Math at the Simon’s Center for Geometry and Physics

IMG_3442I had the very great privilege of being a co-organizer of a workshop held at the Simon’s Center for Geometry and Physics and NYU Stony Brook. This was the workshop on the Symplectic and Algebraic Geometry in the Statistical Physics of Polymers. It was my first time to this campus, and I had a blast with both the math at the workshop AND all the visualization of math in the environment.

My first hint that things were going to be special, was the fantastic Umbilic Torus sculpture found at the end of an avenue of trees between the center and the math department.

IMG_3434The sculpture is by Dr Helamun Ferguson, click  here to find a photo gallery showing the design and construction of the piece. 

The sculpture consists of a space filling curve all over the surface of the sculpture. The sharp curve along the edges is a trefoil knot, winding three times around the central hole (the longitude on the torus) and twice around the sculpture the other way (the meridian on the torus).

The base of the sculpture is a large round granite disk with a 3 sided deltoid mirroring the 3-fold symmetry of the sculpture overhead. The base had to be left to settle for a year, and was greatly loved by the local skate-boarders!


The Simon’s Center itself is in a wonderful airy building, with mathematical themes blended seamlessly in the design. I kept finding treasures as the workshop went on. The most obvious, is the sandstone wall behind the stair case leading up to the cafe on the second floor. It is covered with small math motifs from knots, to physics, to finding the square root of 2.


Even the screens on the side of the first floor lounge are mathematical, with different tilings of the plane illustrated. Just love the artistry of the designs in them.

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Models have arrived!

IMG_3072Our 3D printed math models have arrived in the W&L Mathematics Department ready for the Fall semester. They are in labeled clear plastic containers right over the biz hub in the math work room, so everyone can easily get to them.

We are looking forward to seeing what people do with them in class this coming semester. In particular, if we get any more requests for builds or rebuilds.


MathFest 2015

Our summer build team attended the centennial MAA MathFest Conference in Washington DC. We had a wonderful time attending many talks, the exhibits and the evening entertainment. (Some of our favorites included the talks by Erik Demaine, Noam Elkies, and the Cirque de Mathematiques.)

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Emily and Ryan gave a wonderful (and well attended) talk about their summer work. I spoke about our work in the What Can a Mathematician Do with a 3D printer? session organized by the inspirational Laura Taalman and Ed Aboufadel (below left).  Everyone who brought printed objects got to display them at the front of the room (below right).

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Laura had her MakerBot mini printing before the session started. A small collection of her models was placed in front of it. Our models were right in front of Jason Cantarella’s 3D printed calculus robot Cy. They were very well received by everyone present.

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Here are some of the wonderful models by Christopher Hanusa from Queens College CUNY (left), and Lila Roberts from Clayton State University (right).

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Laura spoke about how she designed and printed the Catalan Wireframe Polyhedra, shown below. We were even lucky enough to each be given one by her! I’ve come away from the session with many good ideas of using 3D printing in the classroom, as well as designing new math models.



PS Printing in powder

One last post about the summer printing. We did end up using the ProJet 260 (gypsum) powder printer. We printed two objects – the solid Strange Bowl, and the Tumor Model. Both had colors added with help from Dave Pfaff. (There is a complicated color bit map involved.) The tumor model’s colors roughly correspond to the distance from the center of the model. While these models are beautiful, they are not as robust as some of our other models. They won’t go into general circulation, but instead will be in our display case.



What a summer!

surfaces-long-3Our summer research project has officially ended. Emily and Ryan have been phenomenal. Together, we’ve designed and 3D printed over 46 math models during the past 9 weeks. Given that our first week was spent working on the math and learning computer programs, we’ve averaged a little over 6 models a week. Phew! 

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We also 3D printed models that other folks designed, which means we’ve well over 55 different models in total.

