# Wolfram Language™

## 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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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]=