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'''''Algebra and Tiling: Homomorphisms in the Service of Geometry''''' is a mathematics textbook on the use of [[group theory]] to answer questions about [[tessellation]]s and higher dimensional [[honeycomb (geometry)|honeycombs]], partitions of the [[Euclidean plane]] or higher-dimensional spaces into congruent tiles. It was written by [[Sherman K. Stein]] and Sándor Szabó, and published by the [[Mathematical Association of America]] as volume 25 of their [[Carus Mathematical Monographs]] series in 1994.{{r|kenyon|walton}} It won the 1998 [[Beckenbach Book Prize]],{{r|beck}} and was reprinted in paperback in 2008.{{r|mainardi}}

==Topics==
The seven chapters of the book are largely self-contained, and consider different problems combining tessellations and algebra.{{r|kenyon}}  Throughout the book, the history of the subject as well as the state of the art is discussed, and there are many illustrations.{{r|mainardi}}

The first chapter concerns a conjecture of [[Hermann Minkowski]] that, in any lattice tiling of a Euclidean space by unit [[hypercubes]] (a tiling in which a [[Lattice (group)|lattice]] of translational symmetries takes any hypercube to any other hypercube) some two cubes must meet face-to-face. This result was resolved positively by [[Hajós's theorem]] in group theory,{{r|kenyon}} but a generalization of this question to non-lattice tilings ([[Keller's conjecture]]) was disproved shortly before the publication of the book, in part by using similar group-theoretic methods.

Following this, three chapters concern lattice tilings by [[polycube]]s. The question here is to determine, from the shape of the polycube, whether all cubes in the tiling meet face-to-face or, equivalently, whether the lattice of symmetries must be a [[subgroup]] of the [[integer lattice]]. After a chapter on the general version of this problem, two chapters consider special classes of cross and "semicross"-shaped polycubes,{{r|kenyon}} both with regard to tiling and then, when these shapes do not tile, with regard to how densely they can be packed. In three dimensions, this is the notorious [[tripod packing]] problem.

Chapter five considers [[Monsky's theorem]] on the impossibility of partitioning a square into an odd number of equal-area triangles, and its proof using the [[p-adic valuation|2-adic valuation]], and chapter six applies [[Galois theory]] to more general problems of tiling polygons by congruent triangles, such as the impossibility of tiling a square with [[Special right triangle|30-60-90 right triangles]].{{r|kenyon}}

The final chapter returns to the topic of the first, with material on [[László Rédei]]'s generalization of Hajós's theorem. Appendices cover background material on lattice theory, [[exact sequence]]s, [[free abelian group]]s, and the theory of [[cyclotomic polynomial]]s.{{r|mainardi}}

==Audience and reception==
''Algebra and Tiling'' can be read by undergraduate or graduate mathematics students who have some background in abstract algebra, and provides a source of applications for this topic. It can be used as a textbook, with exercises scattered throughout its chapters.{{r|walton}}

Reviewer William J. Walton writes that "The student or mathematician whose area of interest is algebra should enjoy this text".{{r|walton}} In 1998, the [[Mathematical Association of America]] gave it their [[Beckenbach Book Prize]] as one of the best of their book publications. The award citation called it "a simultaneously erudite and inviting ex- position of this substantial and timeless area of mathematics".{{r|beck}}

==References==
{{reflist|refs=

<ref name=beck>{{citation|url=https://www.ams.org/notices/199805/comm-maa.pdf|journal=[[Notices of the American Mathematical Society]]|volume=45|issue=5|date=May 1998|title=Beckenbach Book Prize|department=MAA Prizes Presented in Baltimore|page=615}}</ref>

<ref name=kenyon>{{citation|title=Review of ''Algebra and Tiling''|journal=[[Mathematical Reviews]]|first=Richard|last=Kenyon|year=1995|mr=1311249}}, reprinted as {{zbl|0930.52003}}</ref>

<ref name=walton>{{citation|last=Walton|first=William L.|date=December 1995|issue=9|journal=[[The Mathematics Teacher]]|jstor=27969590|page=778|title=Review of ''Algebra and Tiling''|volume=88}}</ref>

<ref name=mainardi>{{citation|title=Review of ''Algebra and Tiling''|journal=MAA Reviews|publisher=[[Mathematical Association of America]]|url=https://www.maa.org/press/maa-reviews/algebra-and-tiling-homomorphisms-in-the-service-of-geometry|date=May 2008|first=Fabio|last=Mainardi}}</ref>

}}

==Further reading==
*{{citation|last=Post|first=K.A.|journal=Mededelingen van Het Wiskundig Genootschap|pages=255–256|title=Review of ''Algebra and Tiling''|volume=41|year=1998}}

[[Category:Tessellation]]
[[Category:Mathematics textbooks]]
[[Category:1994 non-fiction books]]
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