Is it possible to draw this figure using Mathematica?

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}]

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]))}]

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]}]]]