Mathematical art

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Mathematics or art? They probably belong more to the former than to the latter, but they sure look nice. I made them myself with Mathematica (a mathematical software package). More pics and explanation will follow soon. They're relatively large downloads, so only attempt to watch them if you have a broadband connection.

http://www.vrijstaatfierens.nl/plaatjes/logspiralcaustic.gif
http://www.vrijstaatfierens.nl/plaatjes/circlecaustic.gif
http://www.vrijstaatfierens.nl/plaatjes/pearlcaustic.gif

very nice.

Check out fractals; art made through mathematics. Apophosys, Kaos Rei (may be misspelled), there's dozens of free fractal programs. Have fun

Would it not have been easier to do this with Flash?

depends on what you want to do, and how you want to do it. Fractals are visual representations of mathematical equations...

but you could probably do it in Flash...

I don't know if Flash can properly calculate the incidence and reflection angles of the rays of light. It would probably be much more complicated than coding it in Mathematica. This is the Mathematica code for it (for the first one):

r[t_]:=Exp[t/5]
f[t_]:=r[t]Cos[t]
g[t_]:=r[t]Sin[t]
rx[t_]:=Evaluate[D[f[t],t]]
ry[t_]:=Evaluate[D[g[t],t]]
nx[t_]:=Evaluate[-D[g[t],t]]
ny[t_]:=Evaluate[D[f[t],t]]
reflectx[t_]:=(f[t]rx[t]+g[t]ry[t])rx[t]-(f[t]nx[t]+g[t]ny[t])nx[t]
reflecty[t_]:=(f[t]rx[t]+g[t]ry[t])ry[t]-(f[t]nx[t]+g[t]ny[t])ny[t]
norm[n_]:=If[reflectx[n]^2+reflecty[n]^2>0,Sqrt[reflectx[n]^2+reflecty[n]^2],1]
norm2[n_]:=If[nx[n]^2+ny[n]^2>0,Sqrt[nx[n]^2+ny[n]^2],1]
time[t_,n_]:=If[t<=Sqrt[f[n]^2+g[n]^2],0,(t-Sqrt[f[n]^2+g[n]^2])/norm[n]]
a=1.1;

curvecaustic=ParametricPlot[{f[t],g[t]},{t,0,2Pi},PlotRange->{{-a,a},{-a,a}},
Axes->False,PlotStyle->{Thickness[0.02],Background->Black,RGBColor[0,0,1]}];

causticmov=Table[Show[curvecaustic,Graphics[Table[{RGBColor[1,0,0],Line[{{f[n],g[n]},
{(reflectx[n]time[t,n]+f[n]),(reflecty[n]time[t,n]+g[n])}}]},{n,1/100,2Pi,Pi/35}],
PlotRange->{{-a,a},{-a,a}}],
Graphics[Table[{RGBColor[1,1,0],Line[{{0,0},{Cos[n]t,Sin[n]t}}]},
{n,0,2Pi-1/100,Pi/35}],PlotRange->{{-a,a},{-a,a}}],Background->Black],{t,0,4,1/35}];

Export["/Users/***********/Documents/logspiralcaustic.gif",causticmov,ImageSize->600]

Is there a good fractal program for a Mac?

Look at http://www.fractaldomains.com/ fcr example. I myself run Ultrafractal under CrossOver. The former is simpy the best fractal program in existence, whereas the second is a very good Windows emulator for Mac.

See:
http://www.ultrafractal.com/
http://www.codeweavers.com/products/..

Great work! Beautiful constructions in an unconventional setting for art.

Thanks!

I thank you! Perhaps differential geometry is a bit too arid for true art to emerge, but one can certainly draw nice pictures.

UPDATE: Now for some explanation, like I promised. What you're seeing is this:

The blue curve is just what it is: a blue curve, but we imagine it to be made of some reflective material, so a mirror essentially. The yellow straight lines are rays of light emanating from a point source, while the red ones are the rays reflected by the curved mirror, according to the rule "Angle of incidence, w.r.t. the normal, equals angle of reflection. " (Here the normal is defined to be the line perpendicular to the curve's tangent, i.e., the tangent at the point of incidence.) These reflected rays, for most choices of the blue curve, envelope some other curve. This second curve is called the caustic curve of the first one.

The first animation illustrates beautifully that the caustic of a logarithmic spiral is also a logarithmic spiral.

i like expressing mathematics in a visual way.
i often have very satisfying mathematical thoughts that are useless to the rest of the world.
i like to use fractal methodology to design intricate patterns based on mathematical designs. i like writing fractal functions as well as animating them.

here is a function i wrote with some music to accompany it.
please hit the "HQ" button if you do not want it to be shown as a compressed grainy and inferior default.

hmmmm....there seems to be a problem with embedding you tube files on this site for the last few days.

so i will post a link instead.
http://www.youtube.com/user/iamxb70#p/u/21/jHIfuiOE-Jw


i like all manners of mathemtaical expressions that can be represented visually.

here is another. (the accompanying song is crude and i never finished it)
please choose "hq" (high quality) to watch it in.

http://www.youtube.com/user/iamxb70#p/u/1/kRAFGPk-qPM

i have hundreds of ideas for patterns and animations of those patterns that i am yet to write.

here is another (please choose "hq")
http://www.youtube.com/user/iamxb70#p/u/16/Npfx1p6WRE0


it is very hard work to elaborate a pattern with functions to a degree where the patterns can be drilled into for a long way before they reveal the simplistic grid structure of their foundation.

i also have many completed patterns.