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Staff Picks Mathematics Visual Arts Geometry Graphics and Visualization Wolfram Language

[GIF] The Great Mystery

Peter Karpov

Posted 7 years ago

The image consists of multiple layers of circles in a hexagonal grid arrangement. The layer scaling factors are integer powers of the golden ratio. Seamless infinite zoom is achieved by gradually blending in / out the fine / large scale layers. Draw function draws a single animation frame: v1 = {Cos[30 \[Degree]], Sin[30 \[Degree]]}; v2 = {Cos[90 \[Degree]], Sin[90 \[Degree]]}; n = 2; g ={ White, Table[ Circle[i v1 + j v2, 1/2], {i, -3 n, 3 n, 1/2}, {j, -3 n, 3 n, 1/2} ] }; m = 2; Draw[ds_] := Graphics[ {AbsoluteThickness[5/3], Table[ {Opacity[2/3 (m - Abs[i + ds])/m], Scale[g, GoldenRatio^(i + ds), {0, 0}]}, {i, -m, m} ] }, PlotRange -> {{-n, n}, {-n, n}}, Background -> Black, ImageSize -> 800 ];

The Great Mystery

The image consists of multiple layers of circles in a hexagonal grid arrangement. The layer scaling factors are integer powers of the golden ratio. Seamless infinite zoom is achieved by gradually blending in / out the fine / large scale layers. Draw function draws a single animation frame:

v1 = {Cos[30 \[Degree]], Sin[30 \[Degree]]};
v2 = {Cos[90 \[Degree]], Sin[90 \[Degree]]};
n = 2;
g ={
    White,
    Table[
       Circle[i v1 + j v2, 1/2],
       {i, -3 n, 3 n, 1/2}, {j, -3 n, 3 n, 1/2}
    ]
};
m = 2;
Draw[ds_] :=
    Graphics[
       {AbsoluteThickness[5/3],
       Table[
         {Opacity[2/3 (m - Abs[i + ds])/m], 
         Scale[g, GoldenRatio^(i + ds), {0, 0}]}, 
         {i, -m, m}
       ]
       },
       PlotRange -> {{-n, n}, {-n, n}},
       Background -> Black,
       ImageSize -> 800
    ];

POSTED BY: Peter Karpov

7 Replies

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Nural I.

Posted 7 years ago

Hey! Thats amazing! Yeah the hexagon represents an elementary part of our nature, you can find it almost everywhere, in the eyes of bees, even on saturn (the 6th Planet) ^^ and in all what is elementary for life like carbon, here a example from a carbon molecule, recorded with a atomic force microscope (atomic/scanning force microscope, AFM) in the IBM research center in zurich. https://www.weltderphysik.de/gebiet/teilchen/news/2012/chemische-bindungen-sichtbar-gemacht http://science.sciencemag.org/content/337/6100/1326

POSTED BY: Nural I.

Udo Krause

Udo Krause, NOrganization

Posted 7 years ago

Just for clarification: As it stands In[68]:= Grid[ Join[{{"ds", "black", "white"}}, Transpose[ Join[{Range[-5, 5]}, Map[Last, Transpose[ ImageLevels /@ (Binarize /@ (Image /@ (Draw /@ Range[-5, 5])))], {2}]]]]] Out[68]= ds black white -5 640000 0 -4 640000 0 -3 489362 150638 -2 458308 181692 -1 430486 209514 0 430486 209514 1 430486 209514 2 511558 128442 3 581104 58896 4 640000 0 5 640000 0 for `Abs[ds] > 3` the graphics from `Draw[ds]` is uniformly black. An infinite zoom cannot be directly observed. The pictures for $ds \in \{-1,1,1\}$ seem to agree visually. How did $ds$ vary in the GIF animation? Background: I'm interested in a working example of a fractal one can zoom out, with other words iterate backwards behind the starting set or with again other words having the starting set appearing if iterating from negative iteration numbers to iteration number 0. Then going forward, i.e. zoom in. Your example seemed to deliver it, but seems to need some more work to accomplish that.

POSTED BY: Udo Krause

Peter Karpov

Posted 7 years ago

Forgot to mention, ds varies from 0 to 1. It is the fractional part of the scale exponents, so when it reaches 1 we have the same set of scales and a perfect loop. If there was an infinite number of layers, we could change ds from -inf to +inf.