[GIF] Circle - Gecko - triangular tiling transformation inspired by Escher

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Inspired by the work of the Frisian artist M.C. Escher, I decided to make this little animation:

geckotransform.gif

The code is nothing more than linear interpolation between sets of points:

SetDirectory[NotebookDirectory[]];

p1 = {{0.`,0.`},{0.0678`,0.054200000000000005`},{0.1336`,0.09570000000000001`},{0.1831`,0.1257`},{0.2398`,0.1714`},{0.26780000000000004`,0.20850000000000002`},{0.2528`,0.2606`},{0.22760000000000002`,0.3084`},{0.2117`,0.3584`},{0.21930000000000002`,0.41000000000000003`},{0.24550000000000002`,0.4595`},{0.28500000000000003`,0.5056`},{0.34`,0.48260000000000003`},{0.3935`,0.45320000000000005`},{0.4305`,0.43760000000000004`},{0.43820000000000003`,0.39840000000000003`},{0.4303`,0.3698`},{0.3831`,0.3678`},{0.3552`,0.3683`},{0.3925`,0.33180000000000004`},{0.4148`,0.2927`},{0.4339`,0.2671`},{0.49720000000000003`,0.2947`},{0.5356000000000001`,0.33380000000000004`},{0.5789000000000001`,0.3659`},{0.558`,0.4297`},{0.5141`,0.48090000000000005`},{0.5`,0.5`},{0.5`,0.5`},{0.4859`,0.5191`},{0.442`,0.5703`},{0.42110000000000003`,0.6341`},{0.46440000000000003`,0.6662`},{0.5028`,0.7053`},{0.5661`,0.7329`},{0.5852`,0.7073`},{0.6075`,0.6682`},{0.6448`,0.6317`},{0.6169`,0.6322`},{0.5697`,0.6302`},{0.5618000000000001`,0.6016`},{0.5695`,0.5624`},{0.6065`,0.5468000000000001`},{0.66`,0.5174`},{0.7150000000000001`,0.4944`},{0.7545000000000001`,0.5405`},{0.7807000000000001`,0.5900000000000001`},{0.7883`,0.6416000000000001`},{0.7724000000000001`,0.6916`},{0.7472000000000001`,0.7394000000000001`},{0.7322000000000001`,0.7915000000000001`},{0.7602`,0.8286`},{0.8169000000000001`,0.8743000000000001`},{0.8664000000000001`,0.9043`},{0.9322`,0.9458000000000001`},{1.`,1.`}};
p2 = {{1.`,1.`},{1.0396177978506647`,0.8923346254845568`},{1.0553148607198288`,0.8165562085782169`},{1.0612803330660763`,0.7422415758850744`},{1.0593972739777413`,0.6855070651494309`},{1.0410889377634256`,0.6295007686706042`},{0.9985803499841852`,0.5851973901977947`},{0.9483041434655642`,0.5486542747648014`},{0.9100397943346402`,0.495994821587507`},{0.8886126415052703`,0.4220994637695018`},{0.8802712781999131`,0.3410911444732952`},{0.9107680085914569`,0.2922860778674355`},{0.944118750413758`,0.23431580960507237`},{0.9891650545425124`,0.1782359561306078`},{1.0249726000191246`,0.2133888443460414`},{1.0610964405769812`,0.25639761969562114`},{1.0125267563571634`,0.29612575303974287`},{0.9819491132704178`,0.3427388211755879`},{1.042685124568772`,0.3779726220862235`},{1.0923875865214163`,0.4314412022155367`},{1.110695922735732`,0.3701240906516413`},{1.139236037043303`,0.307894872341834`},{1.1672759637805352`,0.24281899830083345`},{1.190159545123539`,0.19293999955865795`},{1.1550875696033072`,0.15058588147025714`},{1.1048113630846865`,0.1033990687684352`},{1.0520783529117537`,0.05681542343084531`},{1.`,0.`}};
rf = RotationTransform[\[Pi]/2, {1, 0}];
p3 = Reverse[rf /@ p2];

colors = {RGBColor[0.9280877328700329, 0.8058790727091572, 0.41541817087124444`],RGBColor[0.5551256603319519, 0.6745729914926235, 0.40725444158653856`]};

