Boundary Considerations, Part II

As promised, today we will look at a close cousin of last week’s surface. A good starting point is the CLP surface of Hermann Amandus Schwarz, about which I have written before.

 

Clp

Up above are four copies of a translational fundamental piece. There are horizontal straight lines meeting in a square pattern, vertical symmetry planes intersecting the squares diagonally, vertical lines through edge midpoints of the squares and horizontal symmetry planes half way between squares at different heights. What more could one want? Well, CLP has genus 3, and we wouldn’t mind another handle.

 

 

Nearsingly

 

There are various ways of doing that, and one of them leads to today’s surface, shown above. For adding a handle we had to sacrifice the vertical straight lines, but all other symmetries are retained. These are, in fact,  essentially the same symmetries we had in last week’s surface, except that there, the squares in consecutive layers were shifted against each other. The similarities go further.

Neardoubly

Again we can ask how things look at the boundary. Pushing the one free parameter the the other limit, gives us again doubly  periodic Scherk surfaces and Karcher-Scherk surfaces. There is a subtle difference (called a Dehn twist), however, how the two types of Scherk surfaces are attached to each other in both cases.

Flat

Finally, as usual, the cryptic rainbow polygons that encode everything. Today, the two fit together along their fractured edges, which has to do with the period condition these surfaces have to satisfy.

 

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