## Books by Doris Schattschneider

 Below we present extracts from reviews of three of Doris Schattschneider's books. We present them in chronological order:

1. M C Escher Kaleidocycles (1977), by Doris Schattschneider and Wallace Walker.

1.1. Review by: Douglas Quadling.
The Mathematical Gazette 62 (421) (1978), 217-219.

The result is a splendid set of models which I found it fascinating to construct from the cardboard nets provided in the package. Least successful are the tetrahedron and the cube, whose faces meet at too sharp an angle to give an adequate idea of the flow of the design; but the octahedron, dodecahedron and icosahedron are a delight to handle, and in each case the symmetries of the design point attention to certain natural symmetries of the solid on whose surface it is imposed. Even more attractive are the kaleidocycles - cleverly dimensioned so that rotation is just possible; the impulse to rotate these is irresistible, and as one does so the dynamic possibilities of Escher's designs - hard to bring out on flat paper - leap out of the model. Marrying Escher to the rotating rings was a stroke of real inspiration. Colouring the models introduces a new and intriguing range of problems, since the colours used in the flat designs cannot be taken over directly into three dimensions. ... The book which accompanies the nets not only introduces the reader to Escher's designs, but also takes him through the various mathematical problems which the authors had to solve. This offers a painless introduction to the regular polyhedra and some of the relationships between them (e.g. the cubes inscribed in the dodecahedron), to symmetry in two and three dimensions, to central and orthogonal projection. ... I thoroughly enjoyed making the models-all the hard work has been done for you, so that all you have to do is fold and glue - and was amazed to see the patterns emerging. If you have a teenage godchild who has a lively curiosity and enjoys mathematics, then your 1978 Christmas present problem is solved.

1.2. Review by: Lee Dembart.
Los Angeles Times (27 May 1988).

When the Dutch artist Maurits C. Escher died in 1972 at the age of 74, he left behind a body of work unique in its imaginative power and creative design. Escher's compelling and fascinating drawings combine mathematics and art in a way never done before or since. Years after I first saw his work, I still look at it in amazement. Escher captures and holds my attention as few other artists can. ... Now Doris Schattschneider and Wallace Walker have produced a remarkable activity book that carries Escher's two-dimensional work into three dimensions, still full of surprises at every turn. ... In the 1950s, Walker, one of the authors of this book, invented "kaleidocycles," which are three-dimensional rings put together in such a way that they can be infinitely rotated in on themselves through a central hole, which can be as small as a point. The word kaleidocycle was coined from the Greek kalos (beautiful) plus eidos (form) plus kyklos (ring). These are extremely interesting forms in their own right. The upshot of all this is that Schattschneider, a mathematician at Moravian College in Bethlehem, Pa., put together the two-dimensional drawings of Escher with the three-dimensional shapes of Walker to create objects of sublime and uncommon beauty. Included in this set is a slim book that explains the theory of Escher and Walker along with 17 models that are easily assembled and glued together to bring the theory to life. "The kaleidoscopically designed geometric forms in this collection are a continuation and extension of Escher's own work," Schattschneider writes. "Your involvement is required also! A casual glance cannot reveal the surprises to be discovered in Escher's prints. So, too, the secrets to be discovered in our models are only revealed by your creating the forms, examining them, and yes, playing with them!"

1.3. Review by: Barbara Cain.
Mathematics Teaching in the Middle School 2 (6) (1997), 443.

M C Escher Kaleidocycles is a collection of seventeen models that begin as two-dimensional adaptations of Escher's prints and become three-dimensional models of the Platonic solids, a cuboctahedron, and eleven kaleidocycles. The kit also includes a fifty-seven-page book containing many Escher illustrations, instructions for assembling the models, notes on geometric solids and kaleidocycles, and an examination of patterns and colour in Escher designs. In the text, Schattschneider defines a kaleidocycle as "a 3-dimensional ring made from a chain of tetrahedra. ... The hinges of the chain allow the ring to be turned through its centre in a continuous motion." With each turn of the beautifully coloured models, Escher designs appear that resemble the tessellating patterns so familiar to students and teachers.

