Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes

116 Pages·1975·7.1 MB·English

by  Thomas Zaslavsky

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Memoirs of the American Mathematical Society
VOLUME 1 • ISSUE 1 • NUMBER 154 (first of 2 numbers)
JANUARY 1975
Thomas Zaslavsky
Facing up to Arrangements:
Face-Count Formulas
for Partitions of Space by Hyperplanes
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island
ABSTRACT
An arrangement of hyperplanes of Euclidean or projective
d-space is a finite set of hyperplanes, together with the induced
partition of the space. Given the hyperplanes of an arrangement,
how can the faces of the induced partition be counted? Heretofore
this question has been answered for the plane, Euclidean 3-space,
hyperplanes in general position, and the d-faces of hyperplanes
through the origin in Euclidean space. In each case the numbers
of k-faces depend only on the incidences between intersections
of the hyperplanes, even though arrangements with the same inter
section incidence pattern are not in general combinatorially
isomorphic. We generalize this fact by demonstrating formulas
for the numbers of k-faces of all Euclidean and projective
arrangements, and the numbers of bounded k-faces of the former,
as functions of the (semi)lattice of intersections of the hyper
planes, not dependent on the arrangement's combinatorial type.
These formulas are shown to be equivalent to Euler1s rela
tions for arrangements. They also lead to generalizations of
familiar planar counting formulas, and to enumerations for the
partitions by a hyperplane of a finite point set in d-space and
for the faces of zonotopes.
The study of Euclidean arrangements yields the first known
enumerative interpretation of Crapo's beta invariant, as well as
a structural decomposition which casts light on the existence
question for bounded faces. We find an algebraic and a geometric
criterion for the existence of a bounded d-face. We also show
that the union of all bounded faces is connected and has a
definite dimension.
AMS(MOS) 1970 Subject Classifications.
Primary 05A15, 05B35, 50B30; secondary 50D20, 52A25.
Key Words and Phrases.
Arrangement of hyperplanes, partition of space, enumeration of faces,
Euler relation, combinatorial Euler number, combinatorial incidence
geometry, matroid, Tutte-Grothendieck invariant, Mobius function of a
lattice, Crapo beta invariant, zonotope, threshold function.
ISBN 0-8218-1854-6
FACING UP TO ARRANGEMENTS:
FACE-COUNT FORMULAS FOR
PARTITIONS OF SPACE BY HYPERPLANES
by
Thomas Zaslavsky
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Received by the editors March 22, 1974.
Research supported by SGPNR grants 71ZZ0604 and 73B1116.
This paper is substantially my doctoral dissertation at the
Massachusetts Institute of Technology. To my thesis advisor
Curtis Greene—who introduced me to arrangements of hyperplanes—
I owe an appreciative "thank you" for his occasional suggestions
and very frequent corrections, his patient willingness to listen
to insanity and inanity, and his constant encouragement and
interest, which helped me through some hard times.
TABLE OF CONTENTS
Section Page
0. Introduction to arrangements 1
PART I. HOW TO COUNT THE FACES OF AN ARRANGEMENT OF HYPERPLANES 9
1. First facts about arrangements 10
A. The lattice and rank of an arrangement. 10
B. The lattice and the geometry of an arrangement. 11
C. The Mobius function and two latticial polynomials. 12
D. Direct sum, induced arrangement, projectivization. 14
2. The main theorems 18
A. The Euclidean case. 18
Theorem A.
B. The projective case. 20
Theorem B.
C. The bounded case. 21
Theorem C.
D. Relative vertices and cross-sections of Euclidean 27
arrangements.
3. Quick proofs (Eulerian method) 30
AB. Proof of the whole-space cases. 31
C. The bounded case and the bounded space. 32
4. The long proofs (Tutte-Grothendieck method) 37
A. Proof of the Euclidean case. 38
B. Proof of the projective case. 42
C. Proof of the bounded case. 44
5. A collocation of corollaries 53
A. The Euler relations proved. 53
B. More counting relations. 55
C. Enumeration in the classical style. 57
v
D. Unbounded faces. 61
J. Back to Buck: arrangements made simple. 64
F. Winder's Theorem and threshold functions. 67
6 Points and zonotopes 71
A. Placing hyperplanes between points. 71
B. The faces of zonotopes. 72
PART II. A STUDY OF EUCLIDEAN ARRANGEMENTS WITH PARTICULAR 75
REFERENCE TO BOUNDED FACES
7. The beta theorem 76
Theorem D.
8. The central decomposition 80
Theorem E.
A. Appendix on spanning sets of coatoms. 8 6
9. The dimension of the bounded space 94
References 97
Index of symbols 100
vi
TABLE OF FIGURES
Figure
0.1. A simple Euclidean arrangement of 4 lines.
0.2. A Euclidean arrangement of 5 planes with 21 regions,
one of them bounded.
0.3. A Euclidean arrangement of 5 lines; and a projective
arrangement of 6 lines.
1.1. An arrangement of lines, a direct sum.
2.1. The simple arrangement of lines P,.
2.2. The non-central arrangement of planes P .
n
2.3. The central arrangement of planes J-..
3.1. An arrangement of lines whose bounded space is not
topologically a ball.
8.1. An arrangement of lines with one-dimensional bounded
8.2. An arrangement of planes with one-dimensional bounded
8.3. An arrangement of planes with planar bounded space.
a. Full view.
b. Sectional view showing the bounded space.
VII
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