# References

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## Escher and His Work

- Bool, F.H.; J.R.Kist, J.L. Locher, F. Wierda (1982). in J.L.Locher:
*M.C. Escher: His Life and Complete Graphic Work*, Essays by B. Ernst, M.C. Escher, Harry N. Abrams. ISBN 0810981130.

Escher's life history, and the complete catalog of all of his work.

- Ernst, Bruno; M. C. Escher [1978] (2007).
*The Magic Mirror of M.C. Escher*, Taschen 25th Anniversary, Taschen. ISBN 3822837032.

Fifteen chapters, each detailing a theme in Escher's work.

- Escher, M.C. [1959] (2001).
*M.C. Escher: The Graphic Work*. Taschen. ISBN 3822858641.

A brief introduction by Escher, then 76 mediocre reproductions of Escher's artwork, along with brief discussions of each. Translated from the 1959 Dutch edition, Grafiek en Tekeningen.

- Escher, M.C. (1983).
*M.C. Escher : 29 Master prints*. Harry N. Abrams. ISBN 0810922681..

Reproductions of 29 Escher prints, and nothing more.

- Escher, M.C.; J. W. Vermeulen (1989).
*Escher on Escher: Exploring the Infinite*, Karin Ford (Translator), Harry N. Abrams. ISBN 0810924145.

A handful of letters, news clippings by or about Escher, and one longer two part lecture. Some illustrations.

- Hofstadter, Douglas (1980).
*Gödel, Escher, Bach: an Eternal Golden Braid*. Basic Books. ISBN 0465026567.

A metaphorical fugue on minds and machines in the spirit of Lewis Carroll.

- Locher, J.L (2000).
*The Magic of M. C. Escher*. Harry N. Abrams. ISBN 0810967200.

Beautiful coffee table book with large color reproductions, fold outs, and some of Escher's sketches and studies.

- Schattschneider, Doris [1990] (2004).
*Visions of Symmetry - Notebooks, Periodic Drawings, and Related Work of M. C. Escher*, 2nd, Harry N. Abrams. ISBN 0810943085.

Definitive guide to Escher's regular division of the plane.

- Schattschneider, Doris. "The Polya-Escher Connection"
*Mathematics Magazine*60 (1987): 293-198

- Schattschneider, Doris. "Escher's Metaphors"
*Scientific American*271 (Nov. 1994): 66-71

- Schattschneider, Doris. "Escher: A mathematician in spite of himself,"
*Structural Topology*15 (1989); reprinted in*The Lighter Side of Mathematics,*, MAA, 1994

- Schattschneider, Doris (2010). "The Mathematical Side of M.C. Escher".
*AMS Notices***57**: 706-718.

- Gardner, Martin (April 1966). "The Eerie Mathematical Art of Maurits C. Escher".
*Scientific American*: 110-121. Reprinted in*Mathematical Carnival.*Washington D.C.: Mathematical Association of America, 1989

## Symmetry and Tessellations

- Grünbaum, Branko; G.C. Shepard (1986).
*Tilings and Patterns*. W.H. Freeman and Company. ISBN 0716711931.

The definitive mathematical text on tessellations, crammed with beautiful black and white illustrations.

- Washburn, Dorothy K.; Donald W. Crowe (1988).
*Symmetries of Culture: Theory and Practice of Plane Pattern Analysis*. University of Washington Press. ISBN 0295970847.

Classification of symmetry groups, with examples from world cultures. The original source of the wallpaper group flowchart.

- Washburn, Dorothy K.; Donald W. Crowe (2004).
*Symmetry Comes of Age: The Role of Pattern in Culture*. University of Washington Press. ISBN 0295983663.

A collection of articles by various authors.

- Stevens, Peter S. (1981).
*Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions*. The MIT Press. ISBN 0262690888.

Large collection of patterns from various sources, organized by symmetry group.

- Weyl, Hermann (1952).
*Symmetry*. Princeton University Press. ISBN 0691080453.

Two lyrical lectures on symmetry in art and science, followed by two much more technical lectures on symmetry groups.

- Pólya, George (1924). "Uber die Analogie der Kristallsymmetrie in der Ebene".
*Zeitschrift fur Kristallographie*. The original source for the classification of the 17 wallpaper symmetry gropus, including an illustration that inspired Escher.

