Whorled plant

The code below creates one leaf.

In[16]:=

  Clear[width,spine,ripple,normalize,tangent,normal,
    binormal]
  Needs["Calculus`VectorAnalysis`"];
  width[z_] = 0.1 z (1-z);
  spine[z_] = {0, z^2, z};
  normalize[v_] = v/Sqrt[v.v];
  tangent[z_] = normalize[spine'[z]];
  binormal[z_] =
    normalize[CrossProduct[tangent[z], {0,1,0}]];
  normal[z_] = CrossProduct[tangent[z], binormal[z]];
  leaf =
    ParametricPlot3D[
      spine[z] + y width[z] normal[z],
    {y,-1,1}, {z,0,1},
    PlotPoints -> {2,30}, ViewPoint -> {2,1,0}];

This leaf must be rotated about the z axis to create a spider-plant type plant.

In[17]:=

  spider =
    Table[Rotate[leaf,theta,{0,0,1}], {theta,0,7Pi/4,Pi/4}];

In[18]:=

  Show[spider];

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