By varying angles and edge lengths of the fundamental piece, one can repeat last week’s construction of the 6-ended Fischer-Koch surfaces and make surfaces with more ends. Above and below you see images of the 10-ended version, together with their twisted friends.
Again one obtains parking garage structures as limits, and the position of the helicoidal axes is indicated below: The case k=3 corresponds to last week’s surfaces, the case k=5 to the ones above. The colored disks represent helicoidal axes, with the color showing the different spins.
What about the case k=4, which should lead to 8-ended surfaces? Daniel Freese has shown that one can untwist the parking garage structures to screw motion invariant minimal surfaces:
and
But these surfaces can not be obtained using the Fischer-Koch construction. Below you see the completely untwisted version with annular ends.
One key difference to the Fischer-Koch surfaces is that opposing ends have opposite normals in Daniel’s surfaces (or differently colored sides, as visible above). If the vertical line was really a straight line, it would be a rotational symmetry line, and opposing ends had the same color.
So things are not always quite what they seem.
The Inner Frame 