- Coxeter, H.S.M (June 1957). "Crystal Symmetry and Its Generalizations".
*The Transactions of the Royal Society of Canada***51**: 1-13.

- Coxeter, H.S.M. (1981). "Angels and Devils".
*The Mathematical Gardener*: 197-209 & Plate IV. Prindle, Weberr & Schmidt. Reprinted as*Mathematical recreations: A Collection in Honor of Martin Gardner*, Mineola, NY: Dover, 1998

- Davis, Dan (1997). "On a tiling scheme from M.C. Escher".
*Electronic Journal of Combinatorics***4**. Article in PostScript format

- Schattschneider, Doris (2002). "The Many Faces of Symmetry in the Works of M.C. Escher".
*Symmetry 2000*: 173-184. London: Portland Press.

- Grünbaum, Branko (2002). "Levels of orderliness: global and local symmetry".
*Symmetry 2000*: 51-61. London: Portland Press.

- Schattschneider, Doris (1997). "Escher's Combinatorial Patterns".
*The Electronic Journal of Combinatorics***4**. (online version)

Escher's "potato stamp" game for making patterns with carved square blocks.

- Grünbaum, Branko (June/July 2006). "What Symmetry Groups Are Present in the Alhambra?".
*Notices of the AMS***43**. (online version)

A careful study of patterns at the Alhambra, intended to show that*not*all 17 groups are present.

- El Said, Isssam; A. Parman (1976).
*Geometric Concepts in Islamic Art*. World of Islam Festival Publishing. ISBN 0905035038.

A manual for constructing traditional Islmaic symmetry and interlace patterns.

## Non-Euclidean Geometry

- Greenberg, Marvin (1980).
*Euclidean and Non-Euclidean Geometries*, 2nd, Freeman. ISBN 0716711036.

Focus on history, axioms, and hyperbolic geometry.

- Stillwell, John (2005).
*The Four Pillars of Geometry*. Springer.

Undergraduate text in mathematics treats projective geometry and perspective among other topics.

- Singer, David A. (1998).
*Geometry:Plane and Fancy*. Springer.

Undergraduate text in mathematics treats Euclidean, spherical, and hyperbolic geometry along with tessellations of all three.

### Circle Limits

- Coxeter, H.S.M. (1979). "The Non-Euclidean Symmetry of Escher's Picture 'Circle Limit III'".
*Leonardo***12**: 19-25, 32..

Describes how Escher created the print.

- Coxeter, H.S.M.. "The Trigonometry of Escher's Woodcut 'Circle Limit III'".
*Mathematical Intelligencer***18**: 42-46.. A reprint was made available by the AMS PDF file See also AMS page with links to images.

- Dunham, Douglas (April 2003). "Creating Repeating Hyperbolic Patterns—Old and New".
*AMS Notices*: 452-455. Algorithms for drawing a hyperbolic tessellation.

- Casselman, Bill (June 2010). How Did Escher Do It? (How did M. C. Escher draw his Circle Limit figures…). AMS website Feature Column.

## Mathematics and Art

- Field, J.V. (2005).
*Piero della Francesca. A Mathematician's Art*. Yale University Press. ISBN 0300103425. - Ernst, Bruno (1987).
*Adventures With Impossible Figures*. Tarquin. ISBN 0906212545. - Ernst, Bruno (2006).
*Impossible Worlds: Adventures With Impossible Objects/Optical Illusions*. Taschen. ISBN 3822854107.

Is this just a repackaging of two other Ernst books? - Emmer, Michele (1993).
*The Visual Mind: Art and Mathematics*. ISBN 026205048X.

Thirty-six articles by mathematicians and artists on visualization, computer graphics, symmetry, and perspective. Most reprinted from the journal Leonardo. - Peterson, Ivars (2001).
*Fragments of Infinity: A Kaleidoscope of Math and Art*. John Wiley & Sons, Inc., 207-218. ISBN 0471165581.

A recreational math book with a shallow but broad coverage of artists working with mathematical ideas. - Molderings, Herbert (2010).
*Duchamp and the Aestrhetics of Chance*. Columbia University Press. ISBN 9780231147620.

