conferenc - american mathematical society · 2019. 2. 12. · 3.1 pseudolines and non-stretchabl e...

Conference Board of the Mathematical Sciences

CBMS Regional Conference Seiles in Mathematics

Nmriber 10

Arrangements and Spreads

Branko Grünbaum

Published for the Conference Board of the Mathematical Sciences

by the American Mathematical Society

Providence, Rhode Island with support from the

national Science Foundation

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

Expository Lectures from the CBMS Regional Conference

held at the University of Oklahoma, Norman, Oklahoma June 21-25, 1971

2000 Mathematics Subject Classification. P r i m a r y 05-XX; Secondary 52 -XX, 55 -XX.

International Standard Book Number 0-8218-1659-4

Library of Congress Catalog Card Number 71-38926

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionOams. org.

© 1972 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http:/ /www.ams.org/

10 9 8 7 6 5 4 10 09 08 07 06 05

CONTENTS 1. Introduetion , , é

2. Arrangements of lines 4 2.1 Isomorphism-types of arrangements 4 2.2 Relations between the numbers of lines, vertices, edges and cells 9 2.3 Vertices of given multiplicity 16 2.4 Numbers of cells of various Mnds 25 2.5 Irrational arrangements 33 2.6 Arrangements associated with sets of points 36 2.7 Other problems and classifications 37

3. Arrangements of pseudolines and arrangements of curves ........................................... 40 3.1 Pseudolines and non-stretchable arrangements 40 3.2 Some resuits on arrangements of pseudolines 45 3.3 Arrangements of simple curves in the Euclidean plane 55 3.4 Generalizations 68

4. Spreads of curves 77 4.1. Definition and properties of spreads 77 4.2 Examples of spreads 81 4.3 Arrangements and spreads 85 4.4 Topological planes 86

References 92

Notes added in proof 112

This page intentionally left blank

92 BRANKO GRÜNBAUM

REFERENCES

The references are arranged alphabetically by the author's name. Each item is followed by a listing (in Square brackets [ ]) of those sections in which that item is mentioned. In as far as such Information is availabie, each item is provided with a reference to a reviewing Journal; MR Stands for "Mathematieal Reviews**» FM for "Jahrbuch über die Fortschritte der Mathematik".

Anonymous 1964 Programma van Jaarlijkse Prifsvragen; Problem No. 6, Nieuw Arch. Wisk. (3)

12 (1964), 64. [2.3]

E. Artin

1940 Coordinates in affine geometry, Notre Dame Math. Colloq. 1940» 15-20. (= Collected Papers (1965), 505-510.) MR 3 p. 179. [4.4]

R. Artzy

1965 Linear geometry, Addison-Wesley, Reading 1965. MR 32 #6274. [4.4]

C. W. AsMey

1944 The AsMey Book of Knots, Doubleday and Co., Garden City, Í . Õ., 1944. [3.4]

N. Balasübramanian 1953 Á theorem on sets ofpoints, Proc. Nat. Inst. Sei. India 19 (1953), 839. MR 15

p. 551. [3.4] W. W. R. Ball

1960 Mathematieal recreations and essayst Revised by H. S. M. Coxeter, MaeMillan Co.,

New York 1960. MR 8 p. 440. [2.3]

R. Baltzer

1885 Eine Erinnerung an Möbius und seinen Freund Weiske, Ber. Sachs. Ges. Wiss.

Leipzig (1885), 1-6. FM 17 p. 518. [3.4]

L. Bankoff

1970 Solution of Problem 130, Math. Mag. 43 (1970), 233. [3.4]

J. G. Basterfield and L. M. Kelly 1968 Á characterization ïf sets of ç points which determine ç hyperplanes, Proe.

Cambridge Phil. Soc. 64 (1968), 585-588. MR 38 #2040. [2.2]

F. Behrend 1938 Ueber die kleinste umbeschriebene und die grösste einbeschriebene Ellipse eines

konvexen Bereiches, Math. Ann. 115 (1938), 379-411. FM 64 p. 731. [4.2]

W. Benz 1960 Über Möbiusebenen, Jber. Deutsch. Math.-Verein. 63 (1960), 1-27. MR 22

#7012. [4.4]

K. H. Berger 1936 Eilinien mit perspektiv liegenden Tangenten-und Sehnendreiecken, S.-B. Heidel·

berg. Akad. Wiss. 1936, pp. 1-11. FM 62 p. 700. [4.2]

