Circle Limit III

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came".

[1] However, as Coxeter demonstrated, there is no hyperbolic arrangement of lines whose faces are alternately squares and equilateral triangles, as the figure depicts.

In the alternated octagonal tiling, the sides of the squares and triangles are hyperbolically straight line segments, which do not link up in smooth curves; instead they form polygonal chains with corners.

In Escher's woodcut, the sides of the squares and triangles are formed by arcs of hypercycles, which are not straight in hyperbolic geometry, but which connect smoothly to each other without corners.

[2] Similar tessellations by lines of fish may be constructed for other hyperbolic tilings formed by polygons other than triangles and squares, or with more than three white curves at each crossing.

[7] Euclidean coordinates of circles containing the three most prominent white curves in the woodcut may be obtained by calculations in the field of rational numbers extended by the square roots of two and three.