planes of rotation in 4D (xy, xz, xw, yz, yw, zw). Click here to download some POV-Ray code. 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 explanation and video
Hopf Fibration, Clifford Torus - 3D prints, by Henry Segerman
Torofluxus - cool toy invented by Jochen Valett, you can buy one here
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
Hyperspheres - If a 1-sphere is a circle, and a 2-sphere is a regular sphere, then what is a 3-sphere? How about a 4.5-sphere? Did you know there is a formula to find the “volume” for any n-sphere?

Knotted Surface - AutoCAD 2000, POV-Ray 3.6.1, 4/24/07
This began as an attempt to animate a similar-looking structure to Bathsheba Grossman’s beautiful Quin Pendant Lamp. To create this image, first I sketched the basic curves for one arm in AutoCAD and then I assembled and rendered it in POV-Ray. Depending on your point of view, this knotted surface can be seen as a dodecahedron with a hole over each edge, or an icosahedron with a hole over each vertex, or an icosahedron with a hole over each edge, or a rhombic triacontahedron with a hole over each face. The left animation shows a homotopy that continuously maps the structure into a sphere with 30 holes. The boundary of each hole loops over itself twice and links with 6 others. This image was featured on the cover of McGraw-Hill's 2012 & 2013 Algebra 2 textbooks. Here is some Mathematica code:
(* runtime: 0.1 second *)
<< Graphics`Shapes` ; alpha = ArcCos[-Sqrt[5]/5];
surface = {{{0.11, 0.35, 1}, {0.16, 0.33, 1}, {0.23, 0.35, 0.99}, {0.3, 0.38, 0.96}, {0.35, 0.43, 0.9}, {0.29, 0.42, 0.8}, {0.22, 0.37,0.7}, {0.14, 0.34, 0.62}, {0.078, 0.296, 0.585}}, {{0, 0, 1}, {0.13, 0.09, 1}, {0.29, 0.22, 0.99}, {0.4, 0.33, 0.95}, {0.41, 0.45, 0.88}, {0.31, 0.47, 0.77}, {0.2, 0.43, 0.65}, {0.08, 0.4, 0.56}, {-0.019, 0.398, 0.526}}, {{0.36, 0, 1}, {0.39, 0.11, 1}, {0.45, 0.23,0.99}, {0.49, 0.35, 0.95}, {0.47, 0.45, 0.86}, {0.36, 0.52, 0.73}, {0.22, 0.5, 0.59}, {0.13, 0.48, 0.48}, {0.07, 0.489, 0.437}}};
arm = Map[Polygon[Flatten[#, 1][[{1, 2, 4, 3}]]] &, Partition[surface, {2, 2}, 1], {2}];
face = Table[RotateShape[Graphics3D[arm], 0, 0, psi][[1]], {psi, 0, 1.6Pi, 0.4Pi}];
Show[Graphics3D[{face, RotateShape[face, 0, Pi, 0], Table[{RotateShape[face, 0, Pi - alpha, psi + Pi/5], RotateShape[face, Pi/5, alpha, psi]}, {psi, 0, 1.6Pi, 0.4Pi}]}]]
Links
Quin Pendant Lamp - a very beautiful lamp by Bathsheba Grossman
Nexus - another one of my favorites by Bathsheba Grossman
Linking Stars - also inspired after Bathsheba Grossman's Quin Lamp, created by Jonathan Johanson using TopMod3d, see also his rotatable model
Icosahedron VI - similar sculpture by Vladimir Bulatov, see also his pendant
Linking Stars - video tutorial how to create these using TopMod3d, by sjo.Johanson
Irregular Detail - an interesting design by Florian Sanwald, hard to figure out
Pocket Art - impressive wood carvings of polyhedral knots, by Bjarne Jespersen, see also his explanation
Life Support - interesting 3D printed model with dodecahedral symmetry, by Stijn van der Linden
TopMod3d - free topological mesh modeling software
Metal Printed Quintrino - by Bathsheba Grossman
Mysterious Star - interesting shape, by Mehrdad Garousi
IQ-light (Interlocking Quadrilaterals) - by Holger StrÝm
Sarah Turner - artist creates lamps from waste plastic drink bottles
Knobbyball - beautiful truncated icosahedron design, by Robert Drummond

Knotted Geodesic Sphere Surface - Mathematica 4.2, POV-Ray 3.6.1, 3/3/11
The pattern on this knotted surface was inspired after Bathsheba Grossman’s beautiful Quin Pendant Lamp.

Links
Starfishberry - a similar concept by Florian Sanwald, here is another variation by Jonathan Johanson

Polyhedrons with Reaction-Diffusion Patterns - new version: C#, POV-Ray 3.6.1, 12/1/11; old version: C++, Mathematica 4.2, 5/24/07
Reaction-Diffusion is a chemical reaction that can resemble stripes or leopard spots (depending what parameters you use). These patterns can be mapped to various polyhedrons using Daniel Walsh's strategy for combining neighboring triangle pairs into rhombuses that can be solved using a skewing transformation. Here is some Mathematica code to simulate a simple periodic pattern using a cellular automaton:
(* runtime: 7 minutes *)
n = 80; w = 0.35; SeedRandom[0]; image = Table[Random[Integer, {0, 1}], {n}, {n}];
Do[image = Table[ID = AD = 0; Do[If[image[[Mod[i + di - 1, n] + 1, Mod[j + dj - 1, n] + 1]] == 1, r = Sqrt[di^2 + dj^2]; If[3 <= r <= 6, ID++]; If[r <= 3, AD++]], {di, -6, 6}, {dj, -6, 6}]; a = AD - w ID; Which[a > 0, 1, a < 0, 0, True, image[[i, j]]], {i, 1, n}, {j, 1, n}]; ListDensityPlot[image, Mesh -> False, Frame -> False], {10}];

Links
Xmorphia - detailed simulation demonstrating varying parameters, by Robert Munafo (originally by Roy Williams)
Gray-Scott Reaction-Diffusion - Java applet by Jonathan Lidbeck
Heated metal alloy simulation - Java applet by Thomas Wanner
Activator-Inhibitor - Wolfram Demonstrations Project
Cahn-Hilliard equation
Here are some platonic solids that I made for my 2 year old son Carl. They were cut and folded from this template of printed photos. Most babies only get to play with 2D shapes, but if these could be baby-proofed, I think they would be a fun way to introduce 3D geometry to kids at an early age. Even most adults are not familiar with many of these solids.


