{"id":34078,"date":"2017-09-07T14:36:39","date_gmt":"2017-09-07T18:36:39","guid":{"rendered":"https:\/\/www.shapeways.com\/blog\/?p=34078"},"modified":"2019-07-01T12:10:57","modified_gmt":"2019-07-01T16:10:57","slug":"shapeways-education-grant-stories-jonathan-gerhards-3d-printed-visualizations-topological-homotopies","status":"publish","type":"post","link":"https:\/\/www.shapeways.com\/blog\/shapeways-education-grant-stories-jonathan-gerhards-3d-printed-visualizations-topological-homotopies","title":{"rendered":"Shapeways Education Grant Stories: Jonathan Gerhard\u2019s 3D Printed Visualizations of Topological Homotopies"},"content":{"rendered":"

Representing James Madison University, Jonathan Gerhard is one of our $1000 <\/i>Education Grant<\/i><\/a> recipients from the Spring 2016 grant cycle. Thanks to Jonathan\u2019s ingenuity and the tangibility of 3D printed objects, we can better understand the lofty concept of algebraic topology.\u00a0<\/i><\/p>\n

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The 3 by 3 Rook\u2019s Graph<\/p><\/div>\n

Do you know the definition of topology<\/i>? Or homotopy<\/i>? Well, you\u2019re 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.<\/p>\n

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<\/a>):\"education<\/p>\n

Despite the literal dimensionality of this area of study, algebraic topologists usually learn the different notions of homotopies from two-dimensional diagrams. Jonathan\u2019s 3D printed work with homotopies is an answer to the 2D status quo, so we\u2019re thrilled that the $1000 Education Grant has enabled him to create 3D printed homotopies we can hold in our hands.<\/p>\n

So, what gave him the idea to incorporate 3D printing into his interest in homotopies?<\/p>\n

\u201cUpon learning about some wildly un-intuitive homotopies (one of my favorites is that a 3-sphere<\/a> 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<\/a> 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.\u201d<\/p><\/blockquote>\n

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With the Education Grant funds in hand, Jonathan got to work designing an array of objects in Fusion360<\/a>, 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.<\/p>\n