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,
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 First-Year 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
- 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.
Tall Hyperboloid of one sheet
- Technical Tools for 3D printing, Joint Math Meetings, MAA IPS Atlanta GA, January 5, 2017. About 30 different objects from my Calculus II, Multivariable Calculus, Geometry & Topology, and Knots & Links collections.
- Unknot Conference III, at Dennison University, July 31 - August 3, 2016. About 30 different objects from my Calculus II, Multivariable Calculus, Geometry & Topology, and Knots & Links collections.
- Illustrating Mathematics, ICERM (Institute for Computational and Experimental Research in Mathematics) at Brown University, June 27 - July 1, 2016. About 25 different objects from my Calculus II, Multivariable Calculus, Geometry & Topology, and Knots & Links collections.
- Mathematics & 3D printing colloquium talk, April 2017 (at Vassar College, and San Diego State University). Download here (11 MB).
- Taping Shape* exhibit (January 30 - September 5, 2016) at the Rueben H. Fleet Science Center in San Diego California.
*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587). (The InforMath project is a partnership between San Diego State University and several museums at the Balboa Park, including the Reuben H. Fleet Science Center.)
- Pair-of-pants and bent pair-of-pants surfaces, with caps and rings.
- Schwarz P triply periodic minimal surface, and soap film frame for the Schwarz P minimal surface.
Schwarz P models
- On Thingiverse
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
Volume: 16 cylindrical shells
- 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^2-x^3 and the x-axis is rotated about the y-axis 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 cross-sections 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 cross-sectional slices.
- 20 Equilateral triangles on a circular base on Thingiverse. A solid has a circular base of radius 1. Parallel cross-sections 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 cross-sectional slices.
- 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.
- 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
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
- Thingiverse Models