Spherical Cube with Reaction-Diffusion Pattern - C#, POV-Ray 3.6.1, 11/25/11
When we tried to map a Reaction-Diffusion pattern to a sphere using spherical coordinates, the integration became unstable at the poles. As an alternate approach, Daniel Walsh proposed a clever formula for mapping the pattern onto a cube that can be projected undistorted onto the sphere. Here is my rendition of a sphere using this approach. You can purchase 3D prints of this here. The 2 inch sphere costs about $10, and the 1 inch sphere costs about $8.

Links
Reaction Seed Lamp 1, Seed Lamp 2, Spiral Lamp, Reaction Vase - beautiful 3D printed models, by Nervous Systems
Neri Oxman - MIT media arts professor explores possibilities of 3D printing
ReactionDiffusionSphere01, ReactionDiffusionSphere03 - by Michael Spaw

Knotted Surface with Reaction-Diffusion Pattern - C#, POV-Ray 3.6.1, 12/9/11
Reaction-Diffusion patterns can also be applied to arbitrary meshes. However, the pattern can become distorted if the mesh faces have corner angles that are too sharp, as you can see here. Maybe I'll try to make a better looking version of this if I have time.


Stereographic Projection of a Loxodrome - new version: C++, POV-Ray 3.6.1, 10/1/08; old version: Mathematica 4.2, 7/18/05
You can purchase this as a poster here and here. 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}]

Links
Triply Orthogonal System and Loxodrome Surface - amazing 3D versions of this concept, by Daniel Piker

Homography of a Logarithmic Spiral - new version: C++, 10/1/08; old version: Mathematica 4.2, 7/18/05
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}}

The photo on the right shows a giant 91 inch poster of this image. Please email me if you are interested in purchasing a giant poster of any of my art creations. You can also purchase smaller posters here.

Links
other double spiral formulas - Cornu spiral (clothoid), tanh spiral
Whirlpools - famous double spiral tessellation by M.C. Escher
Double Spiral - folded out of paper, by David Huffman

Boy’s Surface (Bryant-Kusner Parametrization) - new version: POV-Ray 3.6.1, 6/20/06
old version: Mathematica 4.2, MathGL3d, POV-Ray 3.6.1, 5/24/05
This Boy’s Surface is a one-sided surface that was first parametrized correctly by Bernard Morin. The left animation looks like it’s turning inside-out (although technically that’s impossible because it only has one side). Robert Bryant told me that the parameters (p,q) = (0,1) give this Willmore immersion of RP2 a trilateral symmetry and the parameters (p,q) = (1,0) should give bilateral symmetry. This image was printed on the cover of McGraw-Hill's 2012 Pre-Algebra textbook. Click here to download some POV-Ray code for this image and here for some AutoLisp code. Here is some Mathematica code:
(* runtime: 1 second *)
ParametricPlot3D[Module[{z = r E^(I theta), a, m}, a = z^6 + Sqrt[5]z^3 - 1; m = {Im[z(z^4 - 1)/a], Re[z(z^4 + 1)/a], Im[(2/3) (z^6 + 1)/a] + 0.5}; Append[m/(m.m), SurfaceColor[Hue[r]]]], {r, 0, 1}, {theta, -Pi, Pi}, PlotPoints -> {20, 72}, ViewPoint -> {0, 0, 1}]
POV-Ray also has an internal function for a different parametrization:
// runtime: 50 seconds
camera{location -1.5*z look_at 0} light_source{-z,1}
#declare f=function{internal(8)} isosurface{function{-f(x,y,z,1e-4,1)} pigment{rgb 1}}

Links
Rotatable 3D Boy’s Surface
Steiner Surfaces - POV-Ray animations by Adam Coffman
Another way to obtain other symmetries is to apply a mapping transformation. The n = 1 case is a Klein bottle.

Klein Bottle and Möebius Strip Links
Möebius Strip - a simple one-sided surface, see these twisting and 5 fold animated Möebius strips by Jos Leys
Twin Rail Mobius Pendant - metal Mobius with ball bearings, by Paul King
Hand-Blown Glass Klein Bottles - by Cliff Stoll
Nested Klein Bottles - by Alan Bennett
Möebius strip to Klein bottle transformation - Mathematica code by Ari Lehtonen
Trefoil Knot Minimal Surface - by Nat Friedman

Spidron - folded paper, POV-Ray 3.6.1, 7/28/10
Spidrons were invented by Daniel Erdely in 1979 as a homework assignment for Erno Rubik, inventor of the Rubik's Cube. Spidrons can folded into expandable 3D shapes and can be applied to polyhedrons and tessellations. The original spidron can form a tiling with 3 spiral arms. Click here to download some POV-Ray code. Here is some Mathematica code:
(* runtime: 0.01 second *)
n = 5; elems = Flatten[Table[6i + {{j, Mod[j, 6] + 1, j + 6}, {j + 1, j + 6, Mod[j + 6, 12] + 1}}, {i, 0, n - 1}, {j, 1, 6}], 2];
Do[r = Sqrt[1 - 4dz^2]; dtheta = ArcCos[(1 + 2 Sqrt[3] + 1/(4dz^2 - 1))/4];R = {{Cos[dtheta], -Sin[dtheta], 0}, {Sin[dtheta], Cos[dtheta], 0}, {0, 0, 1}}/Sqrt[3]; nodes0 = nodes = Table[{r Cos[i Pi/3], r Sin[i Pi/3], (2Mod[i, 2] - 1) dz}, {i, 0, 5}]; Do[nodes0 = Map[R.# &, nodes0];nodes = Join[nodes, nodes0], {n}]; Show[Graphics3D[Map[Polygon[nodes[[#]]] &, elems], PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]], {dz, 0.0, 1/3, 0.05}]

Spidron Links
Spidron Relief - folding tessellation animation
Spidron Exhibition
Tile Magic - by Akira Nishihara

