''Quasicrystals and Geometry'' is a book on

quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...

s and

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...

Marjorie Senechal Marjorie Lee Senechal (née Wikler, born 1939) is an American mathematician and historian of science, the Louise Wolff Kahn Professor Emerita in Mathematics and History of Science and Technology at Smith College and editor-in-chief of ''The Mathem ...

, published in 1995 by

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

(). One of the main themes of the book is to understand how the mathematical properties of aperiodic tilings such as the

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...

, and in particular the existence of arbitrarily large patches of five-way rotational symmetry throughout these tilings, correspond to the properties of quasicrystals including the five-way symmetry of their

Bragg peak The Bragg peak is a pronounced peak on the Bragg curve which plots the energy loss of ionizing radiation during its travel through matter. For protons, α-rays, and other ion rays, the peak occurs immediately before the particles come to re ...

s. Neither kind of symmetry is possible for a traditional periodic tiling or periodic crystal structure, and the interplay between these topics led from the 1960s into the 1990s to new developments and new fundamental definitions in both mathematics and crystallography.

Topics

The book is divided into two parts. The first part covers the history of

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...

, the use of X-ray diffraction to study crystal structures through the

s formed on their diffraction patterns, and the discovery in the early 1980s of

s, materials that form Bragg peaks in patterns with five-way symmetry, impossible for a repeating crystal structure. It models the arrangement of atoms in a substance by a

Delone set In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, an ...

, a set of points in the plane or in Euclidean space that are neither too closely spaced nor too far apart, and it discusses the mathematical and computational issues in X-ray diffraction and the construction of the diffraction spectrum from a Delone set. Finally, it discusses a method for constructing Delone sets that have Bragg peaks by projecting bounded subsets of higher-dimensional

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

s into lower-dimensional spaces. This material also has strong connections to

spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...

and

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, deep topics in pure mathematics, but these were omitted in order to make the book accessible to non-specialists in those topics. Another method for the construction of Delone sets that have Bragg peaks is to choose as points the vertices of certain

s such as the

. (There also exist other aperiodic tilings, such as the

pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...

, for which the existence of discrete peaks in the diffraction pattern is less clear.) The second part of the book discusses methods for generating these tilings, including projections of higher-dimensional lattices as well as recursive constructions with hierarchical structure, and it discusses the long-range patterns that can be shown to exist in tilings constructed in these ways. Included in the book are software for generating diffraction patterns and Penrose tilings, and a "pictorial atlas" of the diffraction patterns of known aperiodic tilings.

Audience

Although the discovery of quasicrystals immediately set off a rush for applications in materials capable of withstanding high temperature, providing non-stick surfaces, or having other useful material properties, this book is more abstract and mathematical, and concerns mathematical models of quasicrystals rather than physical materials. Nevertheless, chemist István Hargittai writes that it can be read with interest by "students and researchers in mathematics, physics, materials science, and crystallography".

References

{{reflist, refs= {{citation , last = Cahn , first = John W. , authorlink = John W. Cahn , date = November 1995 , doi = 10.1126/science.270.5237.839 , issue = 5237 , journal =

Science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

, jstor = 2888935 , pages = 839–842 , title = Crystallography expanded , volume = 270, s2cid = 220110430 {{citation , last = Hargittai , first = István , doi = 10.1002/adma.19970091217 , issue = 12 , journal =

Advanced Materials ''Advanced Materials'' is a weekly peer-reviewed scientific journal covering materials science. It includes communications, reviews, and feature articles on topics in chemistry, physics, nanotechnology, ceramics, metallurgy, and biomaterials. Acc ...

, pages = 994–996 , title = Critics on crystals , volume = 9 , year = 1997 {{citation , last = Hayes , first = Brian , authorlink = Brian Hayes (scientist) , date = July–August 1996 , issue = 4 , journal = American Scientist , jstor = 29775727 , pages = 404–405 , title = none , volume = 84 {{citation , last = Kenyon , first = Richard , author-link = Richard Kenyon , journal = Mathematical Reviews , mr = 1340198 , title = none , year = 1996 {{citation , last = Radin , first = Charles , author-link = Charles Radin , date = April 1996 , issue = 4 , journal =

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...

, pages = 416–421 , title = Book Review: Quasicrystals and geometry , url = https://www.ams.org/publications/journals/notices/199604/radin.pdf , volume = 43

External links

*
Quasicrystals and Geometry
' on the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...

Aperiodic tilings Mathematics books 1995 non-fiction books Quasicrystals