Representing James Madison University, Jonathan Gerhard is one of our \$1000 Education Grant recipients from the Spring 2016 grant cycle. Thanks to Jonathanâ€™s ingenuity and the tangibility of 3D printed objects, we can better understand the lofty concept of algebraic topology.Â

The 3 by 3 Rookâ€™s Graph

Do you know the definition of topology? Or homotopy? Well, youâ€™re not alone. Topology is the study of the properties of 3D space that are preserved after physical deformations like stretching or bending. Among the concepts studied by topologists is homotopy.

Jonathan describes a homotopy between two objects as a continuous deformation of one into the other. For example, imagine squeezing a cylinder down until it is just a circle. A cylinder and a circle are one example of a homotopy equivalent. Another classic example is illustrated in the animated gif below (or you can buy the 3D printed version here):

Despite the literal dimensionality of this area of study, algebraic topologists usually learn the different notions of homotopies from two-dimensional diagrams. Jonathanâ€™s 3D printed work with homotopies is an answer to the 2D status quo, so weâ€™re thrilled that the \$1000 Education Grant has enabled him to create 3D printed homotopies we can hold in our hands.

So, what gave him the idea to incorporate 3D printing into his interest in homotopies?

â€śUpon learning about some wildly un-intuitive homotopies (one of my favorites is that a 3-sphere minus a torus is homotopy equivalent to the disjoint union of two solid tori), I had the idea to utilize 3D printing. I had done a project on 3D printing knot invariants with Laura Taalman at JMU my freshman year (final results pictured below), so I thought it was only right to finish off my undergraduate career by doing another 3D printing project with her.â€ť

With the Education Grant funds in hand, Jonathan got to work designing an array of objects in Fusion360, and noted that handing off his 3D printing to Shapeways allowed him to focus on his designs, rather than the troubleshooting experiments that are all too common when working with desktop 3D printers.

â€śI began to design a whole array of objects: The Perko Knots on a praxinoscope, the Rookâ€™s Graph and (in the case of n = 4) the strangely non-isomorphic Shirkhande Graph, all the while making time for some interesting mathematics.â€ť

4 by 4 Rook’s Graph

“The Drum” Shrikhande Graph

Jonathan hopes that, apart from the utility of visualizing homotopies, the 3D prints he created with the Education Grant will be his permanent legacy in the James Madison University math building.

As it turns out, Jonathan was able to impact the field of mathematics outside of his own school walls, too. Jonathan presented his work at the Joint Meetings of the American Mathematical Society and the Mathematical Association of America in Atlanta in January 2017, in the Invited Paper Session on Technical Tools for Mathematical 3D Printing.

We love helping students unlock their full potential with 3D printing, and we are proud to have contributed to Jonathanâ€™s work using Shapeways prints as a tool for grounding abstract mathematical concepts. For a deeper dive into Jonathanâ€™s topological explorations, be sure to check out his personal blog. Plus,dig into our Mathematical Art coverageÂ to discover more from the beautiful intersection of math and 3D printing.

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