## Shortcuts

- My blog: Visions in Math.
- My Thingiverse page.
- Talks and Exhibits …..
- Instructions for making models …..
- Calculus models ….
- Geometrical & Topological models ….
- Mathematics and the Fiber Arts….

## 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, David Bachman , Jason Cantarella, George Francis, and the late Bill Thurston.

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.

**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 pagewhich 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.

*WINRS (Women’s Intellectual Network Research Symposium)*UVA Charlottesville VA, September 15, 2018.- 3D Printing and Illustrating Mathematics talk (10.7 MB)
- About 30 different objects from my Calculus II, Multivariable Calculus, Geometry & Topology, and Knots & Links collections.

*Taping Shape: Why Knot?*exhibit (February 22 – September 30, 2018) at the Rueben H. Fleet Science Center in San Diego California. I co-developed this exhibit with Ashanti Davis, which explores knot theory through a giant trefoil knot made out of packing tape that you can walk inside of, as well as hands-on interactive exhibits.*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.- Cinema 4D and Calculus models talk (10 MB).
- Cinema 4D Interface handout.
- Introduction to Cinema 4D: with basic commands, and tips on fixing meshes.

*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).- 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.
*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.)

- On Thingiverse
- April 11, 2016. Some of my designs feature in glitchpuddings DenneDesigns collection.
- April 7, 2016. Some of my knots feature in glitchpuddings Very Knotty collection.
- March 2016. My quadratic surfaces collection was a featured item on the main Thingiverse page.
- March 2016. On the
*Find People to Follow*list. - As one of the Thingiversity educators.

## Instructions

Please let me know if you have corrections or improvements for these instructions. Thanks!

**Creating models using Mathematica and Cinema 4D**

*Warning: these instructions were last checked in 2016. Some of the software may have changed since then.*

**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.

**Using the Printers**

- MakerBot 2 and 2X Manuals and Guides from the MakerBot website.
- A YouTube clip on how to unclog a MakerBot Replicator 2 nozzle.
- FormLabs Manuals and Guides from the FormLab website.

*Warning: these instructions were last checked in 2016. Some of the software may have changed since then.*

- Our reminder instructions on how to set up and print with the MakerBot 2X printer.
- Our reminder instructions on how to set up and print with the FormLabs Form 1+ printer.

## Calculus

**Calculus II**

*Mathematica*Notebooks- Volumes of Revolution and by Slices
*Mathematica*notebook containing many*Mathematica*Demonstrations of these ideas. - Volume of Revolution
*Mathematica*notebook, giving a specific example of an area between two curves rotated about the line y=1 and y=1.25. - Two Intersecting Cylinders
*Mathematica*notebook.

- Volumes of Revolution and by Slices
- Class Project
- Introduction to Cinema 4D for Calculus II project.
- Calculus II Project: students design 3D printable models of volumes of revolution or volumes by slices.

- 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.

**Multivariate 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- Parametric Curves
- 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 definition 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

Hyperbolic Paraboloid – round- Paraboloid (z = x^2 + y^2) and Elliptic paraboloid (z = x^2/4 + y^2/7) on Thingiverse.
- Ellipsoid 1 (x^2/25 + y^2/16 + z^2/4 = 1) and Ellipsoid 2 (x^2/4 + y^2/6 + z^2/3 = 1) on Thingiverse.
- Hyperbolic paraboloid (x^2 – y^2 = 2z) on Thingiverse.
- Hyperbolic paraboloid – round z= x^2 – y^2 inside x^2+y^2=1) on Thingiverse.
- Hyperboloid of one sheet (x^2 + y^2 – z^2 = 1) on Thingiverse.
- Hyperboloid of one sheet – tall (x^2/4 + y^2/4 – z^2/9 = 1) on Thingiverse.
- Hyperboloid of two sheets (-x^2 – y^2 + z^2 = 1) on Thingiverse.
- Cone (x^2 + y^2 = z^2) on Thingiverse.
- Not a quadratic surface, but still the very interesting Monkey Saddle (z=x^3-3xy^2) on Thingiverse.
- Henry Segerman’s collection of quadratic surfaces on Shapeways.

