## Mathematics with 3D Printing

Back in September I participated in a local Pecha Kucha event on the topic of 3D printing. Of course I talked about interactions of mathematics and 3D printing. The video (well, slides+audio) is now online.

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.

Nice work!

And it’s great to see all the credits and references..

Theo Armour said this on April 27, 2014 at 4:04 am |

[…] Through several steps of examples and iterations, Hohman presents the information needed not just to reprint his fully designed trefoil knot, but to allow users to create their own mathematical 3D printed masterpieces. His project has received some great attention from the mathematical community, including two guest blog posts on Dr. Taalman’s MakerHome blog and a feature on the Sketches of Topology blog post “Mathematics with 3D Printing.” […]

Mathematics/Physics Student Creates 3D Printed Puzzle of Trefoil Knot, Catches Mathematical Community’s Interest - 3DPrint.com said this on December 5, 2014 at 11:37 am |

Wow, these are so beautiful. I’m doing a school presentation on how 3D printing can make math education more beautiful and creative, so your pictures were just what I needed. Thank you so much for sharing these.

TV said this on January 4, 2015 at 11:07 pm |