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The code below

BanchoffChmutov[n_] := ContourPlot3D[ ChebyshevT[n,x]+ChebyshevT[n,y]+ChebyshevT[n,z], {x,-1.3,1.3},{y,-1.3,1.3},{z,-1.3,1.3}, Contours->0.02, AspectRatio->Automatic, Boxed->False, Axes->{False,False,False}, BoxRatios->Automatic, PlotRangePadding->None, PlotPoints->100, ViewPoint->{-2,3,3}];
pic = BanchoffChmutov[10]
Export["C:\\Z10.png", pic, ImageResolution->100, ImageSize->1000, Background->None];

runs fine when 10 is replaced by any of 1,...,9, but it produces an error for 10 and up:

No more memory available. Mathematica kernel has shut down. Try quitting other applications and then retry.

I've stopped almost all other applications, but still it doesn't produce an image.

How can I tell my Mathematica to increase the amount of memory associated to this computation, and also, to calculate longer if necessary, and not just quit. I have 3.24GM of RAM and 3GHz of CPU.

share|improve this question
I'd consider setting MaxRecursion -> 1 first, and then slowly cranking it up for as long as the plot doesn't hang... – J. M. Oct 21 '12 at 4:15
up vote 8 down vote accepted

To understand what happens here, you have to know 2 things: While PlotPoints->n makes that you have n equally distributed sampling points of your plot-range, this alone is not enough. On places where your function varies very much, the algorithm has the chance to subdivide the interval further. MaxRecursion tells you how many times this subdivision can take place.

Therefore, when you try to use ContourPlot without eating all your memory, you should adjust not only the PlotPoint. Here is a version of your function which spares to call ChebyshevT over and over again and lets you send some ContourPlot3D options in:

BanchoffChmutovHal[n_, opts : OptionsPattern[]] :=
 Block[{x, y, z},
  With[{expr = ChebyshevT[n, x] + ChebyshevT[n, y] + ChebyshevT[n, z]},
    expr, {x, -1.3, 1.3}, {y, -1.3, 1.3}, {z, -1.3, 1.3},
    Evaluate[FilterRules[{opts}, Options[ContourPlot3D]]],
    Contours -> 0.02, AspectRatio -> Automatic, Boxed -> False, 
    Axes -> {False, False, False}, BoxRatios -> Automatic, 
    PlotRangePadding -> None, ViewPoint -> {-2, 3, 3}]

Now you can play with the settings and using

GraphicsRow[BanchoffChmutovHal[#, PlotPoints -> 80, MaxRecursion -> 0] & 
 /@ {15, 20}, ImageSize -> 500]

gives you a result after only a few seconds

Mathematica graphics

Setting MaxRecursion to anything other than 0 with PlotPoints even about 20 gives the message about the memory. This tells you not only to raise the memory-level, but it tells you that Mathematica tries to subdivide your space, because it obviously assumes that not all features are cought.

You can never be sure whether all contours are found, but giving Mathematica enough PlotPoints and MaxRecursion is always good. So what can we do if we have not enough memory? You can simply reduce your plot-range. This here took only about 5GB of memory although I used 100 points and a recursion level of 7:

Block[{x, y, z},
 With[{expr = 
    ChebyshevT[20, x] + ChebyshevT[20, y] + ChebyshevT[20, z]},
  ContourPlot3D[expr, {x, 0, 0.5}, {y, 0, .5}, {z, 0, .5},
   PlotPoints -> 100, MaxRecursion -> 7, 
   Method -> {"MaxMemoryUse" -> Infinity},
   Contours -> 0.02, AspectRatio -> Automatic, Boxed -> False, 
   Axes -> {False, False, False}, BoxRatios -> Automatic, 
   PlotRangePadding -> None, ViewPoint -> {-2, 3, 3}]

Mathematica graphics

Here you see at least how the pattern should look like and you can compare it to your graphics created for the whole plot-range.

share|improve this answer
Good explanation! – cormullion Oct 22 '12 at 13:38
@cormullion I tried the last hour PlotPoints->50 and MaxRecursion->4 using 30GB of RAM but it didn't finished. Maybe I start it tonight and let it run a bit longer. – halirutan Oct 22 '12 at 13:40
FYI, the OP has previously posted some background information - but he should still benefit from your answer! – cormullion Oct 22 '12 at 14:21
This is an amazing solution! I do not fully understand what is happening in the first code, but the performance is very good (I've plotted the first 20 surfaces). I hope you'll be willing to give additional explanations, if I later need them. Thank you! – Leon Oct 29 '12 at 2:17

Running this with a value of 10, Mathematica shows me this helpful message:

picture of message

before giving the final result after a minute or so:


which is very pleasing.

On this iMac, I have 12GB RAM, of which the MathKernel has just helped itself to 1.8 GB of real. Mathematica's a bit of a memory hog at times... :)

Edit: With a value of 15 you see this:

with a value of 15

With a value of 20, this:

with a value of 20

share|improve this answer
Hmm, I'm not aiming at just 10, I'd like to go to possibly 16 or 20. What exactly must I write to get results (even if Mathematica needs a few hours to produce them)? How can I persuade her to keep on calculating until it is done? – Leon Oct 21 '12 at 7:53
You may have to buy her a larger house.. – cormullion Oct 21 '12 at 7:55
Umm, my question is precisely about this issue: I would like to not sacrifice plotpoints (quality) but rather the time to produce the images. By the way, could someone with a faster computer produce the images with my code (png, with transparent background, not white, with 1000 pixels height)? I'd really appreciate it, I need it tomorrow. Would it be possible to go up to 16 or 18 or 20? – Leon Oct 21 '12 at 14:57
Many of the points are occluded by points in the front, maybe you could carefully choose a plot range to plot the points that are in view. – s0rce Oct 22 '12 at 1:12
@LeonLampret I hope the images were of use... – cormullion Oct 26 '12 at 12:03

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