If we double the tangle with the red filling (that is, glue it to its reflection across the outer sphere) we obtain the two component unlink.

Getting the unlink from doubling is due to these filled tangles being rational tangles, but nevermind that for now. Instead, ponder this curiosity. Let’s keep the mirrored red filling up top and compare the blue and green down below.

They’re the same except the small change between the blue and green filling. (Taken together, the blue and green arcs bound a twisted rectangle, a “band”). But they’re even closer in another way. Mirror, then rotate:

Mirror the blue guy across the horizontal plane, then rotate about the central vertical axis by a third. And now it’s the same as the green guy.

This generalizes quite nicely.

You might regard the above example as nearly 3-fold. Let’s do 5-fold.

And 7-fold

Note that the red up top got spread apart passing from the 3-fold to the more-fold.

And of course the little twists could be replaced with other tangles.

The (double branched cover version of the) 3-fold version was noted by Bleiler-Hodgson-Weeks in Cosmetic Surgery on Knots. Kazuhiro Ichihara and In Dae Jong recently showed in Cosmetic Banding on Knots and Links that (the double branched covers of) these more-fold generalizations can give counterexamples to one of BHW’s conjectures.

All these examples are bandings from one mess of knotted loops to its mirror image. Is mirroring necessary? (Yep, there are some cheap shots. Anything else?)

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I’d brought along some of the models of mine and others that I’d printed. As you may pick up from the audio, I was tossing them out into the audience. It was all fun and games until someone lost an eye. (The one from that last post has pointy corners.)

My sincerest apologies if I either misrepresented, misattributed, or didn’t mention someone’s work. I don’t have complete knowledge of this blooming field and surely got some things wrong and overlooked some other things. **So by all means, please contribute corrections, updates, or further information in the comments.**

PechaKucha (wikipedia) is a presentation series where the speakers have 20 slides and 2o seconds per slide. Since a PK audience typically has an art/design/architecture bend, I surveyed how 3D printing is being used to represent mathematical ideas. Roughly, the talk progresses from literal representations of mathematics to more abstract and artistic representations. Of course this focused on 3d printed models, but the story easily expands into all sorts of art forms.

Anyways, below is a rough script of what I was at least planning to say together with the slides. If you watch the video, you’ll note that at times I totally strayed far from script. Also I’ve tried to give relevant links for the pictures or models.

1. Thank you for the opportunity to share with you a variety of ways 3D Printing is being used in Mathematics by myself and others. These examples progress through several styles and subjects from more clearly literal prints of math to more suggestive or impressionistic prints.

2. What are we doing with 3D Printing? There’s an idea we give the name CIRCLE, for which we can write an equation, and then obtain an explicit plot. These representations are increasingly more visceral, tangible, physical.

3. In the mid-late 1800’s lots of European Mathematicians –like Clebsch here- were into Algebraic Equations that describe Curves and Surfaces. They could do stuff with equations and they could also draw pictures to explore properties and relationships. But no computers to automagically make pretty pictures.

4. So people would make models out of gypsum and plaster to better communicate. Schilling was quite a prolific sculptor; check out his catalogue from 1903. Some collections of these can be found in museums and math departments.

(*Catalogue Mathematischer Modelle at Archive.org, and I’m not sure where I got this image of the paster model from… but it’s in Angela Vierling-Claassen’s Bridges 2010 article*)

5. These classic algebraic surfaces and their plaster models became some of the earlier mathematical subjects of 3D printing. Two design teams/people made these. They’re using a 3D printer to “plot” a surface; it’s a direct representation of the equation defining the surface.

(*mo-labs.com, formpig.com, Universal Joint on Shapeways*)

6. Graphs of functions are natural steps in printing. You can show how the xy-coordinate system parametrizes the surface, or show the contour lines – notice how the shadows convey yet more information.

7. Basic educational models of the quadratic surfaces can easily (and relatively cheaply) be designed and printed. They are “literal” plots of equations. Touching helps convey senses of curvature of surfaces. These were by mathematicians. Henry Segerman is one of the more prolific and inventive mathematician 3D printing designers out there.

(*segerman.org, Calculus Surfaces on Shapeways*)

8. Moving a small step away from the literal plots we’ve been looking at is this representation of the Cubic Nodal Singularity that I designed with colleague Drew Armstrong. Rather than in the xy plane, the geometry of the curve is emphasized by completing it to the sphere. It has antipodal self intersections, with tangencies to the equator.