H. Bergmann

1969 Die maximale Anzahl von Überschneidungen bei einem Polygon, Arch. Math. 20 (1969), 107-112. MR 40 #7939. [3.4]

ARRANGEMENTS AND SPREADS 9 3

J. Bertrand

1842 Demonstration d'un theoreme de giometrie, J. Math. Pures Appl. (1) 7 (1842), 215-216. [4.2]

A. S. Besicovitch 1948 Measure of asymmetry ofconvex curves, J. London Math. Soc. 23 (1948), 237—

240. MR 10 p. 320. [4.2]

M. Biernacki 1953 Sur quelques properietes des ovales, Ann. Univ. M. Curie-Skfcdowska (Lublin), 7

(1953), 103-112. MR 16 p. 950. [4.2]

W. Blaschke 1916 Kreis und Kugel, Veit, Leipzig 1916. (Reprint: Chelsea, Í . Õ. 1949). FM 64

p. 1109; MR 17 p. 887. [4.2] 1923 Vorlesungen über Differentialgeometrie. II, Grundlehren der Math. Wiss. Vol. 7.

Springer, Berlin 1923. FM 49 p. 499. [4.2] 1924a Eine topologische Kennzeichnung der Kreise auf der Kugel, Hamburg Math. Abh.

3 (1924), 164-166. FM 50 p. 488. [4.4]

1924b Eine Kennzeichnung der Kreise auf der Kugel, Math. Z. 21 (1924), 209-210. FM 50 p. 488. [4.4]

1930 Vorlesungen über Differentialgeometrie, I, Third edition. Springer, Berlin 1930. (Reprinted by Dover, N. Y. 1945.) FM 56 p. 588; MR 7 p. 391. [4.4]

W. Blaschke and G. Bol

1938 Geometrie der Gewebe, Grundlehren der Math. Wiss., Vol. 49. Springer, Berlin 1938. FM 64 p. 727. [4.4]

J. Blazek and M. Koman 1964 "A minimal problem concerning complete plane graphs,ft Theory of Graphs and

its Applications. (Proc. Sympos. Smolenice, 1963), pp. 113—117. Publ. House Czeehoslovak Acad. Sei., Prague 1964. MR 30 #4249. [3.4}

1967 On an extremal problem concerning graphs, Comment. Math. Univ. Carolinae 8 (1967), 49-52. MR 35 #1506. [3.4]

W. E. Bonnice and L. M. Kelly 1971 On the number of ordinary planes, J. Combinat. Theory A l l (1971), 45—53.

[2.6]

R. C. Böse 1935 Á note on the convex oval, Bull. Calcutta Math. Soc. 27 (1935), 55-60. FM

61 p. 1428. [4.2; 4.3]

R. C. Böse and S. N. Roy 1935 Some properties of the convex oval with reference to its perimeter centroid,

Bull. Calcutta Math. Soc. 27 (1935), 79-86. FM 61 p. 1428. [4.2]

M. Bouten and P. de Witte 1965 Á new proof of an inequality of Szekeres, de Bruijn and Erdös, Bull. Soc. Math.

Belg. 17 (1965), 475-483. MR 33 #3170. [2.2]

94 BRANKO GRÜNBAUM

Ê. Á. Brakke 1971 Some new values of Sylvester's function for ç noncollinear points, J, Undergrad.

Math, (to appear). [2.3]

J. G. Brennan 1958 Á property ofa plane convex region, Note 2808. Math. Gazette 42 (1958), 3 0 1 -

302.

M. Brückner

1900 Vielecke und Vielflache, Teubner, Leipzig 1900. FM 31 p. 479. [3.4]

N. G. deBruijn and P. Erdös 1948 On á combinatorial problem, Proc. Nederl. Akad. Wet. 51 (1948), 1277-1279.

MR 10 p. 424. [2.2; 2.3]

G. Brunei 1894 Note sur le nombre de points doubles que peut presenter le perimetre d'un poly-

gone, Mem. soc. sei. phys. et nat. Bordeaux (4) 4 (1894), 273-276. FM 25 p. 873. [3.4]

HL Brunn 1889 lieber Curven ohne Wendepunkte, Ackermann, München 1889. FM 21 p. 815.

