Boy’s Surface

One of the things knocking around in my noggin has been sphere eversions, especially ever since Carter showed me his drawings at LSU. Of course there’s Thurston’s corrugation. And Morin’s surface.

But another method is to pass through a immersion of the projective plane, a non-orientable closed compact surface which the sphere double covers. So this man named Boy came up with a smooth immersion of the projective plane when Hilbert assigned the homework to show it can’t be done. Here are a few pictures of my attempts to make a model of it.

Picture 2

Picture 4

Picture 6

You might notice the 3-fold symmetry. You might compare that with the 4-fold symmetry of Morin’s surface.

There are many approaches to Boy’s surface. Kirby describes an interesting method (pdf). Let’s see how that works.

Start with the octahedron, but punch in alternating faces. Or equivalently take unit diamonds in the coordinate planes and add four triangles in a checkerboard fashion.
Picture 10

Picture 9

This is an immersion of the projective plane. The singularity in the center is the standard triple transverse intersection.

Picture 11 Picture 13

It’s the six corners that cause problems. Topologically, they’re cones on a figure eight curve. And they’re not amenable to smoothing.

Picture 14-2 Picture 14

But if you take these corners in pairs and pull them together….

Picture 15

Picture 18

…you can then undo the badness with a little roll…

Picture 22

Picture 23

Picture 21

At this point every edge can be smoothed. These pictures don’t show it off as well as I would like. Oh well.

One last one for Scott: Boy’s Coffee Cup.

How Surfaces Intersect in Space: An Introduction to Topology

Boy's coffee cup

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~ by Ken Baker on June 17, 2008.

8 Responses to “Boy’s Surface”

  1. [...] I don’t think the reconciling of the non-smoothable cone points in that last post was terribly plain to see. Perhaps if you could manipulate the models yourself, it would be more [...]

  2. John Hughes did a cool vid on Boy’s surface right around the time of Apery’s parametrization. I am trying to finish a pdf version of the first complete rough draft of the eversion. I promised the first view to John Armstrong. My problem this afternoon was my student, Sarah, wrote her thesis in Word (to fulfill grad school regs, and because figure placement in LaTeX is such a bitch. So she had compiled movies, decker sets, and the charts I showed you, page by page. The file with pictures only winds up being a 1/4 gig. I was loading it into Open office to get a pdf version, but it was loading slow.

    Anyway, with luck, I have this draft on Unappologetic tomorrow.

  3. Really?!?! Unless I’m overlooking it, maybe this movie about Boy’s surface should be included on his IMDB page.

  4. Right: Different John Hughes. The one of which I speak is a CS Prof at Brown. I think some of his students did the fabric algorithms for
    Monster’s inc. I have a VCR somewhere in my house of this.

    Thanks for the coffee cup!

  5. Here’s a link to Armstrong’s debut of The Carter-Gelsinger Eversion. When I get more time, we’ll revisit this to show off a few other neat tricks that programs like Rhino3D can do.

  6. I have posted some material on Boy’s surface on my website http://www.maths.ed.ac.uk/~aar/surgery/notes.htm
    (Scroll down to almost the end)

  7. You describe the heptahedral projective plane (H)– the first step in Kirby’s construction of Boy’s surface (B) — as an “immersion” of the projective plane (P^2).

    But although this indeed depicts P^2, it is not an immersion (as Boy’s surface in fact is).

    The problem is those 6 points of H at the apices of the crosscaps. Any continuous mapping

    P^2 -> H

    cannot be locally one-to-one at any of the pre-images of those 6 points, and so
    H can’t be the image of an immersion.

    Sincerely,

    M.P.

    • Thanks for your comment.

      While I do describe the first step in Kirby’s construction (in maroon), it looks like I address the issue you mention and go on to describe how to perturb it all to an immersion. But it’s been a while since I’ve really looked at this… perhaps I’m still missing something?

      For other readers, this heptahedral projective plane is also called a Tetrahemihexahedron

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