Dodeca-Spidroball - POV-Ray 3.6.1, 7/29/10
A spidroball can be formed by applying folded spidrons to the surface of a polyhedron. This example shows a dodeca-spidroball formed by applying spidron tilings with 5 spiral arms to the surface of a dodecahedron. Click here to download some POV-Ray code and here for some AutoLisp code. Here is some Mathematica code:
(* runtime: 0.1 second *)
<< Graphics`Shapes`; nArm = 5; n = 2nArm; depth = 8; alpha = ArcCos[-Sqrt[5.0]/5]; a = Sin[0.5Pi/nArm]^2; b = Tan[alpha/2]; dz = a b/Sqrt[a b^2 + 1]; r = Sqrt[1 - dz^2/a]; z0 = r b Cos[Pi/nArm] + dz; dtheta = ArcCos[(1 + 2 Sqrt[3] + 1/(dz^2/a - 1))/4]; R = (2/3) {{Cos[dtheta], -Sin[dtheta], 0}, {Sin[dtheta], Cos[dtheta], 0}, {0, 0, 1}};
nodes0 = nodes = Table[{r Cos[Pi i/nArm], r Sin[Pi i/nArm], (2Mod[i, 2] - 1)dz}, {i, 0, n - 1}]; Do[nodes0 = Map[R.# &, nodes0]; nodes = Join[nodes, nodes0], {depth}];
elems = Flatten[Table[n i + {{j, Mod[j, n] + 1, j + n}, {j + 1, j + n, Mod[j + n, 2n] + 1}}, {i, 0, depth - 1}, {j, 1, n}], 2];
face = TranslateShape[Map[Polygon[nodes[[#]]] &, elems], {0, 0, z0}];
Show[Graphics3D[{RotateShape[face, 0.3Pi, 0, 0], RotateShape[face,0.3Pi, Pi, 0], Table[{RotateShape[face, -0.3Pi, alpha, theta], RotateShape[face, 0.3Pi, Pi - alpha, theta + 0.2Pi]}, {theta, 0, 1.6Pi, 0.4Pi}]}]]

Spidroball Links
Dodeca-Spidron Ball - by Marc Pelletier and Regina Markus
Spidroball - rhombic triacontahedron
3D Spidron Tessellation - fascinating animation, they should manufacture these as toys
Spidroball Sculpture - large sculpture
Krystyna Burczyk - beautiful origami polyhedrons with no creases
Double Spiral - folded out of paper, by David Huffman
Whishboneahedron - another interesting design by Florian Sanwald
Frabjous - laser-cut wood, by George Hart

7 Arm Sphidron - POV-Ray 3.6.1, 7/21/10
Another one of Daniel Erdely's inventions are sphidrons. Sphidrons are similar to spidrons except they are smooth. I am not exactly sure what defines this surface, but this image uses a spiral wave based on a double spiral. Now, here is the big question: is it possible to "roll" a single piece of paper into a shape that looks something like this, without stretching the paper?

Links
6 Arm Sphidron - YouTube animation, here is another animation
Paper Sphidrons - by Daniel Erdely
Double Spiral - folded out of paper, by David Huffman

"Math Rose" - new version: POV-Ray 3.6.1, 6/21/06
old version: Mathematica 4.2, MathGL3d, 3/5/04
This rose is actually a plot of a single, continuous, parametric math equation. I got this idea while trying to create a visualization of a spiraling spin-lattice relaxation for a physics experiment involving a Nuclear Magnetic Resonance (NMR) spectrometer. You can purchase a 3D printed version of this rose here. Click here to see a larger animation. Click here to see a rotatable 3D version. Click here to download some POV-Ray code for this image and here for some AutoLisp code. See also my Passion Flower. Here is some Mathematica code:
(* runtime: 16 seconds *)
Rose[x_, theta_] := Module[{phi = (Pi/2)Exp[-theta/(8 Pi)], X = 1 - (1/2)((5/4)(1 - Mod[3.6 theta, 2 Pi]/Pi)^2 - 1/4)^2}, y = 1.95653 x^2 (1.27689 x - 1)^2 Sin[phi]; r = X(x Sin[phi] + y Cos[phi]); {r Sin[theta], r Cos[theta], X(x Cos[phi] - y Sin[phi]), EdgeForm[]}];
ParametricPlot3D[Rose[x, theta], {x, 0, 1}, {theta, -2 Pi, 15 Pi}, PlotPoints -> {25, 576}, LightSources -> {{{0, 0, 1}, RGBColor[1, 0, 0]}}, Compiled -> False]

Links
This rose is featured in Ramiro Perez's Incendia software (see his Rose Curve and Rose Gasket)
A Rose for Valentine's Day - adaptation for Wolfram Demonstration Project, by Jeff Bryant
A Rose for Aexion - another rendering of this rose by parrotdolphin

Seifert Surfaces - POV-Ray 3.6.1, 9/24/12
A Seifert surface is a surface whose boundary is a knot or a link. Some Seifert surfaces can also be minimal surfaces. The following Mathematica code was adapted from Jos Leys' PovRay code for a Seifert trefoil:
(* runtime: 1 second *)
<< Graphics`Master`;
StereographicProjection[{x_, y_, z_, w_}] := {x, y, z}/(1 - w);
f[phi_, v_] := Module[{g4 = phi + I vsign Log[1 - v]}, g2 = ((1 + Sin[g4])/(2Sqrt[2]))^(1/3); g3 = Sqrt[(Sin[g4] - 1)/2]; f[k_] := k^3Abs[g3]^2 + k^2 Abs[g2]^2 - 1; k0 = 0.0; k = 1.0; While[Abs[k - k0] > 1.0*^-6, k0 = k;k = k0 - f[k0]/f'[k0]];g2 *= k Exp[I theta]; g3 *= sign k^1.5; StereographicProjection[{Im[g2], Re[g2], Im[g3], Re[g3]}]];
DisplayTogether[Table[ParametricPlot3D[f[phi, v], {phi, -Pi/2, Pi/2}, {v, 0, 1}, PlotPoints -> {20, 10},DisplayFunction -> Identity], {vsign, -1, 1, 2}, {sign, -1, 1, 2}, {theta, 0, 4 Pi/3, 2 Pi/3}], ViewPoint -> {0, 0, 1}]