1.4. Review by: John M Jensen.
The Mathematics Teacher 90 (5) (1997), 413.

'M C Escher: Kaleidocycles' represents a fascinating marriage of the geometry of planar tessellation that Escher used in his work with the properties of these tessellations when applied to the solid geometry of polyhedra. The models are packaged as pre-cut templates to be assembled. When the nets are folded and glued, the solids formed are complex, symmetric, dynamic, and, with the enhancement of Escher's designs, quite attractive. ... The illustrated book that accompanies the models is an excellent source of information about Platonic solids, chains of solids, and Escher's art and design. In all, this package is instructive and engaging and successfully brings to life - and cleverly extends - the fact and fancy of Escher's work.

2. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M C Escher (1990), by Doris Schattschneider.

2.1. Review by: Marjorie Senechal.
Mathematical reviews MR1189799 (93i:52001).

M C Escher (1898-1971) was a graphic artist who insisted that he was not a mathematician, yet whose self-consistent and impossible worlds and regular divisions of the plane both displayed and captured the mathematical imagination. This book is a thorough analysis of Escher's work on plane tessellations, based on his own notebooks of drawings and classification schemes. Though mathematicians have delighted in finding group theory incarnate in Escher's work, it is clear from the notebooks that Escher's mathematical investigations led him in other directions. The development of Escher's ideas is carefully traced, the influence of his work on others, and vice versa, is discussed, and all of the notebook drawings are presented in full colour. Doris Schattschneider has written the Escher book for mathematicians.

2.2. Review by: Douglas J Dunham.
The American Mathematical Monthly 99 (1) (1992), 78-81.

"How did he do it?" is a question raised in Visions of Symmetry that comes naturally to mind when one views the art of M. C. Escher (1898-1972). Most of his work after 1936 had a distinctly mathematical flavor, mainly dealing with periodic patterns of the Euclidean plane. Visions of Symmetry explains "how he did it" - i.e. how he created periodic patterns - both with biographical information and with colour reproductions of Escher's 1941-42 "theory" and "abstract motif" notebooks. The centrepiece of Visions of Symmetry which illustrates what Escher did, is the superb full colour reproduction of all 137 of Escher's periodic patterns from his "regular division drawings" notebooks (plus 14 additional patterns not in the notebooks). The large format of Visions of Symmetry (about 280 x 230 mm) displays Escher's patterns at nearly full scale. Visions of Symmetry is the only book in which all of these patterns are reproduced, dozens of them appearing in print for the first time. Finally, Visions of Symmetry contains notes on all 137 + 14 patterns and a separate chapter that shows how Escher used the patterns in his prints. In addition to the periodic patterns, Visions of Symmetry, contains 200 other Escher illustrations (mostly in colour), three indexes of drawings, a bibliography, and a concordance. For the Escher fan, Visions of Symmetry fills a gap in the literature by showing all of his notebook patterns, answering the question "how did he do it?", and relating the patterns to his prints. For the person interested in tilings and patterns, Visions of Symmetry provides many beautiful examples ... Escher's coloured periodic patterns can even be used to visually illustrate elementary concepts in group theory ...

2.3. Review by: Paul Garcia.
The Mathematical Gazette 75 (473) (1991), 366-367.

This is an awe-inspiring book. It is intended to answer the question, described by the author as "irrepressible", namely, "how did he do it?". Anyone who has looked at Escher's work must surely have wondered how to create such intricate and complex designs. Doris Schattschneider has been fascinated by Escher's work since 1976, and was a co-creator of the well-known Kaleidocycles. She is a professor of mathematics in America, and this new book is clearly written from a mathematician's point of view. ... There is a lot in the book which will be new to even devoted Escher fans, and as a source book it will be invaluable to anyone interested in tilings and patterns. I am very glad to have a copy. But does it answer the question posed on our behalf by Doris Schattschneider, "how did he do it?". Here I must be equivocal; it does, and yet it doesn't. It does, in the sense that it makes clear the underlying mathematics of the patterns and would enable a reasonably good draughtsperson (perhaps) to produce some nice periodic drawings. But it doesn't, in the sense that no amount of mathematics would enable an ordinary mortal to produce a work of art like, say "Metamorphosis II" or "Day and Night". The artist in Escher has injected something into those pictures which transcends the mathematics, and which no mathematics could ever hope to emulate.

