A few of the many triply periodic minimal surfaces can be generated from a quadrilateral slice through a tetrahedron of mirrors, as refined by Surface Evolver. I had the idea that the same concept, applied to one of the analogous tetrahedra that tile spherical 3-space, could result in a pretty model.

To my disappointment, none of the non-prismatic kaleidoscopes — those that generate the six regular polychora — hosts a nondegenerate minimal surface (of this simple form); but each of the duoprism kaleidoscopes gives at least one.

(2018) One could try minimal surfaces with the constraint of partitioning into equal volumes, if one understood how to specify such a constraint in non-Euclidean space. Apparently it needs a metric tensor, whatever that is.

This one is a projection of a design on (half of) the surface of a hypersphere: a ribbon spans two orthogonal great circles, wrapping three times around one and five times around the other.

Kitty teeth did break it at one point, but with redundancy that’s not a serious problem. Having got my pix I’ll keep the new toys away from them henceforth.

Not only did it not fall apart in my hands …

The strand — for it is a Klein bottle woven of a single strand — is thinner than I expected, and has broken in at least two places. Ah well, live and learn; I’ve already redesigned it to be more robust.

I came up with this concept several years ago (I can’t tell exactly when, because I’ve mislaid the original code!) but never posted any of the images until now.

Some of you may like to guess what’s going on before I tell. For more hints, see these patterns’ 254 siblings.

One often hears:

[Libertarianism] can only work if all the conditions of a free market economy are present … things like anyone being able to easily enter any market segment, all consumers having near-perfect knowledge of what they are purchasing, large numbers of sellers selling identical products to large numbers of buyers, etc.

And one is moved to ask whether the political system makes up for such departures from the ideal by adding moral hazard.

Where did the meme came from? My guess is that some introductory economic textbooks contain theorems that rely on those simplifying assumptions, and some students get that far and no further.

I don’t know much about academic economics but I do know that there’s plenty of interest in the negations of those simplifications; for example, Ronald Coase made his name by pointing out the importance of transaction costs, including the cost of overcoming imperfections of knowledge.

What I need is the libertoonian equivalent of the TalkOrigins Archive, containing standard responses to the other side’s tired assertions.

I made some progress on an old project: to make outline fonts based on some favorite old bitmap fonts, by automatic fitting of smooth curves to the sequences of dots.

(The image above is in Scalable Vector Graphic format. If you see nothing, you may – dare I say it – need to update your browser. It works in Firefox 5, Safari 5, Chrome 12, Kindle.) ( . . more . . )