Links
SeifertView - free program for creating knotted surfaces, by Jarke van Wijk and Arjeh Cohen
Lorenz and Modular Flows - by Jos Leys, and Etienne Ghys
Borromean Rings - Seifert surface, by Bathsheba Grossman
Non Singular Knot - by Asher Nahmias

Torus with Knotted Hole - POV-Ray 3.6.1, 7/31/12
Here is a 4D rotation of a trefoil transforming into a torus with a knotted hole. In general, you can continuously turn any 3D object "inside-out" by applying an inverse stereographic projection that maps the 3D object onto the surface of a 4D hypersphere, then rotating it in 4D space, and finally applying a forward stereographic projection to return it back to 3D space.
(* runtime: 11 seconds *)
StereographicProjection[{x_, y_, z_, w_}] := {x, y, z}/(1 - w);
InverseStereographicProjection[p_] := Module[{r2 = Norm[p]}, Append[2 p, r2 - 1]/(r2 + 1)];
Ryw[theta_] := {{1, 0, 0, 0}, {0, Cos[theta], 0, -Sin[theta]}, {0, 0, 1, 0}, {0, Sin[theta], 0, Cos[theta]}};
f[t_] := {(2 + Cos[2 t]) Cos[3 t], (2 + Cos[2 t]) Sin[3 t], Sin[4 t]}/4;
rtube = 1.0/6;
TubePlot[t_, theta_] := Module[{df = f'[t], ddf = f''[t], x, y, z}, z = Normalize[df]; x = Normalize[ddf (df.df) - df (df.ddf)]; y = Cross[x, z]; f[t] + rtube (x Cos[theta] + y Sin[theta])];
ParametricPlot3D[StereographicProjection[Ryw[Pi/2].InverseStereographicProjection[TubePlot[t, theta]]], {t, 0, 2 Pi}, {theta, 0, 2 Pi}, PlotPoints -> {72, 18}, ViewPoint -> {0, -1, 0}]

Links
Knot Complement - 4D rotation of trefoil transforming into torus with knotted hole, by Daniel Piker
Figure 8 Knot Trumpet - 3D print, by Henry Segerman

Dodecahedral Loxodrome - Mathematica 4.2, POV-Ray, 10/28/11
Here is another similar concept to my loxodrome sconce, but with more spirals on it.

“Free Spirit Tessellation” - AutoCAD 2000, AutoLisp, Adobe Photoshop 5.0, 5/12/03
I modeled this tessellation after “Spirit” from Disney’s “Spirit - Stallion of Cimarron” movie. I had to overlap the horse’s hind legs because there was not enough room. Still, I think the rest of it fits together remarkably well.

Maelström Autostereogram - Mathematica 4.2, SISgen 1.58, 10/1/04

This 3D vortex image is hidden in the above picture. To see it, relax your eyes and focus behind the screen. This autostereogram was generated with William Steer’s free autostereogram generating program SISgen. Click here to see an animated version of this picture. I also have a Mathematica-only version of this picture, but it is not as accurate. See also Pascal Massimino’s “Maelstrom” autostereogram.

(* Create Depth Map, runtime: 1 minute *)
r := Sqrt[x^2 + y^2]; z := 0.03Sin[7(2r - ArcTan[x, y])] - 0.1/r - 0.3; phi = Pi/6; ymax = 1.0/Cos[phi];
j1 = 615; i1 = 450; depth = Table[0, {i1}, {j1}];
Do[flag = False; Do[i0 = ip; ip = Round[(i1 - 1 + (j1 - 1) (y Cos[phi] + z Sin[phi]))/2 + 0.2i1] + 1; z2 = z1; z1 = -y Sin[phi] + z Cos[phi]; If[flag, sign = If[i0 < ip, 1, -1]; j = Round[(j1 - 1)(x + 1)/2] + 1; Do[If[0 < i <= i1, depth[[i, j]] = ((i - i0)z1 + (ip - i)z2)/(ip - i0)], {i, i0 + sign, ip, sign}]]; flag = True, {y, ymax, -ymax, -2.0ymax/i1}], {x, -1, 1, 2.0/j1}];
ListDensityPlot[depth, Mesh -> False, Frame -> False, ImageSize -> {j1, i1}, AspectRatio -> Automatic]

(* Create Shift Pattern, runtime: 10 seconds *)
i2 = 450; j2 = 90; SeedRandom[0];
pattern = Map[Hue, -50Abs[InverseFourier[Fourier[Table[Random[], {i2}, {j2}]]Table[Exp[-((j/j2 - 0.5)^2 + (i/i2 - 0.5)^2)/0.025^2], {i, 1, i2}, {j, 1, j2}]]], {2}];
Show[Graphics[RasterArray[pattern],ImageSize -> {j2, i2}, AspectRatio -> Automatic]]

(* Generate Autostereogram, Note: this code is not very accurate, you are better off using SISgen, runtime: 14 seconds *)
f[z_] := 2(14 - 8.7z)/(28 - 8.7z);
g[sign_] := Module[{x=0}, Table[x += sign/f[depth[[i, j]]]; pattern[[Mod[i - 1, i2] + 1, Mod[Round[x] - 1, j2] + 1]], {j, Floor[j1/2], If[sign != 1, 1, j1], sign}]];
Show[Graphics[RasterArray[Table[Join[Reverse[g[-1]], g[1]], {i, 1, i1}]], ImageSize -> {j1, i1}, AspectRatio -> Automatic]]