[4.2; 4.4]

R. C. Bück and E. F. Bück 1949 Equipartition of convex sets, Math. Mag. 22 (1949), 195-198. MR 10 p. 621.

[4.2]

W. Buckel 1953a Eine Kennzeichnung des Systems aller Kreise mit nichtverschwindendem Radius

der euklidischen Ebene, J. Reine Angew. Math. 191 (1953), 13-29. MR 15 p. 149. [4.4]

1953b Eine Kennzeichnung des Systems aller nichtzerfallenden Kegelschnitte der projektiven Ebene, J. Reine Angew. Math. 191 (1953), 165-178. MR 15 p. 149. [4.4]

F. Buekenhout 1971 Inversions in locally affine circular Spaces. I, Math. Z. 119 (1971), 189-202.

[4.4]

F. Buekenhout and J. Doyen

1970 Linear Geometry, Lecture Notes, University of Brüssels, 1970. [2.2; 2.3]

R. J. Bumcrot 1969 Modem projeetive geometry, Holt, Rinehart and Winston, New York 1969.

MR 38 #6450. [4.4]

R. G. Busacker and T. L. Saaty

1965 Finite graphs and networks, McGraw-Hill, New York 1965. MR 35 #79. [3.4]

H. Busemann 1942 Metrie methods in Finsler Spaces and in the foundations ïf geometry, Annais of

Math. Studies No. 8. Princeton University Press 1942. MR 4 p. 109. [4.4]

ARRANGEMENTS AND SPREADS 95

1955 The geometry ofgeodesics, Academic Press, New York 1955. MR 17 p. 779. [4.2; 4.4]

H. Busemann and P. J. Kelly 1953 Projective geometry and projective meines, Academic Press, New York 1953.

MR 14 p. 1008. [4.2]

R. J. Canham

1969 Á theorem on arrangements of lines in the plane, Israel J. Math. 7 (1969), 393-397. MR 40 #7938. [2.3; 2.4]

1971 Ph D. Thesis, University of East Angiia, Norwich 1971. [2.1; 2.3; 3,1; 3.2]

W. B. Carver 1941 The polygonal regions into which á plane is divided by ç straight lines. Amer.

Math. Monthly 48 (1941), 667-675. MR 3 p. 180. [2.1; 2.4]

J. Ceder 1964 On outwardly simple line families, Canad. J. Math. 16 (1964), 1-11. MR 28

#517. [4.2]

1965a Generalized sixpartite problems, Bol. Soc. Matern. Mexic. 9 (1965), 28-32. MR

32 #6318. [4.2]

1965b On á problem of Grünbaum, Proc. Amer. Math. Soc. 16 (1965), 188-189. MR 29 #3413. [4.2]

G. D. Chakerian 1970 Sylvester's problem on collinear points and á relative, Amer. Math. Monthly 77

(1970), 164-167. MR 41 #3305. [2.3; 2.7]

G. D. Chakerian and S. K. Stein

1966 Bisected ehords ofa convex body, Arch. Math. 17 (1966), 561-565. MR 34 #6635. [4.2]

C. M. Christensen 1950 Á Square inscribed in á convex curve, (In Danish) Mat. Tidsskr. B. 1950, 22—26.

MR 12 p. 525. [4.2]

A. B. Coble 1915 Points sets and allied Cremona groups, Trans. Amer. Math. Soc. 16 (1915), 155-

198. FM45p. 234. [2.7]

P. J. Cohen 1967 Decision procedures for real and padie fields, Mimeographed notes, Stanford

University 1967. [2.5]

H. S. M. Coxeter 1942 Review of Melchior [1940], Mathematical Reviews, 3 (1942), 13. [2.3] 1948 Á problem of collinear points, Amer. Math. Monthly 55 (1948), 26-28. MR

9 p. 458. [2.3]

1949 The real projective plane, McGraw-Hill, New York 1949. (Second edition: Cambridge Univ. Press, London 1955). MR 10 p. 729. [2.3]

1961 Introduction to geometry, John Wiley and Sons, New York 1961. MR 23 #A1251. [2.3]

9 6 BRANKO GRÜNBAUM

1962 The Classification ofzonohedra by means of profective diagrams, J. de Math, pures appl. (9) 41 (1962), 137-156. MR 25 #4417. [2.1; 2.3]

H. Croft and R. K. Guy

1971 Research problems in intuitive mathematics, (In preparation.) [2.3; 3.4]

D. W. Crowe 1969 Sylvester's problem on collinear points, Question 10, J. Undergrad. Math. 1

(1969), 133. [2.3]

D. W. Crowe and T. A. McKee 1968 Sylvestern problem on collinear points, Math. Magaz. 41 (1968), 30—34. MR

38 #3761. [2.3]

