Modeling Phyllotaxis
The golden angle is the angle that divides a full angle in two parts as given by the golden ratio.
Exact value of the golden angle.
In[1]:=
FunctionExpand[GoldenAngle]
Out[1]=
Approximate value in radians and degrees.
In[2]:=
N@{GoldenAngle, GoldenAngle/Degree}
Out[2]=
The golden angle can be used to model phyllotaxis patterns.
Out[3]=
Define a phyllotaxis function using GoldenAngle.
In[4]:=
phyllotaxis[step_] :=
Table[Sqrt[i] AngleVector[i GoldenAngle], {i, 0, 1000, step}]
Use Graphics to generate a phyllotaxis pattern from these points.
In[5]:=
Graphics[{PointSize[Medium], Point[phyllotaxis[1]]}]
Out[5]=
Highlight spirals of different steps by using different colors.
In[6]:=
Graphics[{PointSize[Medium], RGBColor[
0.2980392156862745, 0.28627450980392155`, 0.5490196078431373],
Point[phyllotaxis[1]], GrayLevel[1], Point[phyllotaxis[5]]}]
Out[6]=
In[7]:=
Graphics[{PointSize[Medium], LCHColor[
Rational[1, 2],
Rational[1, 2],
Rational[1, 3]], Point[phyllotaxis[1]], CMYKColor[0, 0, 1, 0.05],
Point[phyllotaxis[7]]}]
Out[7]=
Phyllotaxis patterns can also be found in sunflower heads.
In[8]:=
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"], {{0, 200}, {300, 0}}, {0, 255},
ColorFunction->RGBColor],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSize->Automatic,
ImageSizeRaw->{300, 200},
PlotRange->{{0, 300}, {0, 200}}]\)
Define a motif.
In[9]:=
motif = FilledCurve@
BSplineCurve[4 {{0, -.75}, {0, .75}, {1, 0}, {1, 0}},
SplineClosed -> True];
Visualize it.
In[10]:=
Graphics[motif]
Out[10]=
Construct a sunflower head from it.
In[11]:=
Graphics[{EdgeForm[
GrayLevel[1]],
Table[Rotate[Translate[motif, {Sqrt[i], 0}],
i GoldenAngle, {0, 0}], {i, 100, 0, -1}]}]
Out[11]=