Link: Animated Shark Autostereogram - by Fred Hsu

Tetration - Mathematica, 10/8/12
In grade school, we all learned the basic arithmetic operators: multiplication is repeated addition, and exponentiation is repeated multiplication. But not many have learned about hyper operators: tetration is repeated exponentiation, pentation is repeated tetration, and hexation is repeated pentation. For example, consider the tetration ze. When z = 3, we have 3e = eee. But what if z is not an integer, or what if z is complex? Nobody knows how to calculate the tetration in those cases, but we can make a good guess using Taylor series expansions. Shown here is an approximate plot of ze where z is a complex number. This plot uses coefficients from this C++ code by Dmitrii Kouznetsov. Notice that ze transitions into an iterated fractal system as z becomes large. Some people believe the study of tetration may eventually help to unify the branches of mathematics. Here is some Mathematica code:
(* runtime: 37 seconds *)
n = 345; Repair[z_] := If[MachineNumberQ[z], z, $MaxMachineNumber]; Hue2[h_, b_] := Hue[h, Min[1, 2 (1 - b)], Min[1, 2 b]];
SuperExp[z_] := Repair[If[Re[z] <= -1, Log[SuperExp[z + 1]], If[Re[z] > 0, Exp[SuperExp[z - 1]], t = 1; sum = Log[z + 2] + Log[E/2]; Scan[(t *= z/2; sum += # t) &, {1.18353470251664338875, 1.58593285160678321155, 1.36629265207672068172, 1.36264601823980036066, 1.21734246689515424045, 1.10981816083559525765, 0.9667469297476984913, 0.84089872598668435888, 0.71353210966804747617, 0.60168548504001373445, 0.49928574281440518678, 0.41140086629121763728, 0.33506195665178500898, 0.27104779243942234146, 0.21728554054610033086, 0.17311050207880035456, 0.13690016038526570119, 0.10765949732729711286, 0.08413804539743192923, 0.06542450487497340761, 0.05060001212013485322, 0.03895655493977817629, 0.02985084640296329153, 0.02277908979501017117, 0.01730960309240666892, 0.01310389615589767874, 0.00988251130733762764, 0.00742735935367278347, 0.00556296426263720549, 0.00415334478103463346, 0.00309116153137843543, 0.00229387529664008653, 0.00169729976398295653, 0.00125245885041635465, 0.00092172809095368547, 0.00067661152429638357, 0.00049544127485341987, 0.00036192128589181518, 0.00026376927786672476, 0.0001918084004526757, 0.00013917553105723647, 0.00010077412023867018, 0.00007281884753121133, 0.00005251474516228446, 0.00003779882770351268, 0.00002715594536867241, 0.00001947408515177282, 0.00001394059355016322, 9.96213949015693*^-6, 7.1071387229271*^-6, 5.06199803708578*^-6, 3.59960968975399*^-6, 2.55569149787694*^-6, 1.81175810338313*^-6, 1.2824583153843*^-6, 9.0647322737496*^-7,6.3980422418981*^-7, 4.5095738191441*^-7, 3.1741772125007*^-7,2.2312521183625*^-7, 1.5663840476155*^-7, 1.098230101323*^-7, 7.690305934973*^-8, 5.378502675604*^-8, 3.757126131521*^-8, 2.621429405247*^-8, 1.826909956818*^-8, 1.271754463425*^-8, 8.84310192977*^-9, 6.14230041407*^-9, 4.26177146865*^-9, 2.95386817285*^-9, 2.04522503591*^-9, 1.41464900426*^-9, 9.7750884878*^-10, 6.7478454029*^-10, 4.6535930671*^-10, 3.2062550784*^-10, 2.2069891976*^-10, 1.5177557961*^-10, 1.0428189463*^-10, 7.158597119*^-11, 4.90980671*^-11, 3.364531769*^-11, 2.303635851*^-11, 1.575933679*^-11, 1.077213757*^-11, 7.35717912*^-12, 5.02077719*^-12, 3.42362421*^-12, 2.33271256*^-12, 1.58818623*^-12, 1.08046566*^-12, 7.3450488*^-13, 4.9894945*^-13, 3.3868911*^-13, 2.2973789*^-13, 1.5572383*^-13, 1.0548054*^-13,7.139840*^-14}]; sum]]];
Show[Graphics[RasterArray[Table[z = SuperExp[x + I y]; Hue2[Mod[Arg[z]/(2 Pi), 1], Max[0, 0.4 + Log[Abs[z]]/20]], {y, -2.5, 2.5, 10.0/n}, {x, -5.0, 5.0, 10.0/n}]]]]

Links
Tetration - excellent article in Citizendium
Tetration Forum - discussion forum
Tetration.org - informative website, by Daniel Geisler
Who can name the Biggest Number? - arrow notation, chained arrow notation, Busy Beavers
Large Numbers

Stereographic Projection of a Dodecahedron - POV-Ray, 10/10/08
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}];

Kluchikov’s Favorite Isosurface - C++ version: 7/8/09; Mathematica and MathGL3d version: 2/28/05
An isosurface or implicit surface can be thought of as a 3D contour. The original equation for this isosurface is attributed to Alex Kluchikov. I found some beautiful POV-Ray renditions of this surface on Christoph Hormann’s web site so I decided to see if I could render it using my own ray tracer. Isosurfaces are commonly used for scientific visualizations. Here is some Mathematica code:
(* runtime: 25 seconds *)
<< MathGL3d`OpenGLViewer`; x1 = 0.125; y1 = 0.25Sin[2 Pi/3];
Kluchikov[x_, y_, z_, t1_,t2_] := Module[{r = Sqrt[x^2 + z^2] - 1.5, theta = ArcTan[z, x] + Pi/2,t, x2, y2}, t = 8theta/3 + t1; x2 = r Sin[t] + y Cos[t]; y2 = r Cos[t] - y Sin[t]; 0.33(((x2 + 0.25)^2 + y2^2)^(1/64) + ((x2 - x1)^2 + (y2 + y1)^2)^(1/64) + ((x2 - x1)^2 + (y2 - y1)^2)^(1/64)) + 0.01Sin[5theta + t2]];
Scan[MVContourPlot3D[Kluchikov[x, y, z, If[#, 0, Pi/3], If[#, 0, Pi]], {x, -2, 2}, {z, -2, 2}, {y, -0.5, 0.5}, Contours -> {0.945}, PlotPoints -> 100, ContourStyle -> {RGBColor @@ If[#, {0, 0, 1}, {1, 1, 1}]}, MVAlpha -> If[#, 0.5, 1], MVNewScene -> #, MVReturnValue -> None] &, {True, False}]
MVPasteGraphics[];
If you do not wish to use the free MathGL3d package, you can use ImplicitPlot3D or ContourPlot3D, but it doesn’t look as nice:
(* runtime: 80 seconds *)
<< Graphics`ImplicitPlot3D`;
ImplicitPlot3D[Kluchikov[x, y, z, 0, 0] == 0.945, {x, -2.1, 2.1}, {z, -2.1, 2.1}, {y, -0.55, 0.55}, PlotPoints -> 100]
(* runtime: 80 seconds *)
<< Graphics`ContourPlot3D`;
ContourPlot3D[Kluchikov[x, y, z, 0, 0], {x, -2.1, 2.1}, {z, -2.1, 2.1}, {y, -0.55, 0.55}, Contours -> {0.945}, PlotPoints -> 12, ContourStyle -> {EdgeForm[]}]

