The Mathematics of 'Circle Limit III'

M.C.Escher's series Circle Limit I-IV series was inspired by his friendship with the great geometer H.M.S. Coxeter, and Escher's Wikipedia biography mentions this explicitly:

Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."

The paper in question by Coxeter is here, and demonstrates that Escher was a research mathematician in his own right:

Abstract. In M. C. Escher’s circular woodcuts, replicas of a fish (or cross, or angel, or devil), diminishing in size as they recede from the centre, fit together so as to fill and cover a disc. Circle Limits I, II, and IV are based on Poincare's circular model of the hyperbolic plane, whose lines appear as arcs of circles orthogonal to the circular boundary (representing the points at infinity). Suit-able sets of such arcs decompose the disc into a theoretically infinite number of similar “triangles,” representing congruent triangles filling the hyperbolic plane. Escher replaced these triangles by recognizable shapes. Circle Limit III is likewise based on circular arcs, but in this case, instead of being orthogonal to the boundary circle, they meet it at equal angles of almost precisely 80◦.(Instead of a straight line of the hyperbolic plane, each arc represents one of the two branches of an “equidistant curve.”) Consequently, his construction required an even more impressive display of his intuitive feeling for geometric perfection. The present article analyzes the structure, using the elements of trigonometry and the arithmetic of the biquadratic field, subjects of which he steadfastly claimed to be entirely ignorant.

Here's a paper by Douglas Dunham(pdf) that generalizes the fishy model a little further.