mathematics through 3d printing...mathematics through 3d printing how: weekly 3d printed object each...
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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 intoobjects 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
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GMU Math Makerlab: A little background
Research design and design research
Bottom image with Patrick BishopImages with Steve Lucas and Laura Taalman
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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
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Mathematics through 3D printing
Why:
Teach topics that slip through the cracks that “everymajor 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 di↵erent topic
Public display in department’s display case
Weekly presentations: oral, written, blog, and Thingiverseexamples 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 requiredto do their own printing. Thus they learnedabout slicers, manifold and watertightobjects, supports, etc. (not this semester)
Learning assistants helped with setup andprinting.
Lecture each week on the math and thecoding.
Provided with sample code
Provided with an initial step by step tutorialon each softwareexamples 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 15distinct classes is a new result,still under peer review.
There are 17 distinct wallpapergroup symmetries which resultin tilings of the plane
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Is the medium the message?
3D printing may not be the best methodbut too many methods becomes a burden
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Calculus objects: Saddles and surfaces
Learning multivariable calculus involves a lot of algebraicmanipulations, but it all becomes much easier when getting tosee 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 graphof complex functions are four dimensional. We cannot see infour dimensions, so we project to three dimensions. This givesa 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 illusionin the department display caseTake 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 youreyeµ ymirgirojection
Mirror TB peaneeye 0 O projectionplane
as
Two 20 curves30 Curve projects
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Mathematica sample code: Sugihara cylinders
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Di↵erential equations: Strange chaotic attractors
Solutions to 3-dimensional systems of di↵erential equations canbe quite simple, such as a point that never moves, or a periodicmotion that repeats forever. It can also be quite complicated:
Strange the object has interesting fractal shape
Chaotic nearby initial conditions move away with growingtime
Attractors any nearby initial condition has a solutionlimiting to the shape you see.
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Di↵erential equations: Strange chaotic attractors
Strange chaotic attractors are great for printing.
Di↵erential equation methods are built into Mathematica
Smooth curves: easily rendered
With closeness of segments: structurally stable
Made accurate by attraction: numerics reflects theunderlying 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 multivaluedtransformation.
Easily achieved using any CAD system with recursion.
Each layer represents a single iterate of the multivaluedmap
Similar objects exist in three dimensions
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Iterated function systems explained
first 2nditerate iterate
f
Sf fats3
3rd 4th limititerate interate set
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Iterated function systems explained
f s
sf 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 andcoding 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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Di�culties
Huge range of programming background and incomingmathematical 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 payo↵
I learn a lot from my students!
Many coding commands and data structures
Ideas for data visualization
Even math: Student talk on complex graphs outlinedrelationship between sin and sinh as graph projections
Github function collection contains equations for allsuperheros!
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Thanks!
For further information, codes see:
http://gmumathmaker.blogspot.com
Upcoming paper: Modeling Dynamical Systems for 3DPrinting, with Stephen K. Lucas and Laura Taalman,
submitted.
Thanks for your attention!
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