(*Cubic Plane Curves – wikipedia, Nodal Cubic on Shapeways – careful with the choice of material: a couple years ago white strong flexible had a print error but the red did fine. *)

9. Shoen put forth this “triply-periodic” minimal surface in the 70’s that he called the Gyroid. Here’s but a small chunk Bathsheba Grossman is a sculptor that has taken lots of inspiration from mathematics. She took a representation of this surface generated by Brakke, and gave it a texturing.

(*Gyroid-wikipedia, bathsheba.com, Bathsheba’s Gyroid on Shapeways*)

10. The Gyroid has close ties to a particular lattice that has been rediscovered several times – seems to go back to Lavas in the 1930’s. Like the Gyroid, the symmetries of the lattice have somewhat surprising resonance. Geometer and Sculptor George Hart is one of the earlier and prolific trailblazers in using rapid prototyping for math sculptures. Check out his website and the Museum of Math in NYC.

(*georgehart.com, Hart’s Rapid Prototyping, Museum of Mathematics, (10,3)-a*)

11. Rather than translational symmetries, regular polyhedral have “point” symmetry, a spherical style. There here are 3D “shells” of 4D polyhedral. The physical models convey depth.

(*24-cell, 600-cell, 120-cell on Shapeways*)

12. In the same sense Prints of classic 3D Fractals like the Serenpinsky Tetrahedron and the Menger Sponge allow a much more voyeuristic exploration of these objects than vitural models on the computer screen. Here, Hart has sliced the Menger Sponge along a diagonal plane. Hold it in your hands and pull it apart. What does the slice look like?

(*Wahtah on shapeways*, Hart’s Menger Sponge Slice)

13. Here Henry printed a 3D version of the space filling Hilbert Curve. The use flexible material reveals its modular structure, its self organization. Not so immediately apparent on the computer, or if it were done in metal. Erez Aiden has suggested this as a model for how DNA packs tightly into an organized globule.

(*I’m not finding the one of Henry’s I had printed, but here’s a similar one by Treepleks*)

14. Henry and his collaborator Geoffery Irving made a new fractal by cleverly “extruding” the development of a 1D Substitution Fractal in a fractal manner itself. Here’s the Dragon Curve. They’ve made models for several others.

(*Here’s a slightly different one: Developing Dragon Curve on Shapeways*)

15. This is a Legendrian Left-Handed Trefoil. The model suggest structure that is not shown… At every point along the curve, it’s tangent to a little disk. But the orientations of these disks are set by their location in space. This forces a certain twisting of the curve.

(*Legendrian Trefoil on Shapeways*)

16. Here’s two set of points with lines between every pair. It’s structured as the skeleton of a tetrahedron. The cycles in can be rather complex. Viewing a computer model or just picture of the print doesn’t do it justice, there’s a pleasant simple rhythm when you rotate it in your hands.

(*Tetrahedron Join on Shapeways, Earlier Post)*

17. Here’s a tangled mess of four strands between two spheres. For certain ways of filling those spheres you actually get something that unravels into an unknot. This was difficult to draw by hand on paper, the computer model allowed a proper organization, but holding it in my hands made the means of unraveling much more apparent.

(*Berge’s Cosmetic Tangle** on Shapeways*)

18. Here is a joint project with sculptor Lun-Yi Tsai. It was a prototype for a large metal sculpture we had hoped to build. You can see a surface made from circles sweeping around. Join the inner to the outer. For me, that’s a representation of a Klein bottle in a circle of spheres. Many emergent shapes arise…

(*Nesting Spheres** on Shapeways*)

19. Like the last, using math to inspire art, George Hart has this collection of Sand Dollars based on tilings of the hyperbolic plane. Again there’s a fractal sense. He twisted them up in interesting ways, riffing on ideas of torus knots and loxodromes. These models were dyed to evoke depth and life.

(*Scroll down to Echinodermania and follow the links*)

20. What’s ahead? Much more than there’s time for. But for some knots there are nice equations governing how to fill up the negative space with an organized sweep of surfaces. “Milnor Fibrations” Henry Segerman and Saul Schliemier also had this notion and surely it occurred to others. Daniel already had some nice Mathematica scripts to make these pictures. There are some technical hurdles to getting a suitable printable model from it…

(*Dan Dreibel’s visualizations of Milnor’s Fibrations*)

And that’s it.

Actually, as a follow up. The technical issues aren’t so terrible. It’s just a matter of reckoning with how stereographic projection from *S^3* to *R^3* distorts things. If you’re willing to make a model encompassing lots of volume, it’s not such a problem. Of course you just make it mostly hollow. Fred Hohman at UGA under the supervision of David Gay has done just this for the trefoil. Here’s some pictures of what he produced.

