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Алгебра и теория чисел

Моделирование филлотаксиса

Угол золотого сечения - это угол, который делит полный угол на две части, заданные золотым сечением.

Точное значение угла золотого сечения.

In[1]:=
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FunctionExpand[GoldenAngle]
Out[1]=

Приблизительное значение в радианах и градусах.

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N@{GoldenAngle, GoldenAngle/Degree}
Out[2]=

Золотой угол может использоваться для моделирования формы филлотаксис.

Out[3]=

Задать функцию филлотаксис, используя GoldenAngle.

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phyllotaxis[step_] := Table[Sqrt[i] AngleVector[i GoldenAngle], {i, 0, 1000, step}]

Используйте Graphics для генерирования формы филлотаксис из этих точек.

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Graphics[{PointSize[Medium], Point[phyllotaxis[1]]}]
Out[5]=

Выделить спирали различных секций, используя разные цвета.

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Graphics[{PointSize[Medium], RGBColor[ 0.2980392156862745, 0.28627450980392155`, 0.5490196078431373], Point[phyllotaxis[1]], GrayLevel[1], Point[phyllotaxis[5]]}]
Out[6]=
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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]=

Форму филлотаксис также можно увидеть в головке подсолнуха.

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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}}]\)

Определить повторяющийся элемент формы.

In[9]:=
Click for copyable input
motif = FilledCurve@ BSplineCurve[4 {{0, -.75}, {0, .75}, {1, 0}, {1, 0}}, SplineClosed -> True];

Визуализировать его.

In[10]:=
Click for copyable input
Graphics[motif]
Out[10]=

Создaть из него головку подсолнуха.

In[11]:=
Click for copyable input
Graphics[{EdgeForm[ GrayLevel[1]], Table[Rotate[Translate[motif, {Sqrt[i], 0}], i GoldenAngle, {0, 0}], {i, 100, 0, -1}]}]
Out[11]=

Родственные примеры

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