Math & the Art of M.C. Escher: Web Resources

Generally Useful

• The official M.C.Escher website.
• The perfect site for this class has information on symmetry, tessellations, Escher, and much more. It's Totally Tessellated

Technique

• The MOMA's excellent flash demonstration of various printing techniques: What Is a Print?

Symmetry

• Chaim Goodman-Strauss has a great discussion of symmetry and topology at Symmetry and the Shape of Space
• Kali is a program that lets you draw symmetrical patterns based on any of the 17 wallpaper groups, any of the 7 frieze groups, and several rosette groups. There is a Java Version that will run in your web browser. Free downloads of Mac and Windows versions are available from Jeff Weeks' Geometry Games website.
• A more sophisticated shareware program for drawing with symmetry is Peda.com's Tess.
• A page of symmetry group resources, especially good for frieze symmetry.
• Hop's animated 17 Wallpaper Groups.
• Traditional Japanese examples of the wallpaper symmetry groups.
• David Joyce's Wallpaper Groups
• The Symmetry at the St. Louis Cathedral website.

Tessellations

• A good basic reference on polygons and related concepts.
• Science U's Tilings and Tessellations page has basic tiling information, the 17 wallpaper groups (animated!) and aperiodic & Penrose tilings.
• A Java applets for tessellating with quadrilaterals and creating tessellations with other shapes.
• 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.
  The situation for convex hexagons is similar.
• MathWorld has a good definition and pictures of dual tessellations.

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.

Spherical and Hyperbolic Geometry

• Kaliedotile is an excellent program for visualizing tessellations in Euclidean, Spherical, and Hyperbolic Geometry. Free downloads of Mac and Windows versions are available from Jeff Weeks' Geometry Games website.
• Spherical Easel is a web applet for interactive spherical geometry. It lets you draw points, lines, and circles on the sphere, measure triangles, and apply symmetries.
• 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 Institute For Figuring has an excellent discussion of hyperbolic space, including a cool gallery of crochet hyperbolic space models
• NonEuclid is Java software for interactively creating ruler and compass constructions in both the Poincare disk and the upper half-plane models of hyperbolic geometry.
• 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
• Don Hatch has a large collection of hyperbolic tessellations, as well as an excellent java applet that lets you draw them and move them around.
• Douglas Dunham's in-depth analysis and reconstruction of Escher's Circle Limit series, which also has a nice discussion of hyperbolic tessellations.

Depth and Perspective

• Perspective applet that draws a cube in two-point perspective.
• A possible interpretation of Escher's Three Spheres I as a solid model: Three Spheres (side view) .
• Audi commercial with Escher-like impossible scenery.

The Fourth Dimension

• 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

Op Art

• Michael Bach's 60 Optical Illusions. A well done survey of many known visual phenomena.
• Akiyoshi's illusion pages. The state of the art in optical illusion.

Mathematics and Art

• Dartmouth's "Math Across the Curriculum" electronic bookshelf. Terrific resource, featuring:
  • Paul Calter's Geometry in Art & Architecture
  • Pippa Drew and Dorothy Wallace's Pattern
• 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.

• 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.
• Juilan Beever does drawings on pavement that pop into 3D when viewed from the right angle.

More Links

• Jill Britton has written a number of fun books for teaching K-12 mathematics, and has two outstanding links pages with a wide variety of topics. Page 1, Page 2 .