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Mathematical Visualization

Surfaces and volumes from Multivariable Calculus.
Surfaces and volumes from Multivariable Calculus.

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 TaalmanHenry SegermanDavid Bachman Jason Cantarella, George Francis, and the late Bill Thurston.

Various models of topological objects
From Math 341 Topology Fall 2014

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

Spheres approximated by disks.
Spheres approximated by disks.

Knots

Background and Instructions

Thingiverse Models

Three interlocking trefoil knots
Three interlocking trefoil knots

Trefoil Knots

Torus Knots

Collection of torus knots and links
Torus knots and links

Petal knots

Collection of petal knots
Petal knots

Equilateral Stick knots

Knots on the cubic lattice

Collection of knots on the cubic lattice
Knots on the cubic lattice

Other kinds of knots

Seifert Surface
Seifert Surface
 
Volumes from Multivariable Calculus.
Volumes from Multivariable Calculus.

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

Volumes by French Fries
Volumes by French Fries
  • Volume by no French FriesVolume 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.

    Volumes by iterated integrals
    Volumes by iterated integrals
  • Volume: no slicesVolume: 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 \(n=7\).
Quadratic Surfaces
Quadratic Surfaces

Quadratic surfaces

Parametric Curves

Parametric curves
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.

    Steinmetz surfaces
    Steinmetz surfaces
  • 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

Volume: 16 cylindrical shells
Volume: 16 cylindrical shells
  • 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

Calculus II Thingiverse models

Volumes by slices
Volumes by slices
  • 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 bowls
    Strange bowls
  • Strange Bowl: smoothStrange 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

Schwarz P models
Schwarz P models

Topology

Topological objects
  • 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 \(2\pi\).

    Voronoi Klein Bottle
    Voronoi Klein Bottle
  • 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.

    Tall hyperboloid of one sheet.
    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.
  • 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.

    Pair-of-pants models
    Pair-of-pants model
  • 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

Hyperbolic Crochet Coral Reef

Crochet coral reef: Smithsonian
Crochet coral reef: Smithsonian
  • 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.
Crochet coral reef from the Powerhouse exhibit.
Crochet coral reef from the Powerhouse exhibit.