Table Of ContentP roieetíve
P lan es
Profective Planes
Frederick W . Stevenson
U niversity of Arizona
Polygonal Publishing H ouse
W ashington, New Jersey
copyright 1972 by W.H. Freeman and Company
Library oi Congress Cataloging-in-Publication Data
Stevenson, Frederick V.
Projective planes /Frederick W. Stevenson,
p. cm.
Originally published: San Francisco : W.H. Freeman, 1972.
Includes bibliographical references (p. ) and indexes.
ISBN 0-936428-13-9
1. Projective planes. I. Title.
[QA471.S86 1992]
516 ' . 5— dc20 92-7736
CIP
manufactured in the United States of America
by Thomson-Shore
printed on acid-free paper
Polygonal Publishing House
Box 357
Washington, NJ 07882
CONTENTS
Preface ... 1
Part I FUNDAMENTALS
1. Basic Concepts ... 3
2. Examples and Elementary Properties .37
3. Transformations ... 89
4. Groups of Transformations ... 121
Part II DESARGUESIAN PLANES
5. Desarguesian Planes ... 161
6. Pappian Planes ... 204
7. Planes over Division Rings and Fields ., .220
8. Coordinatizing Planes ... 253
Part III NON-DESARGUESIAN PLANES
9. Ternary Rings and Projective Planes ... 267
10. Planes over Planar Rings with Associativity ... 296
11. Planes over Quasifields ... 315
12. Planes over Planar Nearfields ... 341
13. Planes over Semifields ... 366
14. Planes over Alternative Rings ... 374
Bibliography ... 399
Index of Symbols and Notation 404
Index ... 408
PREFACE
This book attempts to examine in depth the consequences of the simple
axiom system that describes the mathematical structure known as the
projective plane. Numerous books have been written on the projective
plane. Pickert’s Projektive Ebenen and, more recently, Dembowski’s
Finite Geometries, provide as complete a coverage of a single mathe
matical subject as can be found in the literature. However, these books
are not textbooks. They are invaluable for the advanced graduate stu
dent but are well beyond the reach of the average undergraduate. Text
books on projective geometry tend to cover a wide range of topics. This
is appropriate because projective geometry provides a natural spring
board for the study of non-Euclidean geometries and even linear alge
bra. Nevertheless, the result is that the projective plane is not studied
for its own sake.
It is generally conceded that an undergraduate majoring in mathe
matics should study in depth at least one area of the subject. Often this
area is algebra—perhaps group theory or ring theory. The projective
plane is also a suitable area for such a concentrated study. Like group
theory and ring theory, it has a simple axiomatic foundation. It also
introduces the student to several basic concepts, the most important
being the transformation group. The projective plane shares a remark
able relationship with algebraic structures of two binary operations,
such as the field, division ring, semifield, nearfield, and quasifield.
Furthermore, its study exposes the student to several other areas of
mathematics, such as combinatorial analysis, linear algebra, and num
ber theory. Finally, there are many unsolved problems in this area;
some are probably workable, others are classics that may never be
solved.
The book is divided into three parts. Part 1 introduces the student
to examples and techniques; Part 2 examines the classical theorems
of Desargues and Pappus; Part 3 is devoted to the study of non-
Desarguesian planes. The first two parts can be taught comfortably in
a one-semester, upper-division course for undergraduates. The whole
book could be covered in a one-semester course at the graduate level.
Perhaps Part 3 and selected portions of Parts 1 and 2 would be ap
propriate for a seminar. Naturally, some sections are more important
than others; for example. Section 2.2, 2.3 (the part dealing with affine
difference sets only), 3.3, 4.6, 5.3, and 6.3 can be omitted without
significantly affecting the student’s understanding of the other sections.
The exercises generally are not of the drill variety; more-difficult ones
are indicated by an asterisk. A careful selection of 100 or less of the
more than 350 exercises would be sufficient for one semester.
I wish to thank my wife, Cheryl, for her invaluable assistance in the
formulation, preparation, and completion of this book.
October 1970 Frederick W. Stevenson
1
CHAPTER
BASIC CONCEPTS
1.1 INTRODUCTION
The field of geometry, perhaps more than any other area of study, has
provided the key ideas in the growth of mathematics as an abstract
discipline. Euclid’s Elements, a treatise written about 300 b.c., repre
sents the first attempt to organize a field of knowledge deductively.
His five postulates for plane geometry served as the outstanding model
of an axiom system for two thousand years. The celebrated fifth postu
late has perhaps been responsible for more controversy than any
other single mathematical axiom; in large part this postulate gave
birth to the idea that an axiom system need not be grounded in con
ceptually familiar ideas.
Euclid’s fifth postulate is best known in the form known as Playfair’s
Axiom: For any line L and point p not on L there exists one and only
one line M parallel to L and passing through p. After centuries of
frustration, attempts to prove the fifth postulate from the four more
“obvious” postulates were abandoned and other postulates were
substituted in its place. For example, in the early nineteenth century.
BASIC CONCEPTS
Hungary’s Bolyai Janos (1802-1860) and Russia’s Nikolai Lobachevski
(1793-1856) developed geometries that allowed more than one line
parallel to a given line to pass through a given point; Germany’s
Bernhard Riemann (1826-1866) developed a geometry with no parallel
lines. Today many different non-Euclidean geometries are being
studied and developed; the projective plane is one such geometry.