This is how you can make this surface in POV-Ray:
// runtime: 7 minutes
camera{location 4*y look_at 0 up y right x} light_source{4*y,1}
#declare x1=1/8; #declare y1=sin(2*pi/3)/4; #declare Sqr=function(X) {X*X}; #declare theta=function{atan2(x,z)+pi/2}; #declare r=function{sqrt(x*x+z*z)-1.5};
#declare T=function(x,y,z,t1) {8*theta(x,y,z)/3+t1};
#declare x2=function(x,y,z,t1) {r(x,y,z)*sin(T(x,y,z,t1))+y*cos(T(x,y,z,t1))}; #declare y2=function(x,y,z,t1) {r(x,y,z)*cos(T(x,y,z,t1))-y*sin(T(x,y,z,t1))};
#macro Kluchikov(t1,t2,c) isosurface{function{0.33*(pow(Sqr(x2(x,y,z,t1)+1/4)+Sqr(y2(x,y,z,t1)),1/64)+pow(Sqr(x2(x,y,z,t1)-x1)+Sqr(y2(x,y,z,t1)+y1),1/64)+pow(Sqr(x2(x,y,z,t1)-x1)+Sqr(y2(x,y,z,t1)-y1),1/64))+0.01*sin(5*theta(x,y,z)+t2)-0.945} threshold 0 accuracy 0.0002 max_gradient 0.75 contained_by{box{-<2.1,0.55,2.1>,<2.1,0.55,2.1>}} pigment{rgbt c}} #end
Kluchikov(0,0,<1,1,1,0>) Kluchikov(pi/3,pi,<0,0,1,0.5>)

Link: Polygonising a scalar field - how to make metaballs, by Paul Bourke

Belousov-Zhabotinsky (BZ) Reaction - Mathematica 4.2, 5/10/07
This periodic cellular automaton simulates a chemical reaction called a Belousov-Zhabotinsky reaction. Here is some Mathematica code:
(* runtime: 24 seconds *)
n = 50; SeedRandom[0]; image = Table[Random[Integer, {0, 255}], {n}, {n}];
Do[image = Table[z = image[[i, j]]; If[z == 255, 0, zlist = Flatten[image[[Mod[i - {0, 1, 2},n] + 1, Mod[j - {0, 1, 2}, n] + 1]]]; a = 9 - Count[zlist, 0]; If[z ==0, b = Count[zlist, 255]; Floor[(a - b)/3] + Floor[b/3], Min[Floor[Plus @@ zlist/a] + 45, 255]]], {i, 1, n}, {j, 1, n}]; ListDensityPlot[image, Mesh -> False, Frame -> False], {200}];

Links
Mathematica notebook - by Takashi Yoshino
Five Cellular Automata program - by Hermetic Systems

Dual of the Geodesic Sphere - calculated in Mathematica 4.2, rendered in POV-Ray 3.6.1, 2/1/11
These geodesic spheres are inspired after Taff Goch's Hex-Pent Geodesic Sphere/Dome. The geodesic dome can be constructed by subdividing the triangular faces of an icosahedron into smaller triangles and projecting the vertices onto a dome. Then if you take the dual of the geodesic dome, you get a polyhedron composed entirely of pentagons and hexagons. But interestingly enough, no matter how many times you subdivided the geodesic dome, it's dual always has 12 pentagons. Another way to find a similar dual is to convert the triangles to hexagons and pentagons by removing central vertices. Buckminster Fuller patented the mathematics of the dome in 1954. Here is some Mathematica code:
(* runtime: 0.02 second *)
Normalize[x_] := x/Sqrt[x.x]; Rx[theta_] := {{1, 0, 0}, {0, Cos[theta], -Sin[theta]}, {0, Sin[theta], Cos[theta]}}; Rz[theta_] := {{Cos[theta], -Sin[theta], 0}, {Sin[theta], Cos[theta], 0}, {0, 0, 1}}; ApplyMatrix[R_, plist_] := Map[R.# &, plist];
phi = (1 + Sqrt[5])/2; phi1 = ArcCos[Sqrt[(3 + 4 phi)/15]]; phi2 = ArcCos[Sqrt[(7 - 4 phi)/15]];
face = Table[{Cos[theta], Sin[theta], (phi + 1)/2}, {theta, Pi/6.0, 1.5 Pi, 2Pi/3}];
icosahedron = Table[{ApplyMatrix[Rz[theta].Rx[phi1].Rz[Pi], face], ApplyMatrix[Rz[theta].Rx[phi2], face], ApplyMatrix[Rz[theta + Pi/5].Rx[Pi - phi2].Rz[Pi], face], ApplyMatrix[Rz[theta + Pi/5].Rx[Pi - phi1], face]}, {theta,0, 1.6Pi, 0.4Pi}];
Subdivide[{p1_,p2_, p3_}] := Module[{p12 =p1 + p2, p23 = p2 + p3, p13 = p1 + p3}, Map[Normalize, {{p12, p23, p13}, {p1,p12, p13}, {p2, p23, p12}, {p3, p13, p23}}, {2}]];
Show[Graphics3D[Polygon /@ Join @@ Subdivide /@ Join @@Subdivide /@ Join @@ icosahedron]]

Links
Buckminsterfullerene (Buckyball) - truncated icosahedron arrangement of carbon atoms, same shape as a soccer ball, see this assembly movie
Sphere Tessellation Icosahedron Subdivision - paper by Gernot Hoffmann
Thomson Problem - Points on a Sphere - Java applet demonstrating minimum energy configuration of electrons on a sphere (Thomson problem), by Cris Cecka
Magnus Wenninger - famous monk for creating origami polyhedrons

Knots Within Knots - Mathematica 4.2, POV-Ray 3.6.1, 3/1/11
These knots are covered with networks of smaller knots.

