''Combinatorics of Finite Geometries'' is an undergraduate mathematics

textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...

finite geometry Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...

Lynn Batten Lynn Margaret Batten (1948 – 28 July 2022 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 ...

in 1986 with a second edition in 1997 ().

Topics

The types of finite geometry covered by the book include

partial linear space A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph. Defi ...

s, linear spaces,

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

s and

affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine ...

s, projective spaces and

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

polar space In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these a ...

generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 ...

s, and partial geometries. A central connecting concept is the "connection number" of a point and a line not containing it, equal to the number of lines that meet the given point and intersect the given line. The second edition adds a final chapter on

blocking set In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane that every line intersects and that does not contain an entire line. The concept can be generalized in several ways. Instead of talking about ...

s. Beyond the basic theorems and proofs of this subject, the book includes many examples and exercises, and some history and information about current research.

Audience and reception

The book is aimed at advanced undergraduates, assuming only an introductory-level of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

and some knowledge of

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

. Its coverage of recent research also makes it useful as background reading for researchers in this area. Reviewer Michael J. Kallaher cites as a "serious shortcoming" of the first edition its lack of coverage of applications of this subject, for instance to the

design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...

and to

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...

. The second edition has a section on applications but reviewer Tamás Szőnyi writes that it needs additional expansion. Because of the many types of geometry covered in the book, the coverage of each of them is, at times, shallow; for instance, reviewer Theodore G. Ostrom complains that there is only half a page on

non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective s ...

s. Additionally, Kallaher feels that

block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of b ...

s should have been included in place of some of the more esoteric geometries described by Batten. Reviewer Thomas Brylawski criticizes the book for "glossing over or ignoring" important results, for overcomplicated proofs, and for missed cases in some of its case analysis. On the other hand, reviewer B. J. Wilson "enjoyed reading this book" and praises it for its "easily followed style", while reviewer R. J. M. Dawson writes that the book "succeeds admirably" in conveying to students "the living, active nature" of this area.

Related books

Other books on related topics include ''Finite Generalized Quadrangles'' by S. E. Payne and J. A. Thas, and ''Projective Planes'' by D. R. Hughes and F. C. Piper.

References

{{reflist, refs= {{citation, first=M., last=Kallaher, journal=

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

, title=Review of ''Combinatorics of Finite Geometries'' (1st ed.), zbl=0608.51006 {{citation, first=T. G., last=Ostrom, title=Review of ''Combinatorics of Finite Geometries'' (1st ed.), journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ...

, year=1987, mr=0842901 {{citation, last=Wilson, first=B. J., date=January 1987, doi=10.1112/blms/19.1.85, issue=1, journal=Bulletin of the London Mathematical Society, pages=85–86, title=Review of ''Combinatorics of Finite Geometries'' (1st ed.), volume=19 {{citation, first=Thomas, last=Brylawski, title=Review of ''Combinatorics of Finite Geometries'' (1st ed.), journal= American Scientist, volume=76, issue=3, date=May–June 1988, page=314, jstor=27855272 {{citation, first=R., last=Dawson, journal=

, title=Review of ''Combinatorics of Finite Geometries'' (2nd), zbl=0885.51012 {{citation, first=Tamás, last=Szőnyi, authorlink= Tamás Szőnyi , title=Review of ''Combinatorics of Finite Geometries'' (2nd ed.), journal=

, year=1999, mr=1474497

External links

''Combinatorics of Finite Geometries'' (1st ed.)
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 ...

Finite geometry Mathematics textbooks 1986 non-fiction books 1997 non-fiction books