Editing M. C. Escher (section)

=== Infinity and hyperbolic geometry ===

[[File:Schattschneider Reconstruction of Escher's Coxeter Diagram.jpg|thumb|left|upright=1.4<!--size for very low image-->|[[Doris Schattschneider]]'s reconstruction of the diagram of hyperbolic tiling sent by Escher to the mathematician [[Harold Scott MacDonald Coxeter|Donald Coxeter]]<ref name=MathSide />]]

In 1954 the International Congress of Mathematicians met in Amsterdam, and N. G. de Bruin organised a display of Escher's work at the Stedelijk Museum for the participants. Both Roger Penrose and [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] were deeply impressed with Escher's intuitive grasp of mathematics. Inspired by ''Relativity'', Penrose devised his [[Penrose tribar|tribar]], and his father, Lionel Penrose, devised an endless staircase. Roger Penrose sent sketches of both objects to Escher, and the cycle of invention was closed when Escher then created the [[perpetual motion]] machine of ''Waterfall'' and the endless march of the monk-figures of ''Ascending and Descending''.<ref name=MathSide />
In 1957 Coxeter obtained Escher's permission to use two of his drawings in his paper "Crystal symmetry and its generalizations".<ref name=MathSide /><ref>{{cite journal |last=Coxeter |first=H. S. M. |title=Crystal symmetry and its generalizations |journal=A Symposium on Symmetry, Transactions of the Royal Society of Canada |volume=51 |issue=3, section 3 |date=June 1957 |pages=1–13}}</ref> He sent Escher a copy of the paper; Escher recorded that Coxeter's figure of a hyperbolic tessellation "gave me quite a shock": the infinite regular repetition of the tiles in the [[Models of the hyperbolic plane|hyperbolic plane]], growing rapidly smaller towards the edge of the circle, was precisely what he wanted to allow him to represent [[infinity]] on a two-dimensional plane.<ref name=MathSide /><ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 4. Mathematical artists and artist mathematicians |url=https://www.ams.org/samplings/feature-column/fcarc-art4 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref>

Escher carefully studied Coxeter's figure, marking it up to analyse the successively smaller circles{{efn|Schattschneider notes that Coxeter observed in March 1964 that the white arcs in ''[[Circle Limit III]]'' "were not, as he and others had assumed, badly rendered hyperbolic lines but rather were branches of equidistant curves."<ref name=MathSide />}} with which (he deduced) it had been constructed. He then constructed a diagram, which he sent to Coxeter, showing his analysis; Coxeter confirmed it was correct, but disappointed Escher with his highly technical reply. All the same, Escher persisted with [[hyperbolic tiling]], which he called "Coxetering".<ref name=MathSide /> Among the results were the series of wood engravings ''Circle Limit I–IV''.{{efn|See [[Circle Limit III]] article for image.}}<ref name=MathSide /> In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter".<ref name=StAndrewsCoxeter>{{MacTutor|title=Maurits Cornelius Escher|id=Escher|mode=cs1}} which cites {{cite book |author=Schattschneider, D. |contribution=Escher: A mathematician in spite of himself |title=The Lighter Side of Mathematics |editor1=Guy, R. K. |editor2=Woodrow, R. E. |publisher=The Mathematical Association of America | location=Washington |year=1994 |pages=91–100}}</ref>