R. H. Crowell and R. H. Fox

1963 Introduction to Knot Theoryf Blaisdell, New York 1963. MR 26 #4348. [3.4]

L. D. Cummings 1932a Hexagonal Systems ofseven lines in á plane, Bull. Amer. Math. Soc. 38 (1932),

105-110. FM 58 p. 676. [2.1] 1932b Heptagonal Systems of eight lines in á plane, Bull. Amer. Math. Soc. 38 (1932),

700-702. FM 58 p. 676. [2.1] 1933 On á method of comparison for straight-line nets, Bull. Amer. Math. Soc. 39

(1933), 411-416. FM 59 p. 610. [2.1] L. Danzer, D. Laugwitz and H. Lenz

1957 Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eihörper einbeschriebenen Ellipsoiden, Arch. Math. 8 (1957), 214-219. MR 20 #1283.

[4.2]

P. Dembowski 1968 Finite Geometries, Ergebnisse der Math. u. Grenzgebiete, Vol. 44. Springer-Ver

lag, New York 1968. MR 38 #1597. [2.7; 4.1]

G. A. Dirac 1951 Collineanty properties ofsets of points, Quart. J. Math. 2 (1951), 221-227.

MR 13 p. 270. [2.3]

H. E. Dudeney 1907 The Canterbury puzzles and other curious problems, W. Heinemann, London

1907. [2.3]

1967 536 puzzles and curious problems, Charles Scribner's Sons, New York 1967. [2.3]

V. Eberhard 1890 Eine Classification der allgemeinen Ebenensysteme, J. Reine Angew. Math. 106

(1890), 89-120. FM 22 p. 553. [2.4]

1891 Zur Morphologie der Polyeder, Teubner, Leipzig 1891. [2.4]

M. Edelstein 1970 Generalizations of the Sylvester problem, Math. Magazine 43 (1970), 250-254.

[2.3]

ARRANGEMENTS AND SPREADS 9 7

H. G. Eggleston

1953 Some properties of triangles as extremal convex curves, J. London Math. Soc. 28 (1953), 32-36. MR 14 p. 896. [4.2]

R. B. Eggleton and R. K. Guy

1971 The crossingmumber ofthe n-cube, (to appear). [3.4]

E. Ehrhart 1955 Une geniralisation du thioreme de Minkowski, C. R. Acad. Sei. Paris 240 (1955),

483-485. MR 16 p. 574. [4.2]

P. D. T. A. Hliott 1967 On the number of circles determined by ç points, Acta. Math. Acad. Sei. Hun-

gar. 18 (1967), 181-188. MR 35 #1793. [2.2; 3.4]

A. Emch 1913 Some properties ofclosed convex curves in á plane, Amer. J. Math. 35 (1913),

407-412. FM 44 p. 561. [4.2] 1915 On the medians of á closed convex polygon, Amer. J. Math. 37 (1915), 19-28.

FM 4S p. 732. [4.2]

P. Erdös

1943 Three point collinearity, Problem 4065. Amer. Math. Monthly 50 (1943), 65. [2.2; 2.3]

1957 Nehäny geometnai problimdrol, Mat. Lapok 8 (1957), 86-92. MR 20 #6056.

[3.4]

1961 Some unsolved problems, Publ. Math. Inst. Hungar. Acad. Sei. 6 (1961), 221—

254. MR31 #2106. [2.3; 3.4]

1971a On á problem of Grünbaum, Canad. Math. Bull, (to appear). [2.2]

1971b Topics in combinatorial analysis, (to appear). [2.3]

G. Ewald 1956a Axiomatischer Aufbau der Kreisgeometrie, Math. Ann. 131 (1956), 354—371.

MR 18 p. 502. [4.4] 1956b Über den Begriff der Orthogonalität in der Kreisgeometrie, Math. Ann. 131

(1956), 463-469. MR 21 #5159. [4.4] 1967 Aus konvexen Kurven bestehende Möbiusebenen, Abh. Math. Sem. Univ. Ham

burg 30 (1967), 179-187. MR 35 #3536. [4.4]

I. Fary 1950 Sur la densite des reseaux de domaines convexes, Bull. Soc. Math. France 78

(1950), 152-161. MR 12 p. 526. [4.2]

L. Fejes Toth 1948 Review of Coxeter [1948], Math. Reviews 9 (1948), 458. [2.3] 1953 Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Berlin 1953.

