# 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.

To print the objects seen on this page, we used the 3D printers in the IQ center: mostly the Ultimaker 3 FDM, Stratasys uPrint SE, and Stratasys F170 3D printer.

Thanks must go to the Washington & Lee students who’ve helped develop these tools: the students of Math 383 Knot Theory in Spring 2023. Also to Emily Jaekle (’16), Ryan McDonnell (’17).

Finally, I’m eternally grateful for all of the technical assistance patiently provided by David Pfaff from the IQ Center at Washington & Lee University.

# Math and 3D printing from the internet

- My own blog Visions in Math, giving an ongoing description of all this neat stuff.
- Laura Taalman has brought 3D printing to so many in the mathematics community. Check out Laura’s
- Thingiverse page,
- Hacktastic blog where she describes her current projects,
- MakerHome blog where she described how she printed one 3D print every day for a year.

- Henry Segerman is known for his mathematical art. Check out Henry’s Shapeways page and Thingiverse page which contain a huge array of truly beautiful mathematical objects and mathematical art.
- David Bachman‘s blog Math/Art gives his meditations on 3D design using Rhino and Grasshopper.
- Jason Cantarella’s Thingiverse page contains some neat math objects as well as a huge list of energy minimizing knots and links.

# Knots

## Background and Instructions

*3D Printed Knots*talk given at the AMS Special Session on Using 3D-Printed and Other Digitally Fabricated Objects in the Mathematics Classroom at the Joint Mathematics Meeting Jan 4, 2024.- Torus knots and links
*Mathematica*notebook showing a torus with meridian and longitude curves, as well as numerous torus knots. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - Instructions for constructing smooth knots in Cinema 4D knowing a parametric equation for the knot.
- Instructions for constructing poygonal knots in Cinema 4D knowing the vertices and edges.

## Thingiverse Models

**Trefoil Knots**

- Summer 2015: (2,3) Torus knot and (3,2) Torus knot on Thingiverse.
- Fall 2014: Three Interlocking Trefoil Knots on Thingiverse.
- Laura Taalman’s Trefoil Menagerie on Thingiverse.

**Torus Knots**

- Spring 2023: T(2,11) torus knot and T(4,16) torus link on Thingiverse.
- Spring 2023: T(2,4), T(2,6), T(2,8) torus links on Thingiverse.
- Spring 2023: T(3,3), T(6,3) torus links on Thingiverse.
- Fall 2014: (5,3) Ribbon Torus Knot on Thingiverse.
- Fall 2014: (3,2) Ribbon Torus Knot on Thingiverse.

**Petal knots**

- Spring 2023: 4_1, 5_2 and 6_1 petal knots on Thingiverse.
- Spring 2023: 3_1, 5_2, 6_2 and 6_3 petal knots on Thingiverse.
- Laura Taalman’s 6_2 Petal knot on Thingiverse.

**Equilateral Stick knots**

- Vertices of these knots can be found in Thomas Eddy’s github repository summarizing superbridge and stick number research by Eddy and Clay Shonkwiler.
- Spring 2023: 3_1 and 4_1 stick knots on Thingiverse.
- Spring 2023: 5_1 and 5_2 stick knots on Thingiverse.
- Spring 2023: 6_1, 6_2, and 6_3 stick knots on Thingiverse.
- Spring 2023: 7_1, 7_2, 7_4, 7_5 and 7_7 stick knots on Thingiverse.
- Spring 2023: 9_49 and K12n630 stick knots on Thingiverse.

**Knots on the cubic lattice**

- Vertices of these knots found on Andrew Rechnitzer’s website on plotting minimal cubic lattices).
- Spring 2023: 3_1 and 4_1 lattice knots on Thingiverse.
- Spring 2023: 5_1 and 5_2 lattice knots on Thingiverse.
- Spring 2023: 6_1, 6_2, and 6_3 lattice knots on Thingiverse.
- Spring 2023: 7_1 and 7_4 lattice knots on Thingiverse.

**Other kinds of knots**

- Spring 2023: 3_1, 4_1 and 7_1 mosaic knots on Thingiverse.
- Spring 2023: The figure 8 knot is achiral models on Thingiverse.
- Fall 2014: 6_2 Knot from SeifertView on Thingiverse.
- Fall 2014: Figure 8 Seifert Surface on Thingiverse.
- 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.

# Multivariable Calculus

- John Zweck (UT Dallas) has a great series of 3D-printable models and active learning projects that accompany them. Click here for more information.
- 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 \([0,2]\times[0,2]\). The second and third models show this volume approximated with rectangular prisms by taking the \([0,2]\times[0,2]\) 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.
- 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 \([0,\pi/2]\times[0,\pi/2]\). 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 \(x=\pi/32, 3\pi/32, … , 15\pi/32\)). This is an illustration of Fubini’s Theorem and interated integration.
**Volumes for double and triple integrals**- Volumes with 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. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - 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 \(\rho=1+1/5 \sin(m\theta)\sin(n\phi)\). This model shows the case where \(m=8\) and \(n=7\).

- Volumes with triple integrals

## Quadratic surfaces

- Quadratic Surfaces
*Mathematica*notebook containing a*Mathematica*demonstration of all quadratic surfaces, and separate examples of individual surfaces. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - 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
*Mathematica*notebook containing examples of parametric curves, knots, and parametric curves arising from the intersection of two surfaces. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - Helix with equation on Thingiverse. Parametrization \(x=\cos t, y=\sin t, z=t\) from \(t=0\) to \(t=5\pi\).
- Spiral on Cone on Thingiverse. Parametrization \(x=t\cos t, y=t\sin t, z=t\) from \(t=0\) to \(t=5\pi\).
- Self-Intersecting Curve on Thingiverse. Parametrization \(x=\cos t, y=\sin t, z=1/(1+t^2)\) from \(t=-2\pi\) to \(t=2\pi\).
- T(2,3) Torus knot and T(3,2) Torus knot on Thingiverse.

## Intersecting cylinders

- Intersecting cylinders
*Mathematica*notebook showing the intersections of 2 and 3 cylinders. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - 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.

# Calculus II

- Volumes of Revolution and by Slices
*Mathematica*notebook containing many*Mathematica*Demonstrations of these ideas. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - Volumes of Revolution (
*Mathematica*notebook, giving a specific example of an area between two curves rotated about the line \(y=1\) and \(y=1.25\). (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - Class Project Fall 2016
- Instructions for Cinema 4D for Calculus II project. (Warning: the software has changed since 2016.)
- Project for Calculus II students design 3D printable models of volumes of revolution or volumes by slices.

## Calculus II Thingiverse models

- Sphere: 10 disks on Thingiverse.
- Sphere: 20 disks on Thingiverse.
- 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.

# Geometry

- 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

- Helicoids
*Mathematica*notebook. (To download file: right click on link, then click “Save Link As”, then under Formal “All Files”.) - 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 \(2\pi\).
- 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 \(2\pi\).
- 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.

# Talks & Exhibited Work

Some of the places my models have been seen.

*3D Printed Knots*talk given at the AMS Special Session on Using 3D-Printed and Other Digitally Fabricated Objects in the Mathematics Classroom at the Joint Mathematics Meeting Jan 4, 2024.*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).

*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.- The
*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.)

# 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.

## 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.
- The Maine Reef: patterns and more.
- The Gainesville Florida Reef: math, patterns and more.
- Don’t know how to crochet? There are plenty of resources out there. You can look online or, even better, go to your local yarn store for help.