## The JVHM Open Book

The open book from the previous post wasn’t quite right. This became quite apparent when tried to color the two sides of a page. Along a “horizontal level” between each pair of binding components there needed to be three Dehn twists. Fortunately, the Dehn twists can be obtained through shearing.

A bit of reworking and rechoosing a fundamental domain gives a pleasant picture.

The three-torus may be viewed as a hexagonal prism with opposite faces identified. Here we show 10 pages of JVHM’s open book for the Stein fillable contact structure on the three-torus in this fundamental domain.

But that’s hard to see…. How about this:

Yeah, these are animated gifs.

After the jump you can see more animations with these prisms tiled. You can also check out a few stills on my Flickr page. Or you can download the SketchUp model directly.

Putting three together so you can see how the pages really tile:

And a few views of a big tiled block, 3x3x3:

Mesmerizing…

Word of caution, some of these are actually mirrors of what they should be.

This is one of the coolest things I’ve seen. Beautifully done.

Jesse Johnson said this on November 25, 2008 at 2:12 am |

I really love these open book animations. They connect very closely with some ideas about flows and surfaces I have been thinking about for a long time, and its the first time I’ve seen this sort of thing visualized.

Im curious about how you made them, are they intuitively ‘sculpted’ entirely in sketchup or is there scripting or other programs involved?

I write programs for experimenting with geometry (mainly Rhino3D and processing) – you can see some of my stuff at http://spacesymmetrystructure.wordpress.com/ and I’d love to have a go at scripting something like these.

Daniel said this on January 5, 2009 at 10:45 pm |

These were all sculpted in sketchup. (Done more or less by hand. The ruby script “curve stitcher” can be quite useful.) I knew what sort of object I was aiming at, guided by the topology. The symmetries present suggested that viewing the three-torus via a hexagonal prism rather than a cube would provide nice regular picture. It actually took a bit more massaging than expected to find this form. Of course in hindsight this one seems obvious.

A page is basically made from rotations and translates of one piece. For the different pages, the relative positions of these pieces are different. Maybe later I’ll do a related post that clarifies some of this construction.

Ken Baker said this on January 6, 2009 at 6:29 pm |

Oh, and to everyone else: Daniel has got some great animations on his blog of stereographic projections of a 3-sphere rotating with objects in it. You’ll see things like a horse before you, but then the 3-sphere rotates so that the point at infinity is inside the horse.

Fun other stuff there too.

Ken Baker said this on January 6, 2009 at 6:37 pm |

Thanks for the clarification about your process – you’ve obviously got very good geometric intuition to picture these things.

Do you know if anyone has studied other open books with different arrangements of straight lines as the bindings?

I imagine there must be some really interesting ones if you use the lines of a 3D weave ( like the green lines at the end of this video : http://vimeo.com/2553639 )

Daniel said this on January 6, 2009 at 9:17 pm |

Regarding straight lines as bindings, the natural spaces for such open books would be torus bundles and the familiar euclidean 3-space, R^3. For open books on torus bundles, Jeremy’s dissertation is the place to start. The open books he looks at have their bindings as curves on the torus fibers.

R^3 admits a greater variety of open books. Though we are forced to consider non-compact pages and non-compact bindings, that makes them interesting in their own right.

One way to get an open book in R^3 with straight lines is to take an open ball chunk of some other open book in which you get a trivial tangle formed from the ball and binding. Perhaps the simplest example has the z-axis as the binding with half-planes as pages.

Another way to get an open book in R^3 with straight lines is to take the, say, universal cover of a torus bundle containing an open book of Jeremy’s sort. That’s what the latter pictures above are suggesting.

There might be a way of taking other spacegroups (crystallographic groups) to produce open books with certain symmetries… (Though I reckon that’d mean you’d have to get some sort of open book structure on orbifolds.)

I wonder if it’s possible to find an aperiodic collection of lines that’s the binding of such a non-compact open book. Might be able to do that by doing a hopf plumbing and then straightening the resulting curves…

Ken Baker said this on January 15, 2009 at 3:49 pm |

Thanks for the reply. I’ll look at Jeremy’s paper some more. Its interesting stuff, though I’m in over my depth here. I’m an architect not a mathematician, and think about these things primarily in a visual/intuitive way. I am studying and trying to develop formal understanding as well, but it takes time. Thats why mathematical images and animations are so nice – they allow a glimpse of ideas beyond my level of formal understanding and make me want to understand more.

The images near the bottom of this page I found recently ( http://comp.uark.edu/~strauss/gyroid/index.htm ) are something like what I had in mind when I mentioned 3D weaves earlier. What I’m wondering is whether there is some book of which that surface is just one page.

Daniel said this on January 21, 2009 at 3:31 pm |

Ah yes, Goodman-Strauss, my mathematical brother. He’s got lots of great visuals in his work. For those who don’t know, he’s been doing a math radio show/podcast for the past five years: Math Factor

A fundamental difference between the gyroid surface and the pages of an open book is that the gyroid has no boundary whereas the binding of an open book is the boundary of each page. Without having given much thought to it, there might be a way of placing curves on the gyroid to chop it up into pages of an open book with those curves as the binding. The basic pieces used in the post might be usable to get something close to this.

Ken Baker said this on January 22, 2009 at 2:07 pm |

Please excuse my ignorance, but surely if there are pages going all the way around the binding, then you can consider each page as joined to the one on the opposite side, to give a surface without a boundary. So locally they would look like helicoids with the binding as the axis.

My latest post probably makes my interest in this a bit clearer

http://spacesymmetrystructure.wordpress.com/2009/02/06/rheotomic-surfaces/

Daniel said this on February 7, 2009 at 10:53 am |

[...] It was asked how one might see the contact Heegaard splitting associated to the JVHM open book on T3. Two pages of an open book form a Heegaard surface that is convex with respect to the induced [...]

Contact Heegaard Splittings « Sketches of Topology said this on September 27, 2009 at 6:18 pm |