ARRANGEMENTS AND SPREADS*

Chapter 1. Introduction

The present survey deals mostly with rather elementary mathematics,—so elementary in

fact that most of its results and problems are (or at least should be) understandable to under-

graduates. It was written out of the eonvietion that many negleeted aspeets of elementary

geometry deserve a wider dissemination, —because of their inherent beauty and interest, and

for the Inspiration and understanding they can impart to our students and to ourselves.

During the last decade or two» it has become an article of faith with many of us that

researeh in mathematics and its teaching are worthwhile only if they deal with very general

and abstract topics. The elements of algebra, set theory, logic, topology, measure theory,

etc., —that is what advanced undergraduates and starting graduate students are being taught

in good schools; it would be foolish to say that this should be changed. What is unfortunate,

and what cripples most of the students for the rest of their professional lives, is that other

topics and points of view are only rarely presented. The students are made to understand

that researeh in, say, finite groups is "important"—no matter how esoteric the problem in-

vestigated,— while topics in the Euclidean plane, for example, are certainly old-fashioned

since the Euclidean plane is a very special strueture. Many aspeets of geometry fare partic-

ularly badly from this attitude; they are even denied the right to exist unless they fit nice

algebraic patterns. In the most ridiculous and extreme aspeets of such tendencies geometry

is equated with a subdiseipline of linear algebra.

As a counterthrust to that trend I would like to recall just a few of the many historical

instances in which the geometric patterns of thought or points of view led not only to

another theorem or two but were crucial in gaining new and revolutionary insights:

(i) The discovery of irrational numbers;

(ii) The invention of calculus;

(iii) The development of the axiomatic method;

(iv) The emergence of algebraic topology, and of functional analysis.

*) The first version of this survey was presented as an invited address at the Annual Spring Meeting

of the Michigan Section of the Mathematical Association of America held on April 4, 1970, at the Wayne

State University in Detroit, Michigan.

Research sponsored in part by the National Science Foundation through the Science Development

Program grant GU-2648, and by the Office of Naval Research through contraet N00014-67-A-0103 -

0003.

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http://dx.doi.org/10.1090/cbms/010/01