We are top-longall currently at the 2015 MAA MathFest conference in Washington DC. Emily and Ryan will be talking about their work in a student research session, and I will be discussing our work in a session on What can a mathematician do with a 3D printer? organized by the inspirational Laura Taalman and Edward Aboufadel.

wedges-longBefore I left to come to MathFest, I had W&L photographer Kevin Remington take some stills of just a few of our models in a professional light box. The results are fantastic. Many of these photos appear in Thingiverse, as well as my web page.

cropped-KRP_5585.jpgThese photos show: a few of the quadratics surfaces we designed and printed; the strange bowl family; some of our ”sliced” volumes; and the part of a helicoid, the “Bulge Head” solid, and Voronoi Klein Bottle.

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3D printing with liquid

The neat thing that we’ve been doing in the past couple of weeks is to use the FormLabs Form 1+ liquid resin printer. It is just so cool!

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StrangeBowl-allThe first objects we printed were the strange bowls (shells, washers and smooth). Previously we tried to print them on the MakerBot 2X, but the sheer number of supports meant the print was not a great success. However, the FormLabs printed them beautifully. We all loved watching the bowls slowly come out of the liquid resin.

We next printed was the Bulge-Head solid. It is one of our favorites!

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top-longFinally, we had great success printing parametric curves and other surfaces with the liquid resin printer.

Klein Bottle

Klein-bottle-printJust a really short post to share our general excitement over having just about completed all of the objects from Multivariable Calculus. We just have a few more to print out. We will spend our remaining week(!) printing out some interesting topological objects – many of these directly from Thingiverse.

We printed one such object today. This is the Voronoi Klein Bottle from MadOverlord on Thingiverse. We printed this on the MakerBot 2X with a raft but no supports. After a moment’s thought one can see that the print succeeds (despite the short horizontal lines on the design) because the Voronoi cells are small enough. Interesting! The black filament also hides a few rough spots on the print.

The Klein bottle is named after Felix Klein (25 April 1849 – 22 June 1925), a German mathematician who saw many connections between Group Theory and Geometry. It is a one-sided surface and is a generalization of a Mobius strip. (In fact, it is topologically equivalent to two Mobius strips glued together along their boundaries.)

There are many fabulous descriptions of this topological object, one of my favorites is The Adventures of the Klein Bottle found on YouTube (from the wonderful folks at the Frei Universitat in Berlin).



Walking down memory lane

In Fall of 2014 I taught Math 341 Introduction to Topology. As part of the class I had the students design and then print a topological object. For most students, this ended up being the highlight of the course.

10482770_4584418265143_4975820982335381144_n-1We spent a week of class in the IQ center under the guidance of David Pfaff. He showed us how objects can be viewed in the stereo 3D lab and gave us a crash course in Cinema 4D. Students then let loose their imaginations and creativity. Many students chose to learn about knots and links, ribbon knots, and Seifert surfaces of knots and links. They produced some wonderful models. Other students chose to create objects with symmetry (like the 20 sided die), or the cube-like Cayley graph.

It turned out that getting the objects that could be 3d printed was hard work! Many objects had not been optimally made (for example with normal vectors pointing inwards). We were fortunate to have David Pfaff’s expertise in sorting out these errors. Eventually all the objects were printed using the IQ center’s ProJet 260. Some of them needed to be printed twice, as they broke when being removed from the printer. Many 3d printed math objects from this class and from Aaron Abrams first year seminar currently reside in the Mathematics Department.


An unexpected bonus

Quadratic_Surfaces We have made some wonderful 3d printed models recently. As an unexpected bonus we have also created some great Mathematica notebooks. These were developed as we first created the shapes in Mathematica, then imported them into Cinema 4D, to finally create the .stl file ready for 3d printing. The notebooks have a mix views of the shapes and also some neat Mathematica Demonstrations that can be used when teaching. They can be found here.

Volumes_TripleIntegrals     Two_intersecting_cylinders


Summer Collaborators

Introducing Emily Jaekle and Ryan McDonnell, two undergraduate students from Washington & Lee University who are doing research with me this summer. Expect to hear from them regularly!