More "Circle Limit III" Patterns

Aus de_evolutionary_art_org
Wechseln zu: Navigation, Suche


Douglas Dunham: More "Circle Limit III" Patterns. In: Bridges 2006. Pages 451–458



M.C. Escher used the Poincar ́e model of hyperbolic geometry when he created his four “Circle Limit” patterns. The third one of this series, Circle Limit III, is usually considered to be the most attractive of the four. In Circle Limit III, four fish meet at right fin tips, three fish meet at left fin tips, and three fish meet at their noses. In this paper, we show patterns with other numbers of fish that meet at those points, and describe some of the theory of such patterns.

Extended Abstract


Used References

[1] Anonymous, On the Cover, Mathematical Intelligencer, 18, No. 4 (1996), p. 1.

[2] H.S.M. Coxeter, Crystal symmetry and its generalizations, Royal Society of Canada, (3), 51 (1957), pp. 1–13.

[3] H.S.M. Coxeter, The Non-Euclidean Symmetry of Escher’s Picture ‘Circle Limit III’, Leonardo, 12 (1979), pp. 19–25.

[4] H.S.M. Coxeter. The trigonometry of Escher’s woodcut ”Circle Limit III”, Mathematical Intelligencer, 18, No. 4 (1996), pp. 42–46. This his been reprinted by the American Mathematical Society at: and also in M.C. Escher’s Legacy: A Centennial Celebration, D. Schattschneider and M. Emmer editors, Springer Verlag, New York, 2003, pp. 297–304.

[5] D. Dunham, Hyperbolic Art and the Poster Pattern:, on the Mathematics Awareness Month 2003 web site:

[6] B. Ernst, The Magic Mirror of M.C. Escher, Benedikt Taschen Verlag, Cologne, Germany, 1995. ISBN 1886155003

[7] M. Greenberg, Euclidean & Non-Euclidean Geometry, Third Edition: Development and History, 3nd Ed., W. H. Freeman, Inc., New York, 1993. ISBN 0716724464

[8] D. Schattschneider, M.C. Escher: Visions of Symmetry, 2nd Ed., Harry N. Abrams, Inc., New York, 2004. ISBN 0-8109-4308-5


Full Text

intern file

Sonstige Links