MR 15 p. 248. [4.2]

A. Forrester 1952 Á theorem on involutory transformations without fixed points, Rroc. Amer. Math.

Soc. 3 (1952), 333-334. MR 14 p. 72. [4.1]

98 BRANKO GRÜNBAUM

J. S. Frame

1945 Review of Robinson [1945], Math. Reviews 6 (1945), 215. [3.4]

G. K. Francis 1969 Null genus realizability criterion for abstract intersection sequences, J. Combinat.

Theory 7 (1969), 331-341. MR 40 #5475. [3.4]

O. Frink

1949 Problem 4325, Amer. Math. Monthly 56 (1949), 423-424. [4.2]

D. Gans 1954 Á circular model of the euclidean plane, Amer. Math. Monthly 61 (1954), 23—

30. MR 15 p. 460. [4.4] 1958 Models of projective and euclidean space, Amer. Math. Monthly 65 (1958),

749-756. MR 20 #4805. [4.4]

1969 Transformations and geometries, Appleton-Century-Crofts, New York 1969. [4.4]

C. F. Gauss 1823 Zur Geometria Situs, Unpublished manuscript. Werke, Band 8, p. 272. Teubner,

Göttingen 1900. [3.4] 1844 Zur Geometrie der Lage, für Zwei Raumdimensionen, Unpublished manuscript.

Werke, Band 8, pp. 282-286. Teubner, Göttingen 1900. [3.4]

M. C. Gemignani 1966 Topological geometries and á new characterization of Rm, Notre Dame J. For

mal Logic 7 (1966), 57-100. MR 34 #8259. [4.4]

E. Gergely 1957 La generalisation de la theorie polaire sur les ovales et les ovaloides, (Romanian;

summaries in Russian and French) Acad. R. P. Romine Fil. Cluj. Stud. Cerc. Mat. 8 (1957), 143-160. MR 20 #3501. [4.4]

1959 Eine Verallgemeinerung der polaren Theorie auf Eilinie und Eifläche, Mathemat-ica (Cluj) 1 (24) (1959), 221-237. MR 23 #A3507. [4.4]

L. W. Green 1963 Auf Wiedersehensflächen, Ann. Math. (2) 78 (1963), 289-299. MR 27 #5206.

[4.4]

B. Grünbaum 1963 Measures of symmetry for convex sets, Proc. Sympos. Pure Math. Vol. 7 (Con-

vexity) (1963), 233-270. MR 27 #6187. [4.2] 1966 Continuous families of curves, Canad. J. Math. 18 (1966), 529-537. MR 33

#4783. [4.1] 1967 Convex polytopes, Interscience, London 1967. MR 37 #2085. [2.1; 2.2; 2.3;

2.4; 2.5; 2.7; 3.2] 1970a Polytopes, graphs, and complexes, Bull. Amer. Math. Soc. 76 (1970), 1131 —

1201. MR 40 #5480. [2.4; 2.7] 1970b The importance ofbeing straight, Proc. 12th Bienn. Internat. Seminar of the

Canad. Math. Congress 1969 (1970), 243-254. [2.4; 3.1; 3.2]


1971 Arrangements of hyperplanes, Proc. Second Louisiana Conference on Combina-torics and Graph Theory. Baton Rouge 1971 (to appear). [2.1; 2.7; 3;1]

H. Guggenheimer 1965 Finite sets on curves and surfaces, Israel J. Math. 3 (1965), 104—112. MR 32

#6326. [4.2]

H. Gupta 1953 Non-concyclic sets ofpoints, Proc. Nat. Inst. Sei. India 19 (1953), 315—316.

MRlSp. 139. [3.4]

R. K. Guy 1960 Á combinatorial problem, Bull. Malayan Math. Soc. 7 (1960), 68-72. [3.4] 1969 "The decline and fall of Zarankiewicz's theorem", Proof Techniques in Graph

Theory, edited by F. Harary. Academic Press, New York 1969. MR 40 #7144. [3.4]

1970 Twenty odd questions in eombinatorics, Proc. Second Chapel Hill Conference on Combinatorial Mathematics and its Applications. Chapel Hill, 1970, pp. 209-237. [3.4]

1971a "Unsolved combinatorial problems", Combinatorial Mathematics and its Applications, edited by D. J. A. Walsh, pp. 121-127. Academic Press, London 1971. [2.2; 3.4]

1971b The crossing number of the complete graph, (to appear). [3.4] 1971c i(Latest results on crossing numbers", Recent Trends in Graph Theory, edited by