The projective plane is an example of a non-Euclidean geometry
having no parallel lines. Its origins are grounded both in the Euclidean
plane and in the familiar plane of common sense. No clear distinction
has been found between these two planes, but if one were found it
would probably rest on the validity of the fifth postulate in the com
mon-sense world. Such familiar examples of parallel lines as railroad
tracks appear to converge as they recede from the observer. If we were
to assume that they eventually meet at some point we could build a
new geometry in the following way. Choose any point p in the Euclid
ean plane and consider all the lines through p. To each of these lines
add an extra point called an ideal point or point at infinity and stipulate
that each different line through p contain a different ideal point. For
any line L not on p, attach the ideal point that belongs to the unique line
through p parallel to L. Thus all the lines in the Euclidean plane have
been assigned an ideal point and families of parallel lines share the
same ideal point. Finally, to ensure that every two points determine a
line, create a new line called the ideal line or line at infinity, made up
of all the ideal points. The resulting plane is a projective plane called
the real projective plane or the extended Euclidean plane.
The use of the word “projective” in the term “projective plane”
arises in connection with the concept of projection mapping. For ex
ample, in the Euclidean plane, let L and M be intersecting lines and let
p be a point not on L or M (see Figure 1). The mapping that takes q on
L to r on M in such a way that p, q, and r lie on a line is called a projec
tion mapping from L to M. If p is considered to be a light source, L is
considered to be a set of points, and M is considered to be a screen,
then r is the shadow cast by q on M. Physically, r would be a shadow
only if q were between p and the screen. This mapping is deficient in
two respects: there exists a point on L that has no image on M, and
there exists a point on M that cannot receive an image from L. These
two points are s and i, respectively: s is the intersection of M' and L
where M’ is the unique line passing through p and parallel to M, and t is
the intersection of L' and M where L' is the unique line passing through
M'
p and parallel to L. In the extended Euclidean plane, the parallel lines
L and L' and M and M' intersect, so the projection mapping becomes a
function with domain L and range M. Such mappings are of prime
importance in the study of projective planes; indeed, the projective
plane can be defined as the study of the properties that are held in
variant under projection mappings as defined here.
Projective geometry is the general study of properties held invariant
under projection mappings in two-space and higher dimensional spaces.
The first major results in the subject, developed in the early seventeenth
century, were due in large part to outstanding mathematicians: France’s
Poncelet (1788-1869) and Brianchon (1785-1864), Switzerland’s
Steiner (1796-1863), and Germany’s von Staudt (1798-1867). Today
projective geometry is an established area of mathematics that is well
suited to serve as a general system from which other geometries can
be developed and that is also of great interest in its own right. This
text explores the special case of projective geometry in two dimensions
— the projective plane.
1.2 TERMINOLOGY AND NOTATION
The following terminology and notation are used in this text: Points
are denoted by lower-case letters (usually p,q,r,s,a,b,c,d)\ lines are
denoted by capital letters (usually L, M, N); sets ^ and denote the
sets of points and lines, respectively; denotes the incidence relation
on points and lines (therefore, ^ Q ^ x Jf); and a plane is represented
by the triple Jf, J^). A basic assumption throughout is that no point
is a line and no line is a point (that is, ^ H oSf = 0). If (p,L) E c/, then
p is “on” L or L “passes through” p. If two or more points lie on the
6 BASIC CONCEPTS
same line, the points are collinear. If two or more lines pass through
the same point, the lines are concurrent. If p, q, r, and s are four points
no three of which are collinear, then the quadruple p,q,r,s is called a
complete four-point or simply a four-point. The order of points p, q, r,
and 5 of a four-point is important for reasons that will become clear
later. Thus, unless otherwise specified, a four-point is an ordered set
of four points. When the text refers to “two points” (lines) or points”
(lines), the points (lines) are assumed to be distinct unless otherwise
noted.
This text departs from the standard textbook notation for points
and lines. Usually, capital letters denote points, lower-case letters
denote lines. The notation here is consistent with Dembowski (1968)
and also has a set theoretical basis since lines are considered to be sets
of points.
Finally, the well-known abbreviation “iff” for “if and only if” is
used in proofs.
1.3 PLANES
This book focuses almost entirely on affine planes and projective
planes. Nevertheless, it is convenient to begin by introducing two
weaker planes, the partial plane and the primitive plane. First, how
ever, the term “plane” should be defined.
Definition 1.3.1 A plane (incidence plane) is a triple
such that S^y <if, and are setSy ^ rio^ = 0, ^UoSf7^0, and
y C ^ X o^.
Notation. Planes are denoted by the letter 2.
Definition 1.3.2 A partial plane is a plane satisfying
Axl: At most one line passes through two points.
Theorem 1.3.3 If L and M are distinct lines in a partial plane 2,
then there is at most one point incident with both lines.
Proof Suppose that there exist two distinct points p and q on both
of the lines. Then (p,L),(p,M),(<y,L),(q^,M) e y, and L^M . This con
tradicts Axl.