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The *join* of two topological spaces and is basically the space of all line segments between every pair of points. As a nice embedded, visceral example of this, a tetrahedron may be viewed as the join of two skew line segments.

Indeed this is the illustration on Wikipedia.

It also arises naturally when you think about grid diagrams of knots. Remember this from way back?

Thinking along those lines, you might consider discretizing it a bit… Instead of taking the join of two entire segments, just take the join of points along each segment.

Here I’ve used . And that gives us the complete bipartite graph naturally embedded as a subset of the tetrahedron. Moreover, any grid number knot is a cycle in this particular embedding of the graph.

While it’s a rather simple object, I’ve nonetheless found it to be a rather pleasing one. Lots of emergent rhythms.

Kinda gotta hold it and spin it around to really appreciate it, so print one out if you’d like. I’ll talk about some of the others models there whenever I eventually get around to it…

Of course I would be amiss without plugging Segerman’s fantastic body of W O R K with 3D printing. If you haven’t seen it before, you’re missing out.

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I thought I’d show the product disk associated to the band. This lets one verify the persistence of fiberedness and work out the resulting monodromy (which I haven’t done myself yet).

Let’s talk through the construction.

The generalized banding is defined by an arc in a Seifert surface with one transverse self intersection and meeting the boundary of the surface in its endpoints. A neighborhood of this blue arc is an annular chunk of the surface. The white tubes are the pieces of the link (the boundary of the surface). The rest of the boundary of this annulus continues on doing whatever it was already doing in the surface.

Now run a band across the surface following the arc, going over itself, and joining the boundary of the old fiber.

A spanning arc of the band gives rise to a product disk. If the side of the original surface we saw was “up” then here we have a product disk going from the red arc to the green one.

Taking the sutured manifold coming from this new banded surface and decomposing it along this product disk leaves us with the sutured manifold coming from the old surface. (Yeah, we could see explicit pictures of this… maybe in a future update.) So if the original Seifert surface was a fiber, then the banded surface will be a fiber too.

Here’s a few more pics of this disk on its own and with parts of the surface.

There’s a few more on the Flickr. Clicking any of these pics should take you there.

A couple of interesting things to note:

1) If the “hole” the original blue arc went around actually bounded a disk in the surface, then this gives us the standard Hopf banding. As I’ve drawn it here, this would result in the positive Hopf band.

2) If the hole doesn’t bound a disk, then it is not a Hopf band as the monodromy it offers is neither right-veering nor left-veering.

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Here’s a sequence of images I drew for a talk I gave about a month ago. I reckon there isn’t much more to say that’s not in the images. Hit flickr for larger versions.

**Twisting along a disk.**

**Twisting along an annulus.**

Of course, you’ll probably want the annulus embedded so that its boundary is not a trivial link…

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That’s the exterior of the trefoil. No, really. I mean, well, it’s a torus embedded as the boundary of the exterior of a trefoil.

Okay, so it’s not how you’d probably choose to draw it. It’s not how I’d first choose to draw it either. Let’s see how I came to it.

Think about the space around the trefoil. The thing’s hollow — it goes on (round and round) forever — and — oh my God

— *it’s full of surfaces*! (That was terrible. My sincere apologies to Clarke.)

Yeah. So…. The exterior of the trefoil is a fibration over the circle with once-punctured tori fibers. Above shows a set of eight fibers, then those fibers animated. Note: On Flickr you have to look at the “original” size to view animated gifs.

Let’s look at it without the torus boundary from some other angles.

Here’s a few of those again, but a bit slower.

You may be getting an idea how I drew this. In each fiber you can see 2 somewhat flat 3-pronged regions that go around in a circle. A bit more tricky to see are the 3 somewhat flat 2-pronged regions that go around in another circle. In fact those two circles link each other once… like the cores curves of a genus one Heegaard splitting, a Hopf link.

This might help you see them, though black background and the translucence perhaps wasn’t the best choice for the gif.

Here’s two views of them from eight fibers at once.

The exterior of any (p,q)-torus knot is fibered over the circle, and each fiber may be viewed as taking p q-pronged disks around one component of a Hopf link, q p-pronged disks around another, and then joining all those prongs with pq bands. Of course you still have to fuss with hooking them up correctly. Then for the further fibers you do it again, rotating those disks around the circles a bit each time.

