When I was little, friends gave me as a birthday present a home made flip book that would show the deformation of the catenoid to the helicoid. That was a lot of work back then when you had to program all the 3D-graphics by hand, including hidden line algorithms. But I liked it to a have a physical object that would allow me to run my own little movie.
Daumenkinos – Thumb Theaters, are they called in German. Today we see some snapshots from a high tec version of such a Daumenkino, attempting to get to the core of Boy’s surface (an immersion of the projective plane), about which I have written briefly before.
The goal is to put a lid on a Möbius strip. The one we start with you will note is not just once twisted, but three times. I don’t know how essential this is to get an immersed projective plane at the end. I suppose it’s not, but makes things easier. Note that the strip has a single boundary curve, as expected.
The first two images show that Möbius strip, growing slowly. Below the first crucial step has happened: The growing strip has created a triple point, and intersection like that of three planes. But there still is only one boundary curve…
We keep growing
Another critical event: The boundary curve emerges completely into free air, i.e. doesn’t pierce through the surface anymore. Now it’s easy to close the lid:
This image (a variation of which I used for many years as a desktop background) is a close-up of the large sculpture below that can be seen at the Mathematical Research Institute in Oberwolfach.
It is a model of the projective plane, a construct that simultaneously extends the Euclidean plane and describes the set of lines through a fixed point in space.
The simplest way to make your own model is via the tetrahemihexahedron,
a polyhedron that seems to take every other triangle from the octahedron and twelve right isosceles triangles to close the gaps left by the removed four equilateral triangles. That, however, is not the only way to look at it. These right isosceles triangles fit together to form three squares that intersect at the center of the former octahedron, in what is called a triple point.
So we truly have a polyhedron with four equilateral triangles and three squares as faces which can be unfolded like so
where arrows and equal letters indicate to glue. From this flattened version we recognize a (topological disk) with opposite points identified, which is yet another abstract model of the projective plane. The tetrahemihexahedron suffers not only under the triple point at the center, but also under six pinch point singularities at the vertices. Maybe it was this model that made Hilbert think that an immersion of the projective plane into Euclidean space was impossible, and having his student Werner Boy work on a proof. Instead, Boy came up in 1901 with an ingenious construction of such an immersion, which has an elegant connection to minimal surfaces.
Robert Kusner constructed a minimal immersion of the thrice punctured projective plane into space, with three planar ends, that you can see above. Applying an inversion, as suggested by Robert Bryant, produces images that are very close to what Boy had in mind.
This explicit parametrization served as the basis for the model in Oberwolfach.