Links
Hexa Tube - sample Mathematica code for hexagonal tilings, by Mark McClure
8nov04b5 - trefoil weave by Robert Scharein

Moiré Pattern - POV-Ray 3.6.1 version: 4/15/07, Mathematica 4.2 version: 1/29/05
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

Spherical Canvas versus Reflective Sphere - POV-Ray 3.6.1, 10/17/07
This image was inspired by Dick Termes' paintings of 3D worlds on a spherical “canvas” called Termespheres. These images have 6 vanishing points as opposed to linear perspective drawings which only have 3 vanishing points. The left picture shows my version of a spherical canvas for my factory scene. This was accomplished by rendering a spherical panorama of the scene in POV-Ray, and then mapping it to a sphere. Click here to download some sample POV-Ray code. The right picture shows a reflective sphere when viewed from the exact same position. As you can see, it looks quite different.

Torus Knots - Mathematica 4.2, MathGL3d, POV-Ray 3.6.1, 10/31/04
This is one continuous torus knotted on itself. The left image shows a (11,3) torus knot and the right image shows a (6,13) torus knot. This image was featured on the cover of McGraw-Hill's 2012 Pre-Calculus textbook. Click here to see a rotatable 3D version. Click here to download an animated screensaver along with C++ source code. Click here to download some POV-Ray code for this image. The following Mathematica code was adapted from Maxim Rytin’s TubePlot code:
(* runtime: 2.6 seconds *)
Normalize[x_] := x/Sqrt[x.x]; p[t_] := {(1 + 0.3 Cos[11t/3])Cos[t], (1 + 0.3 Cos[11t/3]) Sin[t], 0.3 Sin[11t/3]};
f[t_, theta_] := Module[{dp = p'[t], ddp = p''[t], tangent, normal1, normal2},tangent = Normalize[dp]; normal1 = Normalize[ddp(dp.dp) - dp(dp.ddp)]; normal2 = Cross[tangent, normal1]; p[t] + 0.1 (normal1 Cos[theta] + normal2 Sin[theta])];
ParametricPlot3D[Append[f[t, theta], {EdgeForm[], SurfaceColor[Hue[t/(2Pi)]]}], {t,0, 6Pi}, {theta, 0, 2Pi}, PlotPoints -> {360, 15}, Compiled -> False]

POV-Ray has an internal function for torus knots:
// runtime: 8 seconds
camera{location 16*y look_at 0} light_source{16*y,1}
#declare f=function{internal(24)}
isosurface{function{f(x,y,z,6,11,3,0.1,0.5,1,0.1,1,1,0)} max_gradient 2 contained_by{sphere{0,8}} pigment{rgb 1}}

Knots Links
Torus Knot Fun - animated torus knot by Krzystof
Gears Trefoil - by Michael Trott
Tubes and Knots - Mathematica code by Mark McClure
One ring to rule the ball - 3D printed torus knot with ball, by TerraCotta

Silver Spirals - AutoCAD 2000, POV-Ray 3.6.1, 8/19/02
These are just some doodles I made while playing around with AutoCAD.

Silver Noodles - AutoCAD 2000, POV-Ray 3.6.1, 11/30/10
This is another doodle I made while playing around with AutoCAD.

Intersection of 3 Cylinders - POV-Ray 3.6.1, 12/15/08
One thing that has always intrigued me is the intersection of three perpendicular cylinders all meeting at the same point. Notice that the resulting shape is not a sphere. In fact, when viewed from a certain angle, the outline of this shape looks like a hexagon.

Loxodrome - POV-Ray 3.6.1, 10/2/08
A Loxodrome (or Rhumb line) is a spherical spiral. A ship takes this path when it maintains a constant angle with respect to the equator. This image shows a loxodrome intertwined with spirals.

Links
Rhombic Triacontahedron IV - sculpture composed of loxodromes, by Vladimir Bulatov, see also his pendant
Loxodromic Fractal World - by Ramiro Perez

Golden Ratio Logarithmic Spiral Transformation - Mathematica 4.2, 6/21/04
The Golden Ratio f = (1 + sqrt(5))/2 ≈ 1.61803 has an interesting relationship with Fibonacci Numbers. The basic equation for a Golden Spiral is given by r(q) = f2q/p. The interlocking rings pattern in this image was adapted from M. C. Esher’s Snakes. Here is some Mathematica code:
(* runtime: 48 seconds *)
image = Import["C:/Picture.jpg"][[1, 1]]/255.0; imax = Length[image]; jmax = Length[image[[1]]];
n = 275; phi = 0.5 (1 + Sqrt[5]);
Show[Graphics[RasterArray[Table[Module[{x = 2j/n - 1, y = 2i/n - 1, r, theta}, r = x^2 + y^2; theta = ArcTan[y, x]; RGBColor @@ If[r != 0, image[[Floor[imax Mod[2theta/Pi, 1]] + 1, Floor[jmax Mod[theta/Pi + 0.25Log[r]/Log[phi], 1]] + 1]], {0, 0, 0}]], {i, 1, n}, {j, 1, n}]], ImageSize -> n, PlotRange -> {{0, n}, {1, n}}, AspectRatio -> 1]]

Links
Droste Effect Gallery - recursive pictures by Josh Sommers
Seamless Maker - Droste effect software by Hypatiasoft

Complex Map Polar Transformation: f(z) = e2 p z - Mathematica 4.2, 6/14/04
These images were generated by mapping a tessellation to the complex plane, similar to M. C. Esher’s Development II. Here is some Mathematica code:
(* runtime: 80 seconds *)
image = Import["Picture.jpg"][[1, 1]];
n = Length[image]; m = Length[image[[1]]];
Show[Graphics[RasterArray[Table[RGBColor @@ Module[{z = Log[2j/275 - 1 + I (2i/275 - 1)]/(2Pi) + 1.0}, image[[Floor[n Mod[6 m Re[z]/n, 1]] + 1, Floor[m Mod[6 Im[z], 1]] + 1]]/255.0], {i, 1, 275}, {j, 1, 275}]], ImageSize -> 275, PlotRange -> {{0, 275}, {1, 275}}, AspectRatio -> 1]]

