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Richard Palais responded with this image of breather pseudosphere with so few ribs as to be a Kuen surface and the text quoted below.

” Great Question ! !

” The breather surface has a parameter (called bb in the 3D-XplorMath version) that controls the “number” of ribs. Actually, for any choice of bb (in the allowed range 0 < bb < 1) there is really only a single rib that winds around the axis infinitely often while simultaneously winding around a circle in the plane perpendicular to the axis and centered on the axis; bb controls the relative rapidity with which the two windings occur. For certain values of bb, like 0.2, 0.4, 0.6, 0.8, the resulting "rib" is periodic, i.e., when it rotates once around the axis it returns to where it was, and in this case you get a nice clean surface. (Otherwise you get a mess unless you choose a relatively small parameter domain.) The version we use for the 3D-XplorMath icon (which you say has 22 ribs) corresponds to bb = 0.4.

" For some reason I had never looked at what happens when bb is close to 1, although it is natural to wonder about that. So when you asked your question, I put bb = 0.995, and almost jumped out of my seat---for what 3D-XplorMath put on my screen was clearly a piece of the Kuen Surface, and after I adjusted the size of the parameter rectangle appropriately it became the standard version of Kuen. I have named it "BreatherAsKuen". I have never seen it mentioned anywhere that Kuen is a "special case" of Breather, and a quick Google search didn't turn anything up, so perhaps this is a new observation. Anyway, thanks for asking thw question."

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Neil Bickford sent me a link to various versions of the pseudosphere as a short animation.

A video of the pseudosphere being wrapped with varying numbers of ribs arrived from Paul Nylander as well.

On the curvature of surfaces, Wikipedia has a great practical application example for Pizza

http://en.wikipedia.org/wiki/Theorema_egregium

I also looked on
http://demonstrations.wolfram.com

But didn’t see anything that was exactly as you describe above.
Do you have a link to an example on that site?

I’ve got Mathematica Player, so I can run the examples, …even if I
don’t fully understand them all.

Think of it this way. I draw a circle with a fixed radius r. If the circle is in a flat space, it has area pi times r-squared and the circumference is two pi r.

But if the circle is on a sphere, the area of the cap inside it is LESS than pi times r-squared. And the circumference is LESS than 2 pi r. For if you were to stretch the cap out flat, it would get bigger.

If you draw the circle on a saddle then its area is bigger than pr r-squared as the circle is bent two different ways. And its circumference is bigger than two pi r.

So if I have a given fixed radius to work with for my house, it seems I want the innards to be negatively curved.

This said, let’s consider your point about the Flatland igloo. You’ve left the circumference fixed, but you’ve stretched both the area AND the radius. So, I guess one could argue that ANY kind of curved space inside a house of a fixed outer size will add more room. After all, it’s the flat space that’s a “minimal surface.”

So let me reiterate that it’s NOT the case that positive curvature is the ONLY ONE that gives “more area” for a fixed circumference. Negative curvature will also give “more area” within a given fixed circumference.

I particularly like the dodecahedral tessellation of hyperbolic 3-space. And of course the Poincare disk rules.