Is it possible to draw this figure using Mathematica?

Question

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.

Answer 1

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}]

Answer 2

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]))}]

Answer 3

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]}]]]

Answer 4

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).