2.4. Review by: Michele Emmer.
Leonardo 25 (3/4), Visual Mathematics: Special Double Issue (1992), 389.

"How did he do it? ... I still wanted to know how Escher did it. And that is what this book is about." In 1985 at the conference M C Escher: Art and Science, Doris Schattschneider presented a paper entitled "M C Escher's Classification System for his Colored Periodic Drawings," in which she outlined the classification scheme set down in Escher's notebook of 1941-1942. She wrote: "Escher, despite his protests, was in fact a mathematical pioneer, he could even be termed a mathematical researcher." I do not know how many mathematicians will agree with this judgment - not many I suppose - but the really important thing is that not being a mathematician allowed Escher to pursue his own path to the organization of a complex body of knowledge that was important for his creative work. Schattschneider was given permission to reproduce a small number of pages from Escher's original notebooks. In 1976 she saw, for the first time, not only the notebooks, but also Escher's sketchbooks, folio drawings and other archival materials, which were owned by the Escher Foundation, with the Haags Gemeentemuseum serving as custodian. In 1980, the Escher Foundation was dissolved, and most of Escher's work was sold to various parties. Thus his work on the regular division of the plane is now dispersed. If the author's first reason to write this book was to understand how "Escher was able to do it", the second was to attempt to reunite that body of work and tell its story from Escher's point of view.

2.5. Review by: J Kevin Colligan.
The Mathematics Teacher 86 (8) (1993), 695.

This book covers geometry and the art of M C Escher, but it is about creativity and imagination. A pleasure for mathematician and art lover alike, 'Visions of Symmetry' recounts the early influences that directed Escher in his life's work ... The book contains the geometric roots of Escher's work, outstanding graphics, readable type, and large margins for personal notes. This book is exceptionally well orchestrated, readable, and engrossing. For the mathematician, the book offers a view of the geometric underpinnings of Escher's lifelong fascination with regular division of the plane. For those who appreciate Escher at the aesthetic level, it presents a view of a real person and his love affair with his work. Equally at home in a mathematics library, an art department, or on your coffee table, this book is destined to be a classic and the definitive reference on Escher. In a 1960 lecture in England, Escher closed by saying, "Science and art sometimes can touch one another, like two pieces of the jig saw puzzle which is our human life." This book sits on the boundary between mathematics and art, as did Escher. In fact, this book supports the argument that no such boundary exists; rather, the two disciplines coexist and intermingle, enriching both.

2.6. Review by: John Galloway.
New Scientist (2 March 1991).

Many books have been written about Escher's art. None has approached 'Visions of Symmetry' for its scope, scale and sumptuousness. The sheer beauty and ingenuity of the pictures keep you turning the pages as though the book were a collection of detective stories whose plots are brilliantly organised patterns.

2.7. Review by: Editors.
New Scientist Magazine 1848 (21 November 1992).

Inspired by notions of symmetry in the work of mathematicians such as L C Penrose and the non-representational art of Islamic Spain, Maurits Escher 'transformed the spare skeletal beauty of mathematical theory into ornament of the highest order'. Now available in paperback, Doris Schattschneider's 'Visions of Symmetry' will illuminate and entertain a wide audience from mathematicians, chemists to people merely fascinated by his work.

3. M C Escher: Visions of Symmetry (2004), by Doris Schattschneider.

3.1. Review by: Laurence Goldstein.
Print Quarterly 22 (2) (2005), 225-227.

If any text can be called definitive, this one certainly can. It is a revised and expanded edition of Schattschneider's justly acclaimed 'Visions of Symmetry: Notebooks, Periodic Drawings and Related Work of M C Escher' (1990), the expansion consisting of an updated bibliography of books, articles, lectures, letters, notebooks, book illustrations and pamphlets by Escher on the regular division of the plane and comprehensive list of the secondary literature, movies, software and selected websites, together with an afterword containing information on a newly discovered link between Escher and the Hungarian mathematician George Pólya; two types of transition of which Escher was aware, but which are not recorded in his Notebook, a discussion of mathematical questions suggested by Escher's symmetry work; a review of computer programs for generating tessellations; and an account of the artistic legacy of Escher's work on periodic division. While it may be the case that none of the Escher depictions of interlocking shapes and creatures display the genius and ingenuity of Dali's Swans Reflecting Elephants, there is no doubt that the Escher corpus, because of its impressive volume, its steadfast focus, its serious playfulness and its associated documentation has exerted a uniquely significant influence.

