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Sep 11, 2017 | By Tess

Jonathan Gerhard, a student from James Madison University (JMU) in Virginia, has 3D printed a series of objects that physically demonstrate the mathematical concepts of topology and homotopy. Gerhard was awarded a $1,000 Education Grant from 3D printing service Shapeways for his innovative and educational endeavour.

For those of us who aren’t well versed on our mathematical concepts and terms, topology—not to be confused with topography—is the study of physical geometric properties and spatial relations that remain unchanged after a figure has undergone stretching, bending, or other physical deformations.

Homotopy, for its part, is a concept within topology that signifies a continuous deformation from one object into another. Gerhard offers the example of a cylinder being squeezed down into a circle, or a 3D mug being transformed into a ring.

To help illustrate and showcase these mathematical concepts in a more tangible, physical way, Gerhard decided to 3D model and print a series of topologic objects. Traditionally, the concepts had been understood using two-dimensional diagrams and illustrations.

He explains on his blog, “Upon learning about some wildly un-intuitive homotopies (one of my favorites is that a 3-sphere minus a torus is the 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… so I thought it was only right to finish off my undergraduate career by doing another 3D printing project…”

4 by 4 Rook’s Graph

3 by 3 Rook’s Graph

For the project, Gerhard worked with Laura Taalman (aka mathgrrl), an experienced 3D modeler and member of the maker community. 3Ders readers might be familiar with Taalman for her Snowflake Machine, an impressive 3D printed snowflake customizer.

The topology shapes were designed by Gerhard using Fusion360 and were 3D printed in collaboration with Shapeways. “I began to design a whole array of objects: The Perko Knots on a praxinoscope, the Rook’s Graph, and (in the case of n = 4) the strangely non-isomorphic Shirkhande Graph, all the while making time for some interesting mathematics,” he says.

“The Drum” Shrikhande Graph

The Praxinoscope project struck us as especially interesting, as it uses an old-school animation technique to show how each of the 3D printed perko knots are part of the homotopy movement.

In addition to being recognized by his own university and Shapeways, Gerhard also gained some attention from the American Mathematical Society and the Mathematical Association of America for his project. In fact, he was given the opportunity to present his topology-inspired 3D printing project at an event hosted by both groups this past January.

Shapeways is currently inviting likeminded student makers to submit their own innovative 3D printing projects to win one of their $1,000 Education Grants.

 

 

Posted in 3D Printing Application

 

 

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