Rotating the Axial Direction

For full generality in generating superposed foldings, as noted in Thiessen and Means (1980), it is necessary to allow the axis of the second folding to be at any angle to that of the first, as well as allowing initial layers at any orientation. One way to do this in Mathematica is to build a function using the RotationMatrix3D command. However, the standard Geometry`Rotations` package must first be loaded.

<<Geometry`Rotations`;
rot[a_:0,b_:0,c_:0] := 
  ((#).N[RotationMatrix3D[a Degree, b Degree, c Degree]])&

Side remark:The parameters $a$, $b$ and $c$ for RotationMatrix3D are ``Euler angles'' for the required rotation, measured in degrees. This means first an anti-clockwise rotation of angle $c$ about the $z$-axis, followed by an anti-clockwise rotation of angle $b$ about the resulting $x$-axis, followed by an anti-clockwise rotation of angle $a$ about the resulting $z$-axis. Any desired rotation can be obtained this way.

Figure 20: Superposed folding intermediate between Ramsay types 1 and 2, with the second folding at an angle of $45^{\circ }$ to the flow direction of the first.

Show[GraphicsArray[{FoldPlot[{x,y,0}//axiZY[.7,1.5]//rot[0,45,0]
   //axiZX[.5,1.3]//rot[0,-45,0]
     ,{x,-1.5,1.5},{y,-1.5,1.5},PlotPoints->60]
 , FoldPlot[{x,y,-y}//rot[0,45,0]//axiZX[.5,1.3]//rot[0,-45,0]
     ,{x,-1,1},{y,-.65,.65}] }]]

Click on a frame for a larger-sized image.

Figure 20 gives an example of a refolding intermediate between types 1 and 2. Note how two instances of $\mathbf{rot}_{[a,b,c]}$ are needed. The first rotates the coordinate system so that the desired direction becomes the $z$-axis, to use $\mathbf{axiZX}$. After the deformation we must rotate back again, to recover the correct orientation. By choosing the Euler angles appropriately, any of the axial deformation functions $\mathbf{axiXY}$, $\mathbf{axiYZ}$, etc. could be rotated into any other. We choose not to do this here, because it offers no gain in visualising how the foldings superpose.