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

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.
    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.
    • 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.
    *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.)
  • Schwarz P models
  • On Thingiverse
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^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 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
    • Coordinate axes on Thingiverse. This is a set of coordinate axes for 3-dimensional space, where the ends of the axes have been labeled with x, y, and z.
    • Teaching double and iterated integrals
      • Volume by no French Fries, Volume by 16 French Fries, and Volume by 64 French Fries on Thingiverse. The first model shows the volume below z=16-x^2-2y^2, above the xy-plane, and inside the square [0,2]x[0,2]. The second and third models show this volume approximated with rectangular prisms by taking the [0,2]x[0,2] square and subdividing it into 16 and 64 smaller squares. We then create a rectangular prism by choosing the height to be the function value in the center of each square. On all the models the height was scaled down by a factor of four. This is an illustration of the defintion of a double integral.
      • Volumes by iterated integrals
      • Volume: no slices, Volume: slices x constant, and Volume: slices y constant on Thingiverse. The first model shows the volume below the surface z=sin(x)cos(y), above the xy-plane and inside the rectangle [0,pi/2]x[0,pi/2]. The second and third models show the volume approximated by 8 slices, where the x-value, respectively the y-value, has been held constant at the midpoint of each sub-interval (so x=pi/32, 3pi/32, ... , 15pi/32). This is an illustration of Fubini's Theorem and interated integration.
    • Volumes for double and triple integrals
      • Wedge 1 on Thingiverse. Wedge 1 represents the volume enclosed by z=0, x=0, x=y^2, and y+z=1.
      • Wedge 2 on Thingiverse. Wedge 2 represents the volume enclosed by z=0, x=0, y=0, z=1-x^2, and y=1-x.
      • Tetrahedron 1 on Thingiverse. This tetrahedron is defined by equations x+2y+z=2, x=2y, z=0, and x=0.
      • Tetrahedron 2 on Thingiverse. This tetrahedron is defined by equations y=-6, z=0, z=x+4, and 2x+y+z=4.
      • The very interesting Monkey Saddle (z=x^3-3xy^2) on Thingiverse.
      • Bulge Head on Thingiverse. This model is the volume above the xy-plane, outside the unit sphere, and inside the cardioid of revolution rho=1+cos(phi) (in spherical coordinates).
      • Paraboloid-Sphere Intersection on Thingiverse. This model is the volume of intersection between the paraboloid z=x^2+y^2 and the sphere x^2+y^2+z^2=2.
      • Tumor model (spherical cordinates) on Thingiverse. A tumor may be modeled in spherical coordinates by rho=1+1/5 sin(mTheta)sin(nPhi). This model shows the case where m=8 and n=7.
    • Quadratic surfaces
    • Parametric curves
    • Parametric Curves
    • Intersecting cylinders
        Steinmetz surfaces
      • Two Intersecting Cylinders on Thingiverse. This model shows two right circular cylinders of equal radii intersecting at right angles.
      • Two Intersecting Cylinders - halfway on Thingiverse. This model shows half of two right circular cylinders of equal radii intersecting at right angles.
      • Three Intersecting Cylinders on Thingiverse. This model shows three right circular cylinders of equal radii intersecting at right angles.
      • Three Intersecting Cylinders - halfway on Thingiverse. This model shows half of three right circular cylinders of equal radii intersecting at right angles.
      • Steinmetz solid on Thingiverse. The Steinmetz solid is the solid common to two right circular cylinders of equal radii intersecting at right angles.
      • Steinmetz - 3 cylinders on Thingiverse. This Steinmetz solid is the solid common to three right circular cylinders of equal radii intersecting at right angles.

      General volumes and quadratic surfaces
Knots, Topology and Geometry

Knots
Three interlocking trefoil knots
Geometry Topology
Mathematics & the Fiber Arts

Mathematical Knitting and Crochet
  • sarah-marie belcastro has a fantastic page (here) on mathematical knitting and mathematical fiber arts. It's my go-to 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.