The figure is
See the how-to video or a speeded-up GIF.
I believe it should be possible to draw this figure programmatically using some Random function, but I'm rather new to Mathematica, so I could really use some help here.
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asked Mar 23, 2016 at 19:57
AccidentalFourierTransformAccidentalFourierTransform
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1$\begingroup$ This is related. $\endgroup$– J. M.'s missing motivation ♦Commented Mar 24, 2016 at 0:22
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$\begingroup$ Fun question. I still can't get over people spend their free time trying to draw things like this. $\endgroup$– WilliamCommented Mar 24, 2016 at 3:00
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6$\begingroup$ @William, they spend the time because it is, as you say, fun. $\endgroup$– J. M.'s missing motivation ♦Commented Mar 24, 2016 at 3:15
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$\begingroup$ Those are art forms. Mathematica can make a stunning variety of them. Over the past year I have been using Mathematica to design geometric sculptures. Part of the process is looking for attractive 3D graphics which are made up of elements that I can put together in the real world. I recently saw an unanticipated Graphics3D result that I have been able to make in solid form. The result has energetic beauty. $\endgroup$– Ralph DratmanCommented Mar 29, 2016 at 21:04
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4 Answers
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Here's a quick take on it:
Clear[spiralize];
spiralize[p_, d_:10, r_:4, f_:0.8, s_:1, t_:0.005]:=Module[{m,rr=r},
m = Mean @ p[[1]];
Graphics[{EdgeForm[Thickness[t]],FaceForm[White],
NestList[GeometricTransformation[
GeometricTransformation[#,
RotationTransform[rr++s \[Degree],m]],
ScalingTransform[{f,f},m]
]&, p, d]}
]
]
pts = RandomReal[{-1, 1}, {50, 2}];
polys = MeshPrimitives[VoronoiMesh[pts], 2];
Show[spiralize[#, 40, 5, 0.85] & /@ polys]
Play with the parameters:
pts = RandomReal[{-1, 1}, {10, 2}];
polys = MeshPrimitives[VoronoiMesh[pts], 2];
Manipulate[
Show[spiralize[#, d, r, f, s, t] & /@ polys], {{d, 10}, 1, 20,
1}, {{r, 5}, 1, 20}, {{f, 0.85}, 0, 1}, {{s, 1}, 0.1,
3}, {{t, 0.001}, 0, 0.01}]
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3$\begingroup$ wow really cool :) thank you! is there any parameter I can play with to get different results? $\endgroup$ Commented Mar 23, 2016 at 22:44
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$\begingroup$ Yes, try changing the scaling and rotating arguments in the function. $\endgroup$– M.R.Commented Mar 23, 2016 at 23:46
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voronoi[pts_] := ListDensityPlot[Append[#, 0]&/@ pts, InterpolationOrder-> 0,
Frame -> False]
pts = RandomReal[{0, 256}, {20, 2}];
cp = Cases[Normal@voronoi[pts], Polygon[a_, ___] :> Polygon[a], ∞];
cp1 = cp /. Polygon[a___] :> a;
ms = Mean /@ cp1;
Graphics[{EdgeForm[Black], FaceForm[White], cp,
Line /@ Join @@@ (Transpose /@ (MapThread[
Table[BSplineFunction[Join[Join[#1, #1][[i ;; i + 1]], #2]][t],
{i, 1, Length@#1}] &, {cp1, List /@ ms}, 1] /.
a_[t] :> a /@ Range[0, 1, .03]))}]
answered Mar 23, 2016 at 22:28
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$\begingroup$ an arc length parametrization would be better $\endgroup$ Commented Mar 23, 2016 at 22:32
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1$\begingroup$ it looks really nice :) thank you! what parameters could I change to get different results? (such as more/less spirals) $\endgroup$ Commented Mar 23, 2016 at 22:46
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$\begingroup$ @AccidentalFourierTransform The
20inpts = RandomReal[{0, 256}, {20, 2}];is the number of spirals. The spacing is determined byRange[0, 1, .03],so you may try things like(Rescale[Sin[# ^(2)] & /@ Range[0.001, 1, .05]])instead $\endgroup$ Commented Mar 23, 2016 at 22:52
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Here is a slightly different way of going about it:
BlockRandom[SeedRandom[42, Method -> "Rule30CA"]; (* for reproducibility *)
pts = RandomReal[{-1, 1}, {50, 2}]];
With[{h = 1/5 (* offset *), n = 30 (* iterations *)},
Graphics[{FaceForm[], EdgeForm[AbsoluteThickness[1/5]],
NestList[# /. Polygon[p_] :>
Polygon[Transpose[Partition[p, 2, 1, 1], {1, 3, 2}].
