Mathematics Through 3D Printing a GMU capstone class Evelyn Sander Geometry Labs United, July 16, 2020, Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 1 / 27
GMU Math Makerlab: A little background Work with students to turn mathematical concepts into objects you can hold in your hand Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 2 / 27
GMU Math Makerlab: A little background Workshops, camps, math circles, classroom visits, festivals Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 2 / 27
GMU Math Makerlab: A little background Classroom demos, assistive prints Top left and right with Colin Chung and Chloe Ham Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 2 / 27
GMU Math Makerlab: A little background Research design and design research Bottom image with Patrick Bishop Images with Steve Lucas and Laura Taalman Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 2 / 27
Mathematics through 3D printing What: A capstone class to synthesize mathematical knowledge Prerequisites: A proof course and a 300 level math course No textbook: readings of papers, websites Running each fall This and all subsequent models designed by class students Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 3 / 27
Mathematics through 3D printing Why: Teach topics that slip through the cracks that “every major should have seen.” (not unique to math!) Breadth over depth Creativity: No identical creations Patience required Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 3 / 27
Mathematics through 3D printing How: Weekly 3D printed object each on a di ff erent topic Public display in department’s display case Weekly presentations: oral, written, blog, and Thingiverse examples to follow Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 3 / 27
Weekly presentation: blog a Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 4 / 27
Weekly presentation: blog b Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 5 / 27
Weekly presentation: Thingiverse a Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 6 / 27
Weekly presentation: Thingiverse b Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 7 / 27
3D printing specifics Too many pieces of software is overwhelming: OpenSCAD and Mathematica Taking full ownership: Students are required to do their own printing. Thus they learned about slicers, manifold and watertight objects, supports, etc. (not this semester) Learning assistants helped with setup and printing. Lecture each week on the math and the coding. Provided with sample code Provided with an initial step by step tutorial on each software examples to follow Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 8 / 27
OpenSCAD step by step tutorials Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 9 / 27
OpenSCAD step by step tutorials Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 9 / 27
Print topics Two types of tilings of the plane Saddles and surfaces Graphs of complex functions Data visualization Mathematical optical illusions Strange chaotic attractors Iterated function systems Redo a project with an eye to improvement Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 10 / 27
Two types of tilings the plane Irregular convex pentagons Group symmetries There are 17 distinct wallpaper That there are exactly 15 group symmetries which result distinct classes is a new result, in tilings of the plane still under peer review. Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 11 / 27
Is the medium the message? 3D printing may not be the best method but too many methods becomes a burden Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 12 / 27
Calculus objects: Saddles and surfaces Learning multivariable calculus involves a lot of algebraic manipulations, but it all becomes much easier when getting to see what the concepts look like in 3D. Student found a starfish to go atop the starfish saddle Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 13 / 27
Graphs of complex valued functions The complex plane is two-dimensional, meaning that the graph of complex functions are four dimensional. We cannot see in four dimensions, so we project to three dimensions. This gives a sense for what the these graphs look like in 4D. Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 14 / 27
Visualization of data: Individuality! Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 15 / 27
Optical Illusions: Sugihara Cylinders A mathematical optical illusion in the department display case Take a look at each object and its reflection. They’re not the same! Right side up heart Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 16 / 27
Mathematica sample code: Sugihara cylinders Sugihara cylinders your eye mirror ymirgirojection µ Mirror TB peane eye 0 O projection plane as Two 20 curves 30 Curve projects Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 17 / 27
Mathematica sample code: Sugihara cylinders Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 18 / 27
Di ff erential equations: Strange chaotic attractors Solutions to 3-dimensional systems of di ff erential equations can be quite simple, such as a point that never moves, or a periodic motion that repeats forever. It can also be quite complicated: Strange the object has interesting fractal shape Chaotic nearby initial conditions move away with growing time Attractors any nearby initial condition has a solution limiting to the shape you see . Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 19 / 27
Di ff erential equations: Strange chaotic attractors Strange chaotic attractors are great for printing. Di ff erential equation methods are built into Mathematica Smooth curves: easily rendered With closeness of segments: structurally stable Made accurate by attraction: numerics reflects the underlying behavior Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 19 / 27
Mathematica sample code: Chaotic attractor Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 20 / 27
Fractals: Iterated function systems This type of fractal arising from a multivalued transformation. Easily achieved using any CAD system with recursion. Each layer represents a single iterate of the multivalued map Similar objects exist in three dimensions Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 21 / 27
Iterated function systems explained first 2nd iterate iterate f S f fats 3 3rd 4th limit interate iterate set Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 22 / 27
Iterated function systems explained f s s f k fats Theorem Limit set independent of S is Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 22 / 27
OpenSCAD sample code: IFS fractals Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 23 / 27
Successful methods Lecture balanced between mathematical background and coding instructions Time to work in class: Students are happy to act as mentors Clear detailed instructions and rubrics Detailed sample codes Open ended assignments Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 24 / 27
Di ffi culties Huge range of programming background and incoming mathematical knowledge Timeliness with print slots and assignment turn in Mathematica syntax, data set methods, etc. 3D printing is not always the best method Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 25 / 27
The payo ff I learn a lot from my students! Many coding commands and data structures Ideas for data visualization Even math: Student talk on complex graphs outlined relationship between sin and sinh as graph projections Github function collection contains equations for all superheros! Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 26 / 27
Thanks! For further information, codes see: http://gmumathmaker.blogspot.com Upcoming paper: Modeling Dynamical Systems for 3D Printing, with Stephen K. Lucas and Laura Taalman, submitted. Thanks for your attention! Evelyn Sander (GMU) Mathematics Through 3D Printing July 16, 2020 27 / 27
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