3.2. Review by: Editors.
The Journal of Technology Studies 32 (1/2) (2006), 121.

A revision of a classic book that appeared in 1990, this is the most penetrating study of Escher 's work in existence and the one most admired by scientists and mathematicians. It deals with one powerful obsession that preoccupied Escher: what he called the 'regular division of the plane,' the puzzle-like interlocking of birds, fish, lizards, and other natural forms in continuous patterns. Schattschneider explores how he succeeded at this task by meticulously analyzing his notebooks. The work includes many of Escher's masterpieces as well as hundreds of lesser-known examples of his work. The new forward by Douglas Hofstadter and a new epilogue by the author show how Escher's ideas of symmetry have influenced mathematicians, computer scientists, and contemporary artists.

3.3. Review by: Basic Library List Committee.
Mathematical Association of America.
https://www.maa.org/press/maa-reviews/m-c-escher-visions-of-symmetry

Good news! Doris Schattschneider's classic M. C. Escher: Visions of Symmetry is back in print and is better than ever. Taken out of print by the original publisher, W H Freeman, it has been reissued by Harry N Abrams, probably the foremost publisher of art books in the United States. ... There's a new one-page introduction by Douglas Hofstadter, and a new well-illustrated Afterword to bring the reader up-to-date on developments in Escher scholarship since 1990. A lot has happened since then: there have been additional conferences on Escher, some events for the centennial of Escher's birth, and the opening of a new Escher museum in The Hague. I do not plan here to describe at length the content of the first edition since that is known to many readers. Briefly, the book illustrates, with mathematical commentary, Escher's work in which he so ingeniously used symmetry to get his amazing patterns in the plane. For mathematicians, the underlying mathematics is interesting - but I emphasize above the quality of reproduction of the art because ultimately, for most readers, it is not the mathematics that stimulates interest in Escher's work, it is his dazzling ingenuity and use of colour in creating these unlikely configurations. ... It's an impressive piece of scholarship that is extraordinarily beautiful as well. This book is an old friend and it's good to welcome it back in such an elegant and sumptuous form.

3.4. Review by: Roger Penrose.
Times Higher Education (14 January 2005).

In her superbly illustrated and presented book M. C. Escher: Visions of Symmetry - first published in 1990 and now in its second edition with the addition of some valuable new material - Doris Schattschneider makes a thorough and illuminating study of Escher's notebooks, explaining a good deal of the theory that Escher developed independently to explain the different kinds of patterns, with their various symmetries, that appear in the Alhambra designs and in a great deal of Escher's art. ... Schattschneider's book does an excellent job of conveying not only the wonder of Escher's work but also much of its underlying mathematics. It concentrates almost exclusively on the material arising from Escher's "regular divisions of the plane", but there is some discussion of other topics such as Escher's magnificent depictions of hyperbolic (non-Euclidean) geometry and of three-dimensional geometry. Schattschneider also provides fascinating historical insights into what compelled Escher to pursue specific directions. We find, for instance, that the creative impetus that informed his early work seemed to be satisfied largely by realistic depiction of Italian scenery. But he left Italy with the rise of the Fascists, and the flat countryside of his Dutch homeland did not offer him the same visual resources. Thus, he began to turn inwards for inspiration, and he found that mathematics, particularly the symmetry of plane designs, supplied him with what he needed. Finally, this volume provides a great source of Escher reproductions, many of which - so far as I am aware - are not to be found in other books. The reproductions are of fine quality. It is a wonderful experience simply to leaf through the many beautiful designs, to marvel at their cleverness and their artistry, even to appreciate that so many of the repetitions are fundamentally different despite a frequent superficial appearance of similarity - and not worry at all about any of the underlying mathematics.

JOC/EFR November 2017