Heck yes fellas! I'll show off some fun stuff after dinner. Y'all rock, these examples are exactly what I was hoping to find!

With[{f = Compile[{{point, _Real, 1}, {r, _Real, 0}},
            Boole[point.point <= r^2], Parallelization -> True,
            RuntimeAttributes -> {Listable}]},
        dm3d[n_Integer, r_Integer] :=
        f[CoordinateBoundingBoxArray[{-{1, 1, 1}, {1, 1, 1}} Max[1, Quotient[n, 2]]], r]]

Another version of halirutan's method.

What function is this? :)

{-2 x (y - 1/5), x^2 - (y - 1/5)^3} (16 - (x^2 + y^2)^2) + {-y,
   x} (x^2 + y^2)^2

Or if you mean the full command:

(* Valentine flow *)
vstyle = Directive[Darker[Red]; Pink, Thick];
LineIntegralConvolutionPlot[{-2 x (y - 1/5),
    x^2 - (y - 1/5)^3} (16 - (x^2 + y^2)^2) + {-y, x} (x^2 + y^2)^2,
 {x, -3, 3}, {y, -3, 3},
 RasterSize -> 400,
 StreamPoints -> {{{{-0.1, 0.65}, vstyle}, {{0.12, 0.7}, vstyle}}},
 ColorFunction -> (Hue[#5^0.4] &)]

Nice!

We were deriving a Lyapunov function for {-2 x y, x^2 - y^3} in class yesterday, and I thought I'd throw up a picture of what we had constructed. And it looked vaguely heart-shaped. Then I tweaked it a bit.

{-2 x y, x^2 - y^3} looks vaguely familiar.

I can't place it. It's from Simmon's Diff. Eq. with Historical Notes.

I think I recall it as a modification of Hénon-Heiles, but the name of who tweaked it escapes me now.

Yes, it's sort of like H-H. I was thinking of it as a modification of {-2 x y, x^2 - y^2}, with y^2 changed to y^3 to make the origin a stable eq. -- a "textbook" example. in other words.

"Textbook" in the sense of contrived to make the construction of the Lyapunov function fairly straightforward.

(Goodnight, J.M. Nice to chat with you. Bedtime here.)

See you too.

StringTemplate["This is ``. This is ``."][ToString[Style["Arial", FontFamily -> "Arial"], StandardForm], ToString[Style["Source Code Pro", FontFamily -> "Source Code Pro"], StandardForm]]

Seems like CaptureImage doesn't really take any arbitrary function f for ImageCaptureFunction or ImagePreviewFunction

If you define your own function (even if it returns an image) it gives you eg

ImageCapture::erropts: The value ImageAssemble[ImagePartition[#1,Scaled[0.1]]]& specified for the option ImagePreviewFunction is invalid.

aha, I see, they process the function on a single-pixel image to see if it'll work and so if your function can't handle single-pixel images (eg if it resizes below a single pixel) it will break.

@CarlLange That does beg the question of finding a (cheap?) way to test if a function can accept an image.

@J.M.willbebacksoon I would have thought simply checking if the Head was Image...

well, maybe I'm oversimplifying. Even so, I am a bit surprised at this requirement - and even more surprised they don't test the function on a 2x2 image (or 20x20, or the resolution of the camera) rather than 1x1

@CarlLange Image[{3.}, "Real"] is certainly a bad input, but its head will still be Image.

Testing on a 2-by-2 looks sound to me.

Yes, I'll have to report a bug

Also ImageCapture doesn't appear to correctly respond to ImageSize :<

I guess I'll just need to use Refresh of CurrentImage instead.

Ha, also funny. If you have a disabled ImageCapture and run CurrentImage elsewhere in your code, the ImageCapture will re-enable itself

omg, UpdateInterval doesn't appear to have any effect here: Dynamic[Refresh[CurrentImage[], UpdateInterval -> 10]]

it's because CurrentImage already has its own Refresh so it can act "correctly" for shorthand like Dynamic[CurrentImage[]]

terrible

Does anybody have an idea for how to pass UpdateInterval into that internal Refresh? You can see the one I'm talking about with ResourceFunction["PrintDefinitions"][CurrentImage]

passing it to the external Dynamic doesn't work

CurrentImage also doesn't act correctly in response to ImageSize (when you pass ImagePreviewFunction). Wow, what a frustrating afternoon

This seems to be a situation where you can bunch together bug reports, seeing that things that are supposed to mesh together actually don't.

@C.E. Anton seems to have a problem with putting long URLs in PackageData.

@J.M.willbebacksoon that's nice. I was stuck trying with the function Text only.

@J.M.willbebacksoon Thank you for telling me this, I've now corrected it.

@J.M.willbebacksoon @anhnha alternately if you want to maintain composability and avoid the intricacies of having Style-d stuff inside a String:

StringForm[
 "This is ``. This is ``.",
 ToString[Style["Arial", FontFamily -> "Arial"], StandardForm],
 ToString[Style["Source Code Pro", FontFamily -> "Source Code Pro"],
  StandardForm]
 ]

Totally thought there was a 3D version of MatrixPlot Or ArrayPlot (that is not ArrayMesh which does not like to represent grayscale values one might get from an FFT) but I guess I was mistaken? Can anyone clarify? Both of your compiled offerings worked wonders btw @halirutan @J.M.willbebacksoon and I still hope to figure out this method that @b3m2a1 alluded to. :D

@CATrevillian Just use Image3D.

@halirutan I will have to try this, I think that is the one I forget about the most!

@CATrevillian The reason is that you cannot really display a 3d matrix. Image3D uses volume rendering which essentially means that you have translucent voxels and the overall image is created by sending lightrays from the camera through the whole volume. Each lightray accumulates its final coloring by going through all the voxels it passes.

Different values in your 3d matrix have different colors that come from the defined colorscheme.

@CATrevillian Terence Tao is one of the prodigies these days in mathematics if I remember correctly.