M. Capobianco, J. B. Frechen and M. Krolik. Springer-Verlag, New York 1971. pp. 143-156. [3.4]

R. K. Guy and F. Harary 1967 On the Möbius ladders, Canad. Math. Bull. 10 (1967), 493-496. MR 37 #98.

[3.4]

R. K. Guy and T. A. Jenkyns 1969 The toroidal crossing number of Km n, J. Combinat. Theory 6 (1969), 235-

250. MR 37 #5660. [3.4]

R. K. Guy, T. A. Jenkyns and J. Schaer 1968 The toroidal crossing number of the complete graph, J. Combinat. Theory 4

(1968), 376-390. MR 36 #3682. [3.4]

H. Hadwiger 1961 Kleine Studie zur elementaren Stetigkeitsgeometrie, Jber. Deutsch. Math.-Verein.

64 (1951/62), 78-81. MR 25 #5421. [4.1; 4.2] 1971 Ungelöste Probleme Nr. 53, Eiern. Math. 26 (1971), 58. [4.2]

H. Hadwiger and H. Debrunner 1955 Ausgewählte Einzelprobleme der kombinatorischen Geometrie in der Ebene,

Enseign. Math. (2) 1 (1955), 56-89. (French translation by J. Chatelet, En« seign. Math. (2) 3(1957), 35-70.) MR 17 p. 887. [2.3; 3.4]

1960 Kombinatorische Geometrie in der Ebene, Monogr. de FEnseign. Math. No. 2, Universite, Geneve 1960. MR 22 #11310. [2.3; 3.4]

100 BRANKO GRÜNBAUM

Ç. Hadwiger, Ç. Debrunner and V. Klee 1964 Combinatorial geometry in the plane, Holt, Rinehart and Winston, New York

1964. MR 29 #1577. [2.3; 3.4]

H. Hadwiger, H. Debrunner and I. M. Yaglom 1965 Combinatorial geometry of the plane, [In Russian.] Nauka, Moseow 1965.

MR 34 #3428. [2.3; 3.4] H. Hahn

1908 Über die Anordunungssätze der Geometrie, Monatshefte Math. Phys. 19 (1908), 289-303. FM 39 p. 550. [2.5]

M. Hall, Jr. 1967 Combinatorial Theory, Blaisdell, Waltham 1967. MR 37 #80. [2.7]

E. Halsey 1971 Ph. D. Thesis, University of Washington, Seattle 1971. [2.1; 3.1]

P. C. Hammer 1954 Diameters ofconvex bodies, Proc. Amer. Math. Soc. 5 (1954), 304-306. MR

15 p. 819. [4.1]

1963 Convex curves of constant Minkowski breadth, Proc. Sympos. Pure Math. Vol. 7 (Convexity) (1963), 291-304. MR 25 #5421. [4.2]

P. C. Hammer and A. Sobczyk 1953a Planar line families. I, Proc. Amer. Math. Soc. 4 (1953), 226-233. MR 14 p.

787. [4.2] 1953b Planar line families. II, Proc. Amer. Math. Soc. 4 (1953), 341-349. MR 15

p. 149. [4.2] H. Hanani

1951 On the number of straight lines determined by ç points, [In Hebrew]. Riveon Lematematika 5 (1951), 10-11. MR 13 p. 5. [2.2]

1954 On the number of lines and planes determined by d points, Scientif. Publ. Technion Haifa, 6 (1954), 58-63. MR 17 p. 294. [2.2]

F. Harary 1969 Graph Theory, Addison-Wesley, Reading, Mass. 1969. MR 41 #1566. [3.4]

F. Harary and A. Hill 1962 On the number of crossings in á complete graph, Proc. Edinburgh Math. Soc.

(2) 13 (1962-63), 333-338. MR 29 #602. [3.4]

H. Harborth 1971 Über die Kreuzungszahl vollständiger, ç-geteilter Graphen, Math. Nachr. 48

(1971), 179-188. [3.4]

O. Haupt 1965 Verallgemeinerung zweier Sätze über interpolatorische Funktionensysteme, Akad.

Wiss. Lit. Mainz Abh. Math. Natur. Kl. 1965, 239-255. MR 33 #7936. [4.4]

E. Heil 1971 Linienfamilien ebener konvexer Bereiche, (to appear). [4.3]


W. Heise 1970a Eine Definition des Möbiusraumes f Manuscripta Math. 2 (1970), 39—47. MR

41 #7509. [4.4] 1970b Eine neue Klasse von Möbius m-Strukturen, Rend. Ist. di