Fisheye Transformation - Mathematica 4.2, 6/15/04
Here is some Mathematica code:
(* runtime: 26 seconds *)
image = Import["C:/Picture.jpg"][[1, 1]];
n = Length[image]; m = Length[image[[1]]];
Show[Graphics[RasterArray[Table[RGBColor @@ Module[{x = 2j/275 - 1, y = 2i/275 - 1, r, h}, r = 1.0 - x^2 - y^2; h = 0.5(1 - 0.5If[r > 0, Sqrt[r], 0]); image[[Floor[n Mod[5 m h x/n, 1]] + 1, Floor[m Mod[5 h y, 1]] + 1]]/255.0], {i, 1, 275}, {j, 1, 275}]], ImageSize -> 275, PlotRange -> {{0, 275}, {1, 275}}, AspectRatio -> 1]]

RSA Encryption (Rivest, Shamir and Adleman) - Mathematica 4.2, 11/15/06
Here is some Mathematica code to encrypt messages using prime numbers. Anyone with the public key (and n) can encode messages, but only the person with the secret key can decode them:
(* runtime: 0.05 second *)
message = "This is a secret message containing 112 characters. This message will be divided into 4 blocks of 28 characters.";
<< NumberTheory`NumberTheoryFunctions`; SeedRandom[0];
PublicKey = NextPrime[Random[]256^28]; p = NextPrime[Random[]256^14]; q = NextPrime[256.0^28/p]; n = p q; SecretKey = PowerMod[PublicKey, -1, (p - 1)(q - 1)];
ToNumbers[str_] := Map[FromDigits[#, 256] &, Partition[ToCharacterCode[str], 28]];
ToText[nlist_] := StringJoin @@ Map[StringJoin @@ Map[FromCharacterCode, IntegerDigits[#, 256, 28]] &, nlist];
encryption = ToText[Map[PowerMod[#, PublicKey, n] &, ToNumbers[message]]]
ToText[Map[PowerMod[#, SecretKey, n] &, ToNumbers[encryption]]]

If you do not know p and q, you can try to break the code using this method (but it is very slow):
(* runtime: 40 minutes *)
SecretKey = PowerMod[PublicKey, -1, EulerPhi[n]];

Here’s a basic idea how these functions work:
InverseMod[a0_, n0_] := Module[{n = n0, a = a0, x = 0, x1 = 0,x2 = 1, q1 = 0, q2 = 0, i = 1}, While[a != 0, If[i > 2, x = Mod[x1 - x2 q1 + n0^2, n0]; x1 = x2; x2 = x]; q1 = q2; q2 = Floor[n/a]; {n, a} = {a, Mod[n, a]}; i++]; Mod[x1 - x2 q1 + n0^2, n0]];
PowerMod[a_, b_, n_] := If[b == -1, InverseMod[a, n], Module[{c = 1}, digits =IntegerDigits[b, 2]; Scan[If[#[[1]] == 1, c = Mod[c #[[2]], n]] &, Transpose[{digits, Reverse[NestList[Mod[#^2, n] &,a, Length[digits] - 1]]}]]; c]];
GCD[a_, b_] := If[b == 0, a, GCD[b, Mod[a, b]]];

Link: RSA Encryption - Mathematica notebook by Kit Dodson

Spirographs - MacDraw (vector art), 1992?
I made these a very long time ago on my old Macintosh. I don’t remember how to make them anymore.

Mathematica Typesetting Shortcuts
Here is a Mathematica notebook that summarizes some convenient Mathematica typesetting shortcuts. You can type equations quite quickly in Mathematica once you know these shortcuts. I use these shortcuts to type my class notes in Mathematica.
Mathematica Links
Mathematica - excellent technical computing software, easy to make beautiful plots and play equations as sounds
MathGL3d - free Mathematica package for rendering with OpenGL and writing POV-Ray scripts
LiveGraphics3D - software for displaying rotatable 3D Mathematica graphics on the internet
JavaView - another program similar to LiveGraphics3D
Mathematica Information Center - many sample notebooks & packages

Mathematicians
The Thirty Greatest Mathematicians - interesting biographies by James Dow Allen
Leonhard Euler - had a photographic memory, did his greatest work after he went blind, saying "Now I will have less distraction"
Carl Gauss - perhaps the greatest mathematician who ever lived, corrected his family finances at the age of 3
Srinivasa Ramanujan - self-taught mathematician who grew up in poverty and baffled leading mathematicians, developed amazing formulas for pi
…variste Galois - proved there is no quintic formula, died in a duel at age 20 over a woman
Andrew Wiles - solver of Fermat's Last Theorem
John Nash - the movie "A Beautiful Mind" is based on his life

Other Math Links
Paul Nylander's Page - at the Virtual Art Museum
Sphere Eversion - beautiful animation turning a sphere inside-out by Bill Thurston
Penrose Tiling - amazing aperiodic tilings discovered by Roger Penrose, here is a Mathematica notebook by E. Arthur Robinson
Hyperseeing - electronic publication by The International Society of the Arts, Mathematics, and Architecture
Chatin’s Constant - the infamous “incalculable” constant, the probability that a random algorithm halts
Fractional Calculus - fractional derivatives
Discrete Cosine Transform - how JPEGs are made
Strang’s Strange Figures - unexpected patterns in trig functions
Wikipedia article on p
Gödel's Incompleteness Theorem - famous paper that supposedly defeats all hope of formally proving all truth. I can’t say I understand it, but it’s interesting (and controversial).
Chinese Rings - a simple puzzle that can be solved in no less than 18446744073709551616 moves
Number Spiral - interesting pattern of prime numbers
echochrome - M.C. Escher video game
Probability theory paradoxes - Monty Hall problem, Simpson's paradox
The Math Book - by Clifford Pickover, includes some of my math artwork
Rose windows - beautiful stained glass windows found in Gothic churches
Mathematica Art - I especially like the animated GIFs