{1 - h, h}] &,
MeshPrimitives[VoronoiMesh[pts], 2], n]}]]
This version incorporates Rahul's suggestion to randomize the rotation directions:
With[{h = 1/5 (* offset *), n = 30 (* iterations *)},
BlockRandom[SeedRandom[42, Method -> "Rule30CA"]; (* for reproducibility *)
pts = RandomReal[{-1, 1}, {50, 2}];
Graphics[{FaceForm[], EdgeForm[AbsoluteThickness[1/5]],
NestList[# /. Polygon[p_] :>
Polygon[Transpose[Partition[p, 2, 1, 1], {1, 3, 2}].
{1 - h, h}] &,
Map[RandomChoice[{Identity, Reverse}][#] &,
MeshPrimitives[VoronoiMesh[pts], 2], {2}], n]}]]]
answered Mar 24, 2016 at 8:00
J. M.'s missing motivation♦J. M.'s missing motivation
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1$\begingroup$ Maybe it's just a matter of perception, but I like this the best. I think (could be wrong), that the other answers have the lines simply spiraling into the center of the mesh cell, whereas they ought to hit the boundary before turning and heading to the next edge. +1 $\endgroup$– LLlAMnYPCommented Mar 24, 2016 at 8:19
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$\begingroup$ Very perceptive of you! :) That's why I used linear interpolation over each polygon edge instead of scaling + rotating; that other approach will inevitably have corners jutting out of the spiral for some irregular polygons. $\endgroup$ Commented Mar 24, 2016 at 8:22
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$\begingroup$ I wonder if it would be possible to make the edges of the Voronoi mesh less visible. In the original image the polygon edges are barely noticeable because the lines on either side start have a very uniform density, whereas here those edges are quite pronounced. $\endgroup$ Commented Mar 24, 2016 at 9:07
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$\begingroup$ @Martin, the other thing contributing to that illusion is that the polygons were not all rotated in the same direction. That I think takes more work to do. $\endgroup$ Commented Mar 24, 2016 at 9:14
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$\begingroup$ @J.M. Oh, I just noticed that this solution doesn't do that yet. Yeah, that definitely helps with the uniform density, but I think even when the direction on two adjacent polygons is the same, is looks a bit more homogeneous. I'm not entirely sure how to go about that though... I'll see if I come up with anything. $\endgroup$ Commented Mar 24, 2016 at 9:18
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After seeing your awesome contributions I really wanted to do it myself, and I'm pretty happy with the result:
It took me quite a bit of time because I'm very rusty when it comes to progamming. Also, the code is probably highly inefficient, so any suggestion will be very appreciated.
The main idea to genetare this is to first draw some random quadrilaterals:
ClearAll["Global`*"]
a = .25; (*side length*)
c:=.15 RandomReal[{-1, 1}]; (*random shifting*)
d = .15;
n = 3; (*n+1 rectangles in the x direc.*)
m = 2; (*m+1 rectangles in the y direc.*)
s = NestList[{#[[2]],#[[2]]+{a+c,0},#[[2]]+{a+c,a+c},#[[3]],#[[2]]} &,{{0,0},{a+c,0},{a+c,a+c},{0,a+c},{0,0}},n];
AppendTo[s,{#[[2]],#[[2]]+{a,0},#[[2]]+{a,a},#[[3]],#[[2]]}&[Last[s]]];
f[x_] := Module[{k=FoldList[{#1[[2]],#2[[3]],#2[[3]]+{c,a+c},#1[[3]],#1[[2]]}&,{#[[4]],#[[3]],#[[3]]+{c,a+c},#[[4]]+{c,a+c},#[[4]]}&[x[[1]]],Rest@x]},
k[[1,4,1]]=0;
k[[n+2,3,1]]=x[[-1,2,1]];
k];
q = NestList[f,s,m];
Table[q[[-1,j,3,2]]=q[[-1,j,4,2]]=(m+1)a,{j,1,n+2}];
q = Partition[#,2]&/@Partition[Flatten[q],10];
ListPlot[q,Joined->True,Axes->False]
The, I randomly turn some of these quadrilaterals into triangles:
Table[q=ReplacePart[q,i->Sequence@@{q[[i]][[{1,2,3,1}]],q[[i]][[{3,4,1,3}]]}];
,{i,RandomSample[Range[Length[q]],Floor[(n+1)(m+1)/3]]}];
Table[q=ReplacePart[q,i->Sequence@@{q[[i]][[{1,2,4,1}]],q[[i]][[{2,3,4,2}]]}];
,{i,RandomSample[Range[Length[q]],Floor[(n+1)(m+1)/3]]}];
And finally, I generate the spirals inside each polygon:
g[x_]:=Fold[Append[#1,BSplineFunction[#1[[#2]],SplineDegree->1][d]]&,x,Partition[Range[150],2,1]]
ListPlot[g/@q,Joined->True,Axes->False,PlotStyle->Black,ImageSize->Large]
This approach has many flaws compared to the other answers but the most important one is that one has to execute the code many times to get a decent result (because most of the times the polygons overlap).
answered Apr 1, 2016 at 13:20
AccidentalFourierTransformAccidentalFourierTransform
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