Parametric curves - Parametric Curves
- Helix with equation on Thingiverse. Parametrization x=cos(t), y=sin(t), z=t from t=0 to t=5pi.
- Spiral on Cone on Thingiverse. Parametrization x=t cos(t), y=t sin(t), z=t from t=0 to t=5pi.
- Self-Intersecting Curve on Thingiverse. Parametrization x=cos(t), y=sin(t), z=1/(1+t^2) from t=-2pi to t=2pi.
- (2,3) Torus knot on Thingiverse.
- (3,2) Torus knot on Thingiverse.

- 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**

*Mathematica*Notebooks- Torus Knots
*Mathematica*notebook showing a torus with meridian and longitude curves, as well as numerous torus knots.

- Torus Knots
- Thingiverse Models
- (2,3) Torus knot on Thingiverse.
- (3,2) Torus knot on Thingiverse.
- 6_2 Knot from SeifertView on Thingiverse.
- Figure 8 Seifert Surface on Thingiverse.
- Three Interlocking Trefoil Knots on Thingiverse.
- (5,3) Ribbon Torus Knot on Thingiverse.
- (3,2) Ribbon Torus Knot on Thingiverse.

Seifert Surface- Jason Cantarella’s Prime Knots collection on Thingiverse.
- Jason Cantarella’s Prime Links collection on Thingiverse.
- Jason Cantarella’s Composite Knots and Links collection on Thingiverse.

**Geometry**

- Thingiverse Models
Pair-of-pants models
- Pair-of-pants surface, bent pair-of-pants surface, caps and rings.
- Schwarz P surface #1 triply periodic minimal surface.
- Schwarz P surface # 2 triangulated triply periodic minimal surface.
- Schwarz P surface – approximation triply periodic minimal surface.
- Soap film frame for the Schwarz P minimal surface.

**Topology**

*Mathematica*Notebooks- Helicoids
*Mathematica*notebook.

Voronoi Klein Bottle- Helicoids
- Thingiverse Models
- Helicoids: half and full twist on Thingiverse. This is parametrized by equations x(t)=u.Cos(t), y(t)=u.Sin(t), and z(t) = 2t/3, where the parameter u goes between -1 and 1. The parameter t goes between 0 and Pi or 2Pi.
- Helicoid 2 on Thingiverse. This is parametrized by equations x(t)=u.Cos(t), y(t)=u.Sin(t), and z(t) = 2t/3, where the parameter u goes between 0.25 and 1.25. The parameter t goes between 0 and Pi or 2Pi.

- kitwallace has a customizable Mobius strip on Thingiverse. I’ve used it to create Mobius Strip – 5 half-twistsand Mobius Strip – 2 half-twists (the latter is of course an annulus).
- MadOverlord’s Voronoi Klein Bottle on Thingiverse.
- pkafin’s Half Klein Bottle on Thingiverse.

## Mathematics & the Fiber Arts

**Mathematical Knitting and Crochet**

- sarah-marie belcastro has a fantastic page 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**

- Crochet Coral Reef is exhibited at many museums around the world, from the Smithsonian museum to the Powerhouse Museum in Sydney Australia. The exhibits combine hyperbolic geometry, crochet, art and ecology.
- Roanoke Valley Reef: a satellite reef of the hyperbolic crochet coral reef project. The exhibit was held in Roanoke College’s Olin Gallery in January-March, 2013.
- The Institute for Figuring an organization supporting the crochet coral reef project and other math & art projects.
- TED talk by Margaret Wertheim “The beautiful math of coral”.
- The Maine Reef: patterns and more.
- The Gainesville Florida Reef: math, patterns and more.
- Don’t know how to crochet? No worries!
- There are many websites showing you how to crochet; for example this one, this one or this one.
- There are many YouTube lessons on how to crochet; for example this one or this follow up lesson.

Crochet coral reef from the Powerhouse exhibit.