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M. C. Escher. Art and science. Proceedings of the International Congress on M. C. Escher, Rome, Italy, March 26-28, 1985. (English) Zbl 0613.00008

Amsterdam etc.: North-Holland. XIV, 402 p. $ 50.00; Dfl. 140.00 (1986).
This fascinating book contains the following sections: I. Escher and Symmetry: II. Escher, Mathematics and Visual Perception; III. Escher and Geometry; IV. Escher, Cinema and Computer Graphics; V. Escher and the Physical World; VI. Escher and Art; VII. Escher and the Humanities. An article by G. A. Escher, Escher’s son, ”M. C. Escher at work” opens the volume. We will describe some contributions in the first three sections, of a more mathematical nature, but almost all the studies in the volume are interesting in a mathematical perspective too.
In the first section, H. S. M. Coxeter (Coloured Symmetry, p. 15- 33) investigates the way in which Escher colours some symmetrical patterns that fill the plane without any gaps. He shows that the underlying theory involves 3 groups: G (the symmetry group ignoring the distinction of colour), a subgroup \(G_ 1\) (the symmetry group preserving all the colours), and a finite group \(\Gamma\) which permutes the colours. In the ’black and white’ case (when \(\Gamma\) has order 2) and also in nearly all Escher’s patterns using 3 or 4 colours, the subgroup \(G_ 1\) is normal and \(\Gamma =G/G_ 1.\)
Andreas W. M. Dress (The 37 combinatorial types of regular ”Heaven and Hell” patterns in the Euclidean plane, p. 35-45) discusses the complete classification and enumeration of all regular tilings of the Euclidean plane with 2 types of tiles, say black and white ones, such that the symmetry group acts transitively on the black as well as on the white tiles, while no two black and no two white tiles have an edge in common. The various ”Heaven and Hell” designs by Escher are famous examples of such tilings.
Peter Engel (On monohedral space tilings in M. C. Escher’s work, p. 47-51) proposes a local regularity criterion to check the regularity of the plane tilings by M. C. Escher. Application of this criterion in crystallography gives a better understanding of the crystalline state of matter.
Branko Grünbaum (Mathematical Challenges in Escher’s Geometry, p. 53-67) shows that, although Escher was not a mathematician in this professional sense, in his investigations of plane tilings he anticipated several lines of inquiry that have since been developed into mathematical theories. Moreover, several of his works lead to problems in tiling theory which are only now starting to be explored, and to which we still have no complete answers.
Caroline H. MacGillavry (Hidden Symmetry, p. 69-80) shows that, when studying symmetry aspects in M. C. Escher’s graphic work, it is essential to distinguish between the two-dimensional symmetry of the print itself and the symmetry of the three-dimensional object it represents, and that of the image the print evokes in the mind.
Doris Schattschneider (M. C. Escher’s classification system for his coloured periodic drawings, p. 82-96) considers that the system of creation and classification of coloured periodic drawings developed by M. C. Escher and recorded by him in 1941-42 show him to be a mathematical pioneer. His notebooks, together with over 150 colour drawings give evidence of his surprisingly comprehensive research in the field which today is called ”colour symmetry”.
Marjorie Senechal (Escher designs on surfaces, p. 97-109) refers to the interesting patterns Escher designed for the cylinder, the Möbius band, and the sphere, in addition to his well-known tesselations of the Euclidean and hyperbolic planes. His patterns raise the intriguing mathematical problem of determining the symmetrical patterns that are possible for these and other non-planar surfaces.
G. C. Shephard (What Escher might have done, p. 111-122) remarks that Escher was intrigued by what he called ”regular divisions of the plane” (isohedral tilings) and their colourings. Both artistically and mathematically the most satisfactory colourings are those which are technically known as ”perfect”. This ”perfection” is explained and examples are given, pointing out that there are several attractive possibilities that Escher might have explored. - In the second section, Bruno Ernst (Escher’s impossible figure prints in a new context, p. 125-134) considers that the international study of impossible figures in the past decade has lead to a better understanding of the intrinsic qualities of the three impossible figure prints which Escher made. The essence of impossible figures is made clear and set in a historical context. After some remarks about ”Ascending and Descending” and ”Waterfall”, the author focuses in ”Belvedere” to show the elegant way in which Escher played with the impossible cuboid.
Richard L. Gregory (Puzzles of pictures as untouchable objects, p. 135-142) refers to some phenomena of visual perception, especially illusions of ”ambiguity” and ”paradox”. To understand and assess this aspect of Escher’s work, it is important to appreciate that any picture is a very peculiar kind of object. It is odd because, as a sheet of paper or canvas bearing lines, and perhaps regions of colour, it exists not only in the normal three dimensional world of objects that we can touch and handle; but also as bearing perceptual evidence of some other, untouchable world.
Roger Penrose (Escher and the visual representation of mathematical ideas, p. 143-157) shows that in the art of M. C. Escher can be found many illustrations of abstract mathematical ideas, such as symmetry groups, periodic and non-periodic tilings of the Euclidean and Lobachevski planes, self-referential logical concepts and paradoxes, impossible (but locally possible) objects with their relations to global (i.e. holistic) mathematical concepts.
In the third section, Kodi Husimi (Rolling a tetrahedron on the plane to produce periodic patterns of symmetry P2 and drawing dragon curves as backbones of Escher figures, p. 181-186) proposes a method of producing two-dimensionally repeated patterns on the plane. The technique is used to invent new Escher-type figures. Haresh Lalvani (Metamorphosis and cycle in curved space structures, p. 187-193) discusses relevant examples from Escher’s work. Arthur L. Loeb (Polyhedra on the work of M. C. Escher, p. 195-202) discusses how polyhedra of lower symmetry, for instance cubic, are placed by Escher inside solids of higher symmetry, usually icosahedral. Jean Pedersen (Braiding Escher models, p. 203-210) discusses the works of M. C. Escher that involve the regular (convex and non-convex) polyhedra. J. F. Rigby (Butterflies and snakes, p. 211-220) observes that in Butterflies Escher apparently tries to conceal the systematic framework on which the design is based; but the manner in which the drawing is coloured reveals clearly an underlying framework of circles that must intersect at right angles. J. M. Wills (Polyhedra in the style of Leonardo, Dali and Escher, p. 231-236) introduces polyhedra as drawn by several artists, which have additional algebraic and combinatorial properties and which are closely related to the five Platonic solids and the four Kepler-Poinsot-star polyhedra.
Reviewer: S.Marcus

MSC:

00B25 Proceedings of conferences of miscellaneous specific interest

Biographic References:

Escher, M. C.