This course will be an opportunity to explore how various aspects of mathematics can be visualized by physical and virtual models, often in quite beautiful ways, and conversely how mathematics can be used as a tool in designing beautiful physical and virtual models.

Here are some examples that we may look at, but some choice of topics will be guided by the interests of the class members.

- Symmetry: How many different kinds of wallpaper patterns are there? How can we recognize them?
- Polyhedra: What are they, and what role do they play in art, chemistry, etc.?
- Proofs without words: How can we "see" such formulas as the sum of the first n odd integers is n squared?
- The fourth dimension: How can we visualize it?
- The Mandelbrot set: Some of us have seen it many times. What is it?
- Fractals: What are they, and how are some fractal images generated?
- Animations: What is some mathematics underlying animated films such as those produced by Pixar?

The prerequisite for the course is facility with algebra and geometry, and a willingness to learn more as needed. Calculus is not a prerequisite, but if you know it, you will find use for it! Knowledge of a particular computer programming language is not a prerequisite, but we will be learning how to use some computer programs that require some attention to logic and detail, so patience in this regard is likely to be needed.

At the end of the course it would be nice to host an exhibition!

- Homework #1
- Homework #2
- Alvey source file
- Alvey png image
- Arnold source file
- Arnold png image
- Bernhard source file
- Bernhard png image
- Fenz source file
- Fenz png image
- Foster source file
- Foster png image
- Hadaidi source file
- Hadaidi png image
- Jury source file
- Jury png image
- Kidwell source file
- Kidwell png image
- Lawless source file
- Lawless png image
- Mattingly source file
- Mattingly png image
- McKie-Miller source file
- McKie-Miller png image
- Reyes-Cabrera Blender source file
- Reyes-Cabrera png image
- Ricketts source file
- Ricketts png image

- Practicing with Videos

- Art and Music
- Beyond the Third Dimension
- Bridges The Bridges Organization focuses on connections between art and mathematics
- Bridging the Gap Between Math and Art Bridges Festival: Mathematical Connections in Art, Music and Science
- Discovering the Art of Mathematics
- Erik Demaine Computer scientist, mathematician and artist
- European Society for Mathematics and the Arts
- FlowingData "FlowingData explores how statisticians, designers, data scientists, and others use analysis, visualization, and exploration to understand data and ourselves."
- The International Society of the Arts, Mathematics, and Architecture
- Journal of Humanistic Mathematics
- Manny's Math Art "This FB page is for digital + conventional artwork of any kind that demonstrates mathematical and aesthetic qualities."
- Math and Art: The Good, the Bad, and the Pretty Course at Franklin and Marshall College
- Math and the Art of M.C. Escher
- Mathematical Imagery
- Mathematics and Art Wikipedia article
- Mathematics and Art Feature column of the American Mathematical Society
- Mathematics and Art -- So Many Connections Article by Doris Schattschneider
- Mathematics in Art and Architecture
- The Museum of Math New Museum of Mathematics in New York
- Topics in Mathematics: Art and Music Resource list from the Math Archives
- The University of Kentucky Center for Visualization and Virtual Environments
- Virtual Math Museum
- A Virtual Space-Time Travel Machine
- Visual Insight American Mathematical Society blog on visualizing mathematics
- Visualizing Math Tumblr
- Visual Mathematics

- Useful Information
- Additional Information
- Design Considerations Handout
- Polyhedra Still Life Thumbnails
- Feedback Samples

- POV-Ray -- Raytracing software
- Official Website
- Hall of Fame
- Download Page
- POV-Ray Unofficial for Mac
- Examples (from a previous course)
- More Examples (from a previous course)
- Some POV-Ray Textures and Colors
- POV-Ray Predefined Colors (Click on a color to see its name)
- More Examples
- Making Videos
- Puzzle Source File
- Puzzle Movie
- Parabolic trajectory file
- Parabolic trajectory movie
- Ripple file (uses isosurface)
- Ripple movie
- Blob file
- Blob movie
- Falling sphere file (uses switch)
- Falling sphere movie
- Spheredrops file (uses a macro)
- Spheredrops movie

- ImageToAVI For assembling a sequence of images into a movie

- SketchUp Free 3D modeling software
- Blender Powerful modeling software
- OpenSCAD Free 3D modeling software
- Documentation. See also the examples available to you when you run the program.
- Some Examples

- Autodesk 123D Collection of apps for modeling, including:
- 123D Catch Create 3D models from a set of 2D photos
- 123D Design

- GeoGebra Algebra and geometry
- Manual and tutorials
- GeoGebra Tube
- iPad app
- Planet animation GeoGebra file

- Sage General computation
- Wingeom Geometry (PC only)

- Thingiverse Models to download and print on 3D printers
- MakerBot MakerWare Software to prepare products for 3D printing
- Printrbot
- Shapeways
- MeshLab
- Meshmixer
- Smithsonian X 3D Three-D representations of artifacts to download and print

- Dancing and Fractals
- Mandelbrot Set
- Mandelbrot set Wikipedia article
- Zooming in on the Mandelbrot set Video
- mandelbrot.ggb (GeoGebra file)
- mandelbrot2.ggb (GeoGebra file)

- Fractals with POV-Ray
- Documentation
- Julia set example Source file
- Julia set example png file
- Mandelbrot set example Source file
- Mandelbrot set example png file

- Games and Puzzles
- Some Games (And there are iPad apps for some of these games)
- Some Game Boards
- Game of Set iPad App
- Game of Hex (or Nash) iPad App
- Game of Gale iPad App
- Game of Interlocked iPad App
- Mathematics of Juggling

- Music
- Animusic Virtual musical instruments
- Change Ringing. See also the book by Dorothy Sayers, The Nine Tailors

- Origami and Math
- Sonobe Origami Polyhedra
- Sonobe Assembly Guide
- Modular Origami
- Modular Origami Diagrams
- Paper Models of Polyhedra
- The Fold and Cut Problem
- MIT Course on Geometric Folding Algorithms: Linkages, Origami, Polyhedra With notes, slides, and videos
- Erik Demaine
- Metamorphosis of the Cube
- How to Fold it
- Rose Pavilion
- Hyperbolic Paraboloid
- Triangular Spiral Tilings

- Poly
- Quasicrystals and Aperiodic Tilings
- Shadows of the Intangible
- Virtual Polyhedra
- Zometool
- Many-Dimensional Geometry
- Mark Newbold's Java Stuff
- Polyhedra and Polytopes
- The Regular Convex Four-Dimensional Polytopes
- Viewing Four-Dimensional Objects in Three Dimensions
- Visualizing the Hypercube

- Interlocked iPad App in which you take puzzles apart

- Patterns
- iOrnament iPad App
- iOrnament Examples (Go to iOrnament-->gallery and iOrnament-->world exhibition)
- Superimpose
- Scott Kim's Experiencing Symmetry
- Symmetry Notes from MA111 Including Border Pattern Classification (last section)
- More Patterns
- Frieze Patterns
- Jack Schmidt's Border Pattern Maker

- Tensegrity Wikipedia article
- Tensegrity
- Kenneth Snelson
- Building Tensegrity Models
- Soda Straw Tensegrity Structures
- Tensegrity Wikispace