Sphere Spirals

1956–1960

The Penroses and “impossible figures”

“House of Stairs” (1951) attracted the interest of the mathematician Roger Penrose and his father, the biologist Lionel Penrose; in 1956 they published a paper on impossible objects and later sent Escher a copy. He replied, admiring their continuously rising flights of steps, and enclosed a print of “Ascending and Descending” (1960); he went on to use the Penrose triangle repeatedly, notably in “Waterfall” (1961).

1957–1959

Coxeter and “Circle Limit”

In 1957 the geometer Donald Coxeter, with Escher’s permission, used two of his drawings in a paper on crystal symmetry and sent him a copy; the figure of a hyperbolic tessellation, with tiles shrinking rapidly towards the edge of a circle, “gave him quite a shock” — here was a way to represent infinity on a plane. He called his experiments “Coxetering”; they gave the series of wood engravings “Circle Limit I–IV”, and in 1959 Coxeter concluded that they were accurate to the millimetre.

1958–1959

Books on tessellations and on his own graphic work

In 1958 came the book “Regelmatige vlakverdeling” (“Regular Division of the Plane”), on the systematic build-up of mathematical designs, and in 1959 “Grafiek en tekeningen” (“Graphics and Drawings”), in which Escher himself commented on 76 works. After reading it in 1959 Albert Flocon began a correspondence with the artist and wrote, with André Barre, the book “La Perspective curviligne”.

August – October 1960

Lectures on symmetry: Cambridge, Ottawa, MIT

In 1959 Escher met the chemist and crystallographer Caroline MacGillavry and at her invitation gave a lecture on symmetry at the international crystallographic conference in Cambridge in August 1960. On 29 August he sailed for Vancouver, and in October lectured in Ottawa and at the Massachusetts Institute of Technology.