Math and the Art of M.C. Escher

A copy of the Syllabus

Our Schedule of in-class activities. This site will be a good reference to see what has been assigned, what reading you should have done / need to do, etc.

Here is a list of topics that can be covered.

A tentative schedule for Fall 2004. This is a tentative plan of the material we plan to cover. We may deviate from the plans slightly if deemed necessary.

Links to Named Prints of MC Escher

Some software we use:

Tess is a program we use in class. According to their page: " With Tess, you can quickly create attractive symmetric planar illustrations. While you draw, Tess will automatically maintain the symmetry group you have chosen; 11 rosette, all 7 frieze, and all 17 wallpaper groups are included." The link allows you to download a demonstration version of the program.

We also use The Geometer's Sketchpad. The link will allow you to download a demo version of the program. Note that a student version of the program is $39.95.
A quote: "Sketchpad is a dynamic construction and exploration tool that enables students to explore and understand mathematics in ways that are simply not possible with traditional tools—or with other mathematics software programs. With a scope that spans the mathematics curriculum from middle school to college, The Geometer’s Sketchpad® brings a powerful dimension to the study of mathematics."

To explore Spherical Geometry and the Platonic solids we use Kaleidotile. At the same page we also find the Torus and Klein Bottle games. These are nice to use when exploring glide translations, as well as when you want to explore the topology of the torus and the Klein bottle. All of these programs are available from Jeff Week's page.

To explore Spherical Geometry we use the Spherical Easel. The link provides access to the Java Applet that allows you to explore spherical drawing program.

To explore Hyperbolic Geometry we use NonEuclid. The link provides access to the Java Applet that allows you to explore the Poincare Disk Model of Hyperbolic space.

Some resources on the web:

(With thanks to Bryan Clair)

Symmetry

Symmetry groups webpage at the Journal of Online Mathematics. Has examples of all plane symmetry groups (Rosette, Frieze, Wallpaper).
David Joyce's Wallpaper Groups
Traditional Japanese examples of the wallpaper symmetry groups

Tessellations

A good basic reference on polygons and related concepts.
The official M.C.Escher website has most of the symmetry drawing online. Go to Galleries, then click on the symmetry drawings.
Science U's Tilings and Tessellations page has basic tiling information, the 17 wallpaper groups (animated!) and aperiodic & Penrose tilings.
A Java applet for creating tessellations.
Tessellations by pentagons have an interesting history. There are currently 14 known ways to tessellate with convex pentagons, but there might be more ways still undiscovered. Here's a brief history of the discoveries, and the home page of Marjorie Rice.
MathWorld has a good definition and pictures of dual tessellations.
Jos Leys crated dome Wacky Wallpapers. He has also done more geometric Tilings and some Penrose Tilings (Aperiodical).

Self-Similarity and Fractals

Escher and the Droste effect - an excellent site with an in-depth analysis of Escher's Picture Gallery.
Fractal Geometry online course from Yale University. Full of great stuff, especially the Panorama of Fractals and Their Uses.
Fractals by Jos Leys. This site is called "The Kleinian Galleries". More can be found under Hyperbolic Tilings. These are more geometric.
The fractal nature of the art of Jackson Pollack is described in Jackson Pollock - Mathematician by Jennifer Quellette.
What a fractal dimension is is explained in Jackson's fractals.

Spherical and Hyperbolic Geometry

There's a cool java applet that lets you draw triangles on a sphere, then rotate the sphere. It also talks about parallel transport, if you're interested. It's EggMath: Spherical Geometry
An online reference for The Geometry of the Sphere.
A page with nets to make Paper Models of Polyhedra, in particular the Platonic solids.
A page with java applets that display the five platonic solids and duality
The Cabri Geometry hyperbolic geometry website has an in depth discussion of hyperbolic geometry with lots of cool Java applet demonstrations.
Hyperbolic geometry Java applet lets you draw and drag pictures in the Poincare disk model.
David Joyce's Hyperbolic Tessellations
Douglas Dunham's in-depth analysis and reconstruction of Escher's Circle Limit series, which also has a nice discussion of hyperbolic tessellations.
Spherical Tessellations by Jos Leys.

The Fourth Dimension

Flatland: A Romance of many Dimensions by Edwin A. Abbott can be read online.
Robert Heinlein's short story And He Built a Crooked House.
Hypercube visualizations: Rotation applet and 4D game. Nice rotation pictures. Cross your eyes for a stereoscopic visualization/rotation applet.
Spirits, Art, and the Fourth Dimension

Mathematics and Art

Paul Calter's Dartmouth course Geometry in Art & Architecture.
A brief history of perspective, with illustrations.
The Math Association of America's Mathematics and Art page features interesting articles by both mathematicians and artists.
The Dutch Art and Math page has a good galerij.

Knot Theory

Jos Leys has a page of Celtic Knots.
A website devoted to Knot Theory.
Another good source is this page: Untangling the Mathematics of Knots.

Artists

George W. Hart is a New York artist who does geometric sculpture. Also see his Pavilion of Polyhedreality and the Encyclopedia of Polyhedra
Jos Leys has a fantastic collection of computer generated geometric art using fractals, including excellent examples of hyperbolic geometry, impossible figures, and tessellations.
Helaman Ferguson is well known sculptor with mathematical themes.
Crockett Johnson is well known for his children's books, but also created a series of geometric paintings.
Makoto Nakamura uses tessellations in his artwork.
Istvan Orosz's work involves visual paradoxes.
Dick Termes does paintings on spheres.

Student Work:

Tessellations Fall 2004

Tessellations from previous semesters

Non-euclidean Geometry

The 4-th Dimension

Student work submitted to other professors (in related courses):
Dr. Clair: MT A 124 Tessellation Projects
Dr. Clair: MT A124 Student Art Projects
Dr. Blyth: MT A 125 Project on the 4th dimension and Fractals/Chaos

EXAMS

Here are some examples of exams from pevious years:

Studyguide Exam I
Exam I

Studyguide Exam II
Exam II

Studyguide Final
Final

Seminar : Mathematics and the Art of M.C. Escher (a SLU 2000 course)

Brochure for Mathematics and the Art of M.C. Escher

Mathematics and Art (This page is under construction)

Symmetry drawings