Elizabeth Denne
Department of Mathematics
Washington & Lee University
202 Robinson Hall




Shortcuts 

Background 
This webpage is dedicated to sharing some of the ways key ideas in mathematics may be visualized.
Everything on the page should be freely accessible everyone. You'll find
Mathematica notebooks, as well as 3D printable objects that can be used in the classroom.
I've been inspired by the likes of
of Laura Taalman,
Henry Segerman,
Jason Cantarella,
George Francis, and the late Bill Thurston.
From Math 341 Fall 2014
We used the 3D printers in the IQ center and the Mathematics Department: Series 1 Pro, uPrint SE, FormLabs 1+, MakerBot Replicator 2 and 2x, Afinia H480, and ProJet 260 printers. The 3D printers in the Mathematics Department were originally purchased for the 2013 Jockey John Robinson FirstYear seminar titled The Shape of Space taught by Professor Aaron Abrams.
Thanks must go to the Washington & Lee students who've helped develop these tools: Emily Jaekle ('16), Ryan McDonnell ('17).
Thanks also to the Washington & Lee Summer Research Scholars Program which funded their research.
Finally, I'm eternally grateful for all of the technical assistance patiently provided by David Pfaff from the
IQ center at Washington
& Lee University.

Found on the internet 
Blogs
 My own blog Visions in Math, giving an ongoing description of all this neat stuff.
 Laura Taalman's blog Hacktastic is where she
describes her current projects. MakerHome blog is where she
described how she printed one 3D print every day for a year.
 David Bachman's blog Math/Art gives his meditations on 3D design using Rhino and Grasshopper.
 Henry Segerman's webpage is
always worth a look.
Spheres approximated by disks.
3D printable objects
 Thingiverse is a huge resource of downloadable 3D printable objects.
 Shapeways is an online shop which allows folks to upload their 3D designs and have them printed. You can also buy 3D prints of many of the featured designs.
 My own Thingiverse page which has the mathematical
objects from this page and more.
 Laura Taalman's Thingiverse page contains
a huge array of mathematically inspired objects, and also other more whimsical objects.
 Henry Segerman's Shapeways page and Thingiverse page contain a huge array of truly beautiful mathematical objects and mathematical art.
 Jason Cantarella's Thingiverse page contains
some neat math objects as well as a huge list of energy minimizing knots and links.

Talks & Exhibited work 
Some of the places my models have been seen.

Instructions 
Please let me know if you have corrections or improvements for these instructions. Thanks!
"Bulge Head" Solid
Creating models using Mathematica and Cinema 4D
 Cinema 4D interface handout, giving handy short cuts and language to describe C4D.
 Introduction to Cinema 4D: A guide to getting started in Cinema 4D, with basic commands and tips on fixing meshes.
 Our collective wisdom on designing and printing is here: Trouble Shooting Guide.
 Instructions on how to import a Mathematica object into Cinema 4D.
 Instructions on how to put equations on solids in Cinema 4D.
 To make something like the three volumes with defining equations (shown below right), first make the solid in Mathematica, then import it into Cinema 4D, and finally add equations.
 Instructions on how to download Magic Merge and use it in Cinema 4D.
 Instructions on how to construct a volume by disc method, by cylindrical shell method, and by general slices (Calculus II) in Cinema 4D.
 Instructions on how to construct a volume demonstrating the slices used in iterated integrals (Multivariable Calculus) in Cinema 4D.
 Instructions on how to create quadratic surfaces (Multivariable Calculus) in Cinema 4D.
 Instructions on how to put text along a spline and an extruded parametric curve (like a knot) in Cinema 4D.
 Instructions on how to create a knot in Cinema 4D.
Three volumes with defining equations
Using the Printers

Calculus 
Calculus II
Volume: 16 cylindrical shells
 Mathematica Notebooks
 Class Project
 Thingiverse models
 Sphere: 10 disks on Thingiverse.
 Sphere: 20 disks on Thingiverse.
Volumes by slices
 Volume: 16 cylindrical shells on Thingiverse. The area between the function y=2x^2x^3 and the xaxis is rotated about the yaxis creating a volume of revolution. This model shows this volume approximated by 16 cylindrical shells. (The 16th shell in the center has zero volume so is not included in the print model!)
 10 Equilateral triangles on a circular base on Thingiverse. A solid has a circular base of radius 1. Parallel crosssections perpendicular to the base are equilateral triangles. This solid is approximated by 10 equilateral triangular prisms. This approximation illustrates how the volume of the solid is found using an integral of the crosssectional slices.
 20 Equilateral triangles on a circular base on Thingiverse. A solid has a circular base of radius 1. Parallel crosssections perpendicular to the base are equilateral triangles. This solid is approximated by 20 equilateral triangular prisms. This approximation illustrates how the volume of the solid is found using an integral of the crosssectional slices.
Strange bowls
 Strange Bowl: smooth, Strange Bowl: 16 cylindrical shells, and Strange Bowl: 16 washers all on Thingiverse. The area between y=x and y=x^2 is rotated about the line y=1.25. This creates a volume of revolution which looks a bit like a bowl, but with a conical interior and a big hole in the bottom. This volume is shown, along with an approximation by 16 washers and 16 cylindrical shells. Note that the 16th washer and 16th shell do not appear on the models. (Near the bottom of the bowl, the shape is so flat that they are disconnected from the others.)
 Volumes of Hanoi on Thingiverse by Laura Taalman.
3D model for illustrating a popular calculus concept: volumes of solids of revolution, approximated by cylindrical shells and washers.
Multivariable Calculus
 Mathematica Notebooks
 Parametric Curves Mathematica notebook containing examples of parametric curves, knots, and parametric curves arising from the intersection of two surfaces.
 Quadratic Surfaces Mathematica notebook containing a Mathematica demonstration of all quadratic surfaces, and separate examples of individual surfaces.
 Volumes by Triple Integrals Mathematica notebook containing the Mathematica code for Wedge 1 & 2, Tetrahedron 1 & 2, the intersection of a paraboloid and a sphere, and a model of a tumor.
 Intersecting Cylinders Mathematica notebook showing the intersections of 2 and 3 cylinders.
Volumes by French Fries
 Thingiverse models

Knots, Topology and Geometry 
Knots
Three interlocking trefoil knots
 Mathematica Notebooks
 Torus Knots Mathematica notebook showing a torus with meridian and longitude curves, as well as numerous torus knots.
 Thingiverse Models
Geometry
 Thingiverse Models
Pairofpants models
Topology
 Mathematica Notebooks
Voronoi Klein Bottle
 Thingiverse Models
Topological objects

Mathematics & the Fiber Arts 
Mathematical Knitting and Crochet
 sarahmarie belcastro has a fantastic page
(here)
on mathematical knitting and mathematical fiber arts. It's my goto page for information
on this.
 Crochet Coral Reef is exhibited
at many museums and art galleries around the world. See below.
Hyperbolic Crochet Coral Reef
Left image from the Smithsonian exhibit, right image from
the Powerhouse exhibit.

