艾雪在1936年訪問了在西班牙格拉納達與阿罕布拉之後,就對平面鑲嵌有了極大的興趣[4][5]並從他1937年的作品《變形I》開始,人類和動物鑲嵌圖就成為他作品的材料之一[5]。直到1958年艾雪寄了一封信給H. S. M.考克斯特,艾雪寫道他的靈感來自於考克斯特的文章《晶體對稱性及其應用》(Crystal Symmetry and its Generalizations),他開始了一系列圓極限作品的創作[3][4]。考克斯特的圖描繪了由30°-45°-90°直角三角形[註 3](羅氏三角形)完成的雙曲平面鑲嵌,該鑲嵌可被解釋為描繪鏡射和鏡射準線的基本域[6]。創作了圓極限III的隔年,艾雪又以(6,4,2)三角群的八角化六階正方形鑲嵌再創作了圓極限IV——天堂和地狱,為圓極限系列的最後一件作品。
^ 4.04.1Emmer, 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.
^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.
^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.
^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.