I’ve done this here with p=2 and q=3 to get our trefoil exterior. I made sure each pronged disk clocked around an appropriate amount, used Rhino’s “Blend Crv” function to make the edges of the bands, and then the “Network Srf” to actually make the band. Really, someone with more time should be able to write a program that automates this construction for any (p,q) torus knot or link with however many fibers. Of course there’s a few degenerate situations and other annoyances to reckon with…

And I’d be amiss if I didn’t mention that one could instead view these fibrations as Milnor fibrations. Daniel Dreibelbis has some produced some lovely representations of these fibrations with Mathematica. Here’s his fibration of the trefoil:

Maybe someone could tinker with his code to move the point at infinity in the stereographic projection onto the trefoil. Then you could have Mathematica automagically fiber the trefoil exterior!

Totally let me know if you do!

Let’s round off this looooong overdue post with a bit more eye candy.

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They’re both strongly invertible. Let’s quotient the first link by its strong inversion to get a tangle. Then we’ll isotop that tangle around and eventually take its double branched cover to get the second.

Notice that this homeomorphism swaps the red and blue meridians and longitudes.

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What’s below are some pictures suggesting Seifert fibrations of *S^1 x S^2*, the circle of spheres. View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.

A *Seifert fibration* of a 3-manifold is a filling of the 3-manifold with circles so that around each point a teeny tiny enough chunk looks like it’s filled with parallel lines. Here these circles get chopped into intervals. The interval going through the north pole connects up to give one circle, and so does the interval through the south pole. We call these circles the singular fibers.

All the other intervals connect up with a fixed number of others to form circles. These are the regular fibers. Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.

Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the *S^1*‘s of the *S^1 x S^2*.

In the three examples above, red fibers are shown at regularly spaced latitudes (along fixed longitudes) going from the north pole to the south pole. The yellow fibers are copies rotated to other longitudes.

Let’s look at just the red ones as the orders of the singular fibers increase.

Neat, I guess.

While we’re here. Have some eye candy. There’s more in this set.

I think all these below come from the order 1.

(Oh, and these pictures are a result of experimenting with Grasshopper for Rhino and some non-photorealistic rendering.)

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(Hmmmm…. I guess I had transparency in the original background. Oh well.)

The projective measured lamination space of the twice punctured torus is homeomorphic to the three-sphere, . That’s a result of Thurston. Parker-Series [1], continuing work of Keen-Parker-Series [2], gives a triangulation of consisting of 28 tetrahedra, 56 faces, 39 edges, and 11 vertices. Each vertex may be represented by a particular (isotopy class of an) unoriented simple closed curve while each tetrahedron is spanned by the projective weightings of a Birman-Series -train track [3].

Two of these vertices correspond to natural North and South Poles of the three-sphere as they never occupy the same tetrahedron and the remaining 9 vertices with the 14 faces among them form a polyhedral two-sphere separating these poles.

We are thus able to display this triangulation of in by omitting the North Pole, placing the South Pole at the origin, choosing an embedding of the polyhedral enclosing the origin, and extending the remaining edges and faces radially. (Actually when shown in these pictures I used a round sphere while the vertices of the polyhedral sphere defined the radial edges.) At each vertex other than the poles we center a copy of with its corresponding simple closed curve; at a representative point within each tetrahedron we center of copy of with its corresponding fundamental train track. To keep the parametrization of each copy of consistent we first view as being skewered along a radial axis with one puncture heading towards the North Pole and the other towards the South Pole. Then we ask the model (using the Face Me function in SketchUp… looks like it’s now called Always Face Camera) to rotate each copy of along its axis so that its “front” maximally faces the viewer. This allows us to dynamically rotate the model while inferring the parametrization of each copy of by its radial orientation.

[1] J. R. Parker and C. Series, The mapping class group of the twice punctured torus, Groups: topological, combinatorial and arithmetic aspects}, London Math. Soc. Lecture Note Ser. 311 (2004), 405–496.

[2] L. Keen, J. R. Parker and C. Series, Combinatorics of simple closed curves on the twice punctured torus, Israel J. Math. 112 (1999), 719–749.

[3] J. S. Birman and C. Series, Algebraic linearity for an automorphism of a surface group, J. Pure and Applied Algebra 52 (1988), 227–275.

Generic train tracks for the tetrahedra:

Simple closed curves for the vertices (sans the poles):

The four pics of the full triangulation that make up the first picture:

The list of tetrahedra and vertices labeled with the Parker-Series notation:

And of course clicking on the pictures will take you to Flicker where you can view somewhat higher resolution versions.

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Through the double branched cover, we get the fibration of the solid torus by disks.

Here’s a few more views of these.

Simple as they are, there’s a pleasant rhythm.

There are a couple more on the Flickr (just click any of these pics). You have to view the “original” size for the gifs to animate.

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