## The Fourth Dimension

- Henderson, Linda D (1993).
*The Fourth Dimension and Non-Euclidean Geometry in Modern Art*. Princeton University Press.

The definitive source on the influence of 4D on the cubists and other artists of the period.

- Banchoff, Thomas (1990).
*Beyond the Third Dimension*. Scientific American Library.

- Robbin, Tony (2006).
*Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought*. Yale University Press. ISBN 0300110391.

Discusses the influence of 4D on cubists, especially Picasso, the projection model, and spacetime physics.

### Fiction

- Abbott, Edwin A. (1880).
*Flatland: A Romance of Many Dimensions*, Dover Thrift Edition. ISBN 048627263X. Freely available online, for example at http://www.alcyone.com/max/lit/flatland/ or see wikipedia:Flatland for a list.

- Heinlein, Robert A (1941). "
*... And he built a crooked house ...*".*Astounding Science Fiction*.

Short science fiction story, available online at http://www.scifi.com/scifiction/classics/classics_archive/heinlein/heinlein1.html .

- L'Engle, Madeleine (1962).
*"A Wrinkle In Time"*.

## Recreational Math Books With Broad Coverage

- Mankiewicz, Richard (2000).
*The Story of Mathematics*. Princeton University Press.

This recreational math book has many relevant chapters, including "The Renaissance Perspective", "New Geometries", "Catching Infinity" (fractals), and "Maths and Modern Art."

- Maor, Eli (1987).
*To Infinity and Beyond*. Birkhauser.

Recreational math book with sections "Maurits C. Escher - Master of the Infinite","Tiling the Plane", and more.

## Other

- Smit, B. de, Lenstra Jr, H.W., "The Mathematical Structure of Escher's Print Gallery" In
*Artful Mathematics: The Heritage of M. C. Escher*in AMS Notices 2003. PDF file from the AMS notices

## Web Sites

### Escher Related

- The Official Escher website On this website you can find information about the use of M.C. Escher's work, a short biography, news, bibliography, links and some fun stuff like a Virtual Ride through some of his works.

- Escher in het Paleis Website of the museum in The Hague, the Netherlands, dedicated to M.C. Escher.

- M.C.Escher Mindscapes Page dedicated to Escher from the National Gallery of Canada.

- Escher and the Droste effect Website from Leiden University in the Netherlands. The website aims to visualize the mathematical structure behind Escher's Print Gallery. In this print there is a blank space in the middle and several methods are shown for completing the picture.

- Escher in the Classroom Website by Jill Britton. Great resoource for symmetry and tessellations. Materials accompany her book "Symmetry and Tessellations" (Aimed at Middle School)

### Geometry Related

- Jeff Weeks' Geometry Games Jeff Weeks’ Topology and Geometry Software. This includes the torus and klein-bottle games, Kali, Kaleidotile and more.

- Spherical Easel Spherical Easel is a program for creating interactive diagrams in spherical geometry.

- Non-Euclid A Java based program that allows one to explore hyperbolic geometry.

### Tessellations

- Tilings and Tessellations from ScienceU This site has basic tiling information, the 17 wallpaper groups (animated!) and aperiodic & Penrose tilings.

- "Complement Tiling" From The Wolfram Demonstrations Project

- "Wallpaper Groups." by Weisstein, Eric W. From MathWorld--A Wolfram Web Resource.

### Art Related

### Assorted Topics

- VisMath. The Art and Science Electronic Journal of ISIS-Symmetry.
- Fractal Geometry. An online course from Yale University. Full of great stuff, especially the Panorama of Fractals and Their Uses.
- Jackson's Fractals. An article explaining the fratal nature of Jackson Pollock's art. Includes an explanation of fractal dimension.
- Math Awareness Month - Mathematics and Art (2003). This website features an Escher-inspired hyperbolic tessellation, essays by artists and mathematicians, and other resources. See for instance:

- Penrose Stairway by Weisstein, Eric W. From MathWorld--A Wolfram Web Resource.
- MATC Electronic Bookshelf. Interdisciplinary materials including:
- Geometry in Art & Architecture by Paul Calter.
- Pattern by Pippa Drew and Dorothy Wallace.