ClearAll[GetLines, MakeScene]
GetLines[\[Beta]_] := 
 Module[{\[Alpha], goal1, goal2, goal3, goal, lenp},
  If[0 <= \[Beta] <= 0.5,
   \[Alpha] = 2 \[Beta];
   
   lenp = Length[p1] + Length[p2] + Length[p3];
   goal = CirclePoints[{0.66, 0.33}, {0.33, 3.97}, lenp];
   
   {goal1, goal2, goal3} = FoldPairList[TakeDrop, goal, (Length /@ {p3, p2, p1})][[{3, 2, 1}]];
   Polygon[Join @@ {\[Alpha] p1 + (1 - \[Alpha]) Reverse[
         goal1], \[Alpha] p2 + (1 - \[Alpha]) Reverse[
         goal2], \[Alpha] p3 + (1 - \[Alpha]) Reverse[ goal3]}]
   ,
   \[Alpha] = 2 (\[Beta] - 0.5);
   goal1 = Subdivide[0, 1, Length[p1] - 1];
   goal1 = {goal1, goal1}\[Transpose];
   
   goal2 = Subdivide[1, 0, Length[p2] - 1];
   goal2 = Thread[{1, goal2}];
   
   goal3 = Subdivide[1, 0, Length[p3] - 1];
   goal3 = Thread[{goal3, 0}];
   Polygon[Join @@ {(1 - \[Alpha]) p1 + \[Alpha] goal1, (1 - \[Alpha]) p2 + \
\[Alpha] goal2, (1 - \[Alpha]) p3 + \[Alpha] goal3}]
   ]
  ]
MakeScene[\[Alpha]_] := Module[{in, shape},
  in = GetLines[\[Alpha]];
  shape = {in, Rotate[in, \[Pi], {0.5, 0.5}]};
  shape = Riffle[colors, shape];
  shape = Rotate[shape, #, {0, 0}] & /@ Range[0, 3 \[Pi]/2, \[Pi]/2];
  shape = Translate[shape, Tuples[{-2, 0, 2}, 2]];
  shape
  ]

To animate it using manipulate use:

Manipulate[Graphics[MakeScene[\[Tau]], PlotRange -> 2.5], {\[Tau], 0, 1}]

And to output the animation I used:

n=150;
ClearAll[Nonlineartime]
Nonlineartime[t_]:=0.5LogisticSigmoid[25(t-0.2)]+0.5LogisticSigmoid[25(t-0.75)]
Plot[Nonlineartime[t],{t,0,1}]
ts=Table[Nonlineartime[t],{t,Subdivide[0.0,1,n]}];
ts[[{1,-1}]]={0.0,1.0};
imgs=Table[Rasterize[Graphics[MakeScene[t],PlotRange->2.5,ImageSize->400],"Image"],{t,ts}];
Export["geckotransform.gif",imgs~Join~Reverse[imgs],"DisplayDurations"->0.03]
17 Likes
2

Spectacular! Now I finally understand what process Escher was using to make these tilings. Thank you!

2 Likes
3

2015-11-02_10-07-11.png

  • another post of yours has been selected for the Staff Picks group, congratulations !

We are happy to see you at the top of the “Featured Contributor” board. Thank you for your wonderful contributions, and please keep them coming!

2 Likes
4

Glad you like it!

5

Very nice! The simplicity of the idea makes it all the more appealing.

2 Likes
6

As others mentioned it is great and instructional. It look like an XKCD version too…

1 Like
7

Hi Sander,

Very cool !

I play a little bit your code, export a image format.

Escher_02.png
Escher_02.png3024Ă—360 198 KB

ImageRotate[ImageAssemble[Partition[Image@Table[Graphics[MakeScene[\[Tau]],PlotRange->{{-2.5,2.5},{0,2}}],{\[Tau],0,1,.05}],1]],Pi/2]

It looks more like Escher’s Metamorphosis style in his time ( without internet and GIF).

4 Likes
8

That is really neat! Thanks for sharing! Now I feel I should do more of these kind of transformations!

9

I think about two things we can explore for Escher in WL.

  1. Micheal Trott blogs a image transform also. For example, some points and lines on lizard need to be appear and disappear.
    we can use his method, probably.

2.Escher’s Tessellation style, see wiki.
Escher’s design method, see this link GIF.
I think Escher start from a simple square, then base on some rules he get a lizard profile, that can be made for tessellation.

3 Likes
10

nice ! +1

1 Like