Mathematics Through 3D Printing...Mathematics through 3D printing How: Weekly 3D printed object each on a di↵erent topic Public display in department’s display case Weekly presentations:

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 intoobjects you can hold in your hand



Workshops, camps, math circles, classroom visits, festivals



Classroom demos, assistive prints

Top left and right with Colin Chung and Chloe Ham



Research design and design research

Bottom image with Patrick BishopImages with Steve Lucas and Laura Taalman


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



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



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


Weekly presentation: blog a


Weekly presentation: blog b


Weekly presentation: Thingiverse a


Weekly presentation: Thingiverse b


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


OpenSCAD step by step tutorials


OpenSCAD step by step tutorials


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


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


Is the medium the message?

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


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


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.


Visualization of data: Individuality!


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


Mathematica sample code: Sugihara cylinders

Sugihara cylinders

mirror youreyeµ ymirgirojection

Mirror TB peaneeye 0 O projectionplane

as

Two 20 curves30 Curve projects


Mathematica sample code: Sugihara cylinders


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.


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


Mathematica sample code: Chaotic attractor


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


Iterated function systems explained

first 2nditerate iterate

f

Sf fats3

3rd 4th limititerate interate set


Iterated function systems explained

f s

sf k fats

Theorem Limit set is independentofS


OpenSCAD sample code: IFS fractals


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


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


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!


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!


Documents

Mathematics Through 3D Printing...Mathematics through 3D printing How: Weekly 3D printed object each on a di↵erent topic Public display in department’s display case Weekly presentations: