Monday, 30 September 2013

Nice pattern obtained from rolling a small square inside a larger one


Nice pattern obtained from rolling a small square inside a larger one

Image via Dribbble
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6 comments:

  1. Math Doubts30 September 2013 at 02:59

    Fantastic pattern and Hats off to the creator of this pic.

    ReplyDelete
  2. Asif Ahmed30 September 2013 at 04:38

    what code or tool was used to generate this?

    ReplyDelete
  3. Corina Marinescu30 September 2013 at 04:42

    See the code below Asif Ahmed 
    1.
    RollSub[p_, dir_, \[Theta]_] := Module[{sq, rp},
    2.
       sq = {p, p + dir, p + dir + RotationMatrix[Pi/2].dir,
    3.
         p + RotationMatrix[Pi/2].dir};
    4.
       sq = Map[RotationMatrix[-\[Theta]].(# - (p + dir)) + (p + dir) &,
    5.
         sq];
    6.
       sq = Map[# &, sq]
    7.
       ];
    8.
    Roll[p_, dir_, \[Theta]_] :=
    9.
      Module[{n = Floor[2 \[Theta]/Pi], sq0, sq},
    10.
       sq0 = RollSub[p, dir, 0];
    11.
       sq = Nest[RotateLeft[RollSub[#[[1]], dir, Pi/2]] &, sq0, n];
    12.
       RollSub[sq[[1]], dir, \[Theta] - n Pi/2]
    13.
       ];
    14.
    RollAround[\[Theta]_] := Module[{p, dir, th, n, m},
    15.
      n = Mod[Floor[\[Theta]/Pi], 4] + 1;
    16.
      p = {{0, 0}, {3, 0}, {3, 3}, {0, 3}}[[n]];
    17.
      dir = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}[[n]];
    18.
      th = \[Theta] - Pi (n - 1);
    19.
      m = Mod[Floor[2 \[Theta]/Pi], 4] + 1;
    20.
      Nest[RotateRight, Roll[p, dir, th],
    21.
       1/4 (-3 - (-1)^m + 6 m) + 2 Floor[(n + 1)/2]]
    22.
      ];
    23.
    Manipulate[
    24.
     Show[Graphics[{
    25.
        EdgeForm[Directive[Thick, RGBColor[0.1, 0.2, 0.1]]], FaceForm[],
    26.
        Rectangle[{0, 0}, {3, 3}],
    27.
        EdgeForm[Thick],
    28.
        Polygon[RollAround[Mod[\[Theta], 4 Pi]]]
    29.
        }, PlotRange -> {{-0.1, 3.1}, {-0.1, 3.1}}],
    30.
     
    31.
      ParametricPlot[
    32.
       RollAround[t], {t, If[\[Theta] < 4 Pi, 0, Mod[\[Theta], 4 Pi]],
    33.
        If[\[Theta] < 4 Pi, Max[\[Theta], 0.01], 4 Pi - 0.01]},
    34.
       PlotStyle -> RGBColor[0.450385`, 0.157961`, 0.217975`]]
    35.
      , ImageSize -> 320]

    36.
     , {\[Theta], 0, 8 Pi - 0.01}]


    Raw data:

    RollSub[p_, dir_, \[Theta]_] := Module[{sq, rp},
       sq = {p, p + dir, p + dir + RotationMatrix[Pi/2].dir,
         p + RotationMatrix[Pi/2].dir};
       sq = Map[RotationMatrix[-\[Theta]].(# - (p + dir)) + (p + dir) &,
         sq];
       sq = Map[# &, sq]
       ];
    Roll[p_, dir_, \[Theta]_] :=
      Module[{n = Floor[2 \[Theta]/Pi], sq0, sq},
       sq0 = RollSub[p, dir, 0];
       sq = Nest[RotateLeft[RollSub[#[[1]], dir, Pi/2]] &, sq0, n];
       RollSub[sq[[1]], dir, \[Theta] - n Pi/2]
       ];
    RollAround[\[Theta]_] := Module[{p, dir, th, n, m},
      n = Mod[Floor[\[Theta]/Pi], 4] + 1;
      p = {{0, 0}, {3, 0}, {3, 3}, {0, 3}}[[n]];
      dir = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}[[n]];
      th = \[Theta] - Pi (n - 1);
      m = Mod[Floor[2 \[Theta]/Pi], 4] + 1;
      Nest[RotateRight, Roll[p, dir, th],
       1/4 (-3 - (-1)^m + 6 m) + 2 Floor[(n + 1)/2]]
      ];
    Manipulate[
     Show[Graphics[{
        EdgeForm[Directive[Thick, RGBColor[0.1, 0.2, 0.1]]], FaceForm[],
        Rectangle[{0, 0}, {3, 3}],
        EdgeForm[Thick],
        Polygon[RollAround[Mod[\[Theta], 4 Pi]]]
        }, PlotRange -> {{-0.1, 3.1}, {-0.1, 3.1}}],
     
      ParametricPlot[
       RollAround[t], {t, If[\[Theta] < 4 Pi, 0, Mod[\[Theta], 4 Pi]],
        If[\[Theta] < 4 Pi, Max[\[Theta], 0.01], 4 Pi - 0.01]},
       PlotStyle -> RGBColor[0.450385`, 0.157961`, 0.217975`]]
      , ImageSize -> 320]
     , {\[Theta], 0, 8 Pi - 0.01}]

    ReplyDelete
  4. Corina Marinescu30 September 2013 at 04:56

    Matlab is fun =)

    ReplyDelete
  5. Jeremiah Cullison30 September 2013 at 04:57

    Objects are so kool.

    ReplyDelete
  6. Kelvin Mead30 September 2013 at 15:28

    Welcome to the Spirograph of the 80s

    ReplyDelete

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