Bäcklund transforms of the pseudosphere: Intuitive description? Is it possible to describe what are all the Bäcklund transforms of the pseudosphere
at a high, intuitive level? The shapes in the aggregate set of all the transforms?
Typically one sees images of very pseudosphere-like solutions
,
but I was surprised by this image:

          


          

(Image from Tim Hoffmann:
"a Bäcklund transform of the ... pseudosphere.")


I admit I have not penetrated the Bobenko & Pinkall paper, which
might explain this image:


"Discrete Surfaces with Constant Negative Gaussian Curvature and the Hirota Equation" (J. Diff. Geom., 1996, 43, pages 527--611).
  (ProjectEuclid PDF download.)

 A: Bäcklund transforms are certain operations that allow the construction of solutions to certain very special (integrable) systems of PDEs; this much is explained on the wikipedia page you linked and there are some more details for the case of sine-Gordon / pseudospherical surfaces / K-surfaces in lots of places, including pages 5-9 of this short note "Geometry of Solitons" by Terng and Uhlenbeck. 
In the case of pseudospherical surfaces, the Bäcklund transforms are realizations of "line congruences" mapping the nets of asymptotic lines of one pseudospherical surface to another; for every pseudospherical surface, the Bäcklund transforms are parametrized by an angle $\theta$ and a unit tangent vector of the surface that is not in a principal direction. Bäcklund transforms have a very nice property called "Bianchi permutability": if we have two surfaces $M_1$ and $M_2$ which are both Bäcklund transformations of a starting surface $M_0$, then there exists a unique surface $M_{12}$ which is a Bäcklund transformation of both $M_1$ and $M_2$. 
The particular surface that you show is a particular discretization of what seems to be called "Kuen's surface" (mathworld,virtual math museum).
The Bobenko and Pinkall paper is primarily about this particularly nice ("discrete integrable") discretization of pseudospherical surfaces. While Bäcklund transformations were originally discovered for smooth surfaces, it turns out that the discrete perspective also sheds some light on their existence. Bobenko and Suris have written a nice book "Discrete Differential Geometry" which explains the following attractive viewpoint on Bäcklund transformations of surfaces (see the introduction of either the book or this less complete pre-publication version on the arxiv or these lectures of Bobenko, published in Lect. Notes Phys. 644, 85-110 (2004)):

Imagine first that we create a sequence of surfaces $M_{j}$ by performing repeated Bäcklund transformations on a surface $M_{0}$. (The picture on the right shows two consecutive surfaces in this sequence). Note that each point on $M_j$ is related to a point on $M_{j-1}$ and $M_{j+1}$ so we 
 can think of this sequence as a "semi-discrete" 3-dimensional space, where one dimension is discrete and two dimensions are continuous. 
It turns out that there is a much more symmetrical picture if we replace these smooth surfaces by the discretizations in Bobenko and Pinkall's paper (see also 4.2 of Bobenko and Suris's book (2.5 in the arxiv version) or the introduction to Bobenko's lectures). Then we instead have a 3D grid of points (see left side of the above figure), and amazingly the equation that 4 points around a "quad" within one of the discrete surfaces satisfies is identical to the equation that 4 "neighboring" points satisfy if two of them are neighboring points in $M_j$ and the other two are the corresponding points on $M_{j+1}$ (or $M_{j-1}$)! (See one of the "transverse" quads in the figure above.) [These equations are not so complicated, they amount to saying that opposite edges of the quads must have the same length; this, of course, is reminiscent of a Chebyshev net]. Thus we can think of the sequence of pseudospherical surfaces above as one of these 3D grids where we've taken a "continuum limit" in two directions; the "Bäcklund" direction is the one which remained discrete.
This picture extends to higher dimensions too. Using Bianchi permutability, one can imagine defining an infinite "2D grid" of smooth pseudospherical surfaces $M_{j,k}$ all related to a starting one by a sequence of Bäcklund transformations. In the discrete case, this comes from a 4D grid of points all satisfying the same equations around quads. (In Bobenko's lectures linked above, he actually derives the discretization of K-surfaces from geometrical properties satisfied by points in such a Bianchi grid).
