Mathematics Through 3D Printing a GMU capstone class Evelyn Sander - - PowerPoint PPT Presentation

Mathematics Through 3D Printing

a GMU capstone class Evelyn Sander

Geometry Labs United, July 16, 2020,

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GMU Math Makerlab: A little background

Work with students to turn mathematical concepts into

• bjects you can hold in your hand

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GMU Math Makerlab: A little background

Workshops, camps, math circles, classroom visits, festivals

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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

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Mathematics through 3D printing

How: Weekly 3D printed object each on a different topic Public display in department’s display case Weekly presentations: oral, written, blog, and Thingiverse examples to follow

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Weekly presentation: blog a

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Weekly presentation: blog b

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Weekly presentation: Thingiverse a

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Weekly presentation: Thingiverse b

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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

• bjects, 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

• n each software

examples to follow

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OpenSCAD step by step tutorials

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OpenSCAD step by step tutorials

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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

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Two types of tilings the plane

Irregular convex pentagons Group symmetries That there are exactly 15 distinct classes is a new result, still under peer review. There are 17 distinct wallpaper group symmetries which result in tilings of the plane

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Is the medium the message?

3D printing may not be the best method but too many methods becomes a burden

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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

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Graphs of complex valued functions

The complex plane is two-dimensional, meaning that the graph

• f 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.

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Visualization of data: Individuality!

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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

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Mathematica sample code: Sugihara cylinders

Sugihara cylinders

mirror

youreye

µ

ymirgirojection

Mirror TB

peane

eye 0 O

projectionplane

as

Two 20 curves 30 Curve projects

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Mathematica sample code: Sugihara cylinders

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Differential equations: Strange chaotic attractors

Solutions to 3-dimensional systems of differential 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.

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Differential equations: Strange chaotic attractors

Strange chaotic attractors are great for printing. Differential 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

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Mathematica sample code: Chaotic attractor

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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

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Iterated function systems explained

first 2nd iterate iterate f

S f fats

3

3rd 4th

limit

iterate

interate

set

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Iterated function systems explained

f s s

f k

fats

Theorem Limit set

is independentofS

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OpenSCAD sample code: IFS fractals

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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

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Difficulties

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

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The payoff

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!

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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!

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