# 圓極限III

《圓極限III》, Circle Limit III, 1959

## 靈感

(6,4,2)三角群雙曲鑲嵌給了艾雪靈感

## 注釋

1. ^ strings of fish shoot up like rockets from infinitely far away
2. ^ fall back again whence they came
3. ^ 30°-45°-90°直角三角形是一種雙曲幾何圖形，並非平面幾何的圖形，其為一種施瓦茨三角形
4. ^ the fish move "perpendicularly to the boundary"
5. ^ 該頂點與八階三角形鑲嵌的頂點的頂點同構，即階為八個不重疊正三角形的公共頂點

## 參考文獻

1. ^ 1994 M. C. Escher《Circle Limit III》, 1959, CAordon Art-Baarn-Holland
2. Escher, as quoted by Coxeter (1979).
3. Coxeter, H. S. M., The non-Euclidean symmetry of Escher's picture 'Circle Limit III', Leonardo, 1979, 12: 19–25, JSTOR 1574078.
4. Emmer, Michele, Escher, Coxeter and symmetry, International Journal of Geometric Methods in Modern Physics, 2006, 3 (5-6): 869–879, MR 2264394, doi:10.1142/S0219887806001594.
5. Schattschneider, Doris, The mathematical side of M. C. Escher (PDF), Notices of the AMS, 2010, 57 (6): 706–718 [2014-06-17], （原始内容存档 (PDF)于2015-02-23）.
6. ^ An elementary analysis of Coxeter's figure, as Escher might have understood it, is given by Casselman, Bill, How did Escher do it?, AMS Feature Column, June 2010 [2014-06-17], （原始内容存档于2014-07-14）. Coxeter expanded on the mathematics of triangle group tessellations, including this one in Coxeter, H. S. M., The trigonometry of hyperbolic tessellations, Canadian Mathematical Bulletin, 1997, 40 (2): 158–168, MR 1451269, doi:10.4153/CMB-1997-019-0.
7. ^ Dunham, Douglas, More “Circle Limit III” patterns, The Bridges Conference: Mathematical Connections in Art, Music, and Science, London, 2006 (PDF), [2014-06-18], （原始内容存档 (PDF)于2014-07-14）.
8. ^ Coxeter, H. S. M., The trigonometry of Escher’s woodcut Circle Limit III, M.C.Escher’s Legacy: A Centennial Celebration, Springer: 297–304, 2003, doi:10.1007/3-540-28849-X_29.
9. ^ Conway, J. H., The orbifold notation for surface groups, Groups, Combinatorics & Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165, Cambridge: Cambridge Univ. Press: 438–447, 1992, MR 1200280, doi:10.1017/CBO9780511629259.038. Conway wrote that "The work Circle Limit III is equally intriguing" (in comparison to Circle Limit IV, which has a different symmetry group), and uses is it as an example of this symmetry group.
10. ^ Herford, Peter, The geometry of M. C. Escher's circle-Limit-Woodcuts, Zentralblatt fü Didaktik der Mathematik, 1999, 31 (5): 144–148, doi:10.1007/BF02659805. Paper presented to the 8th International Conference on Geometry, Nahsholim (Israel), March 7–14, 1999.
11. ^ Escher, M. C., M. C. Escher: The Graphic Work, Taschen: 10, 2001 [2014-06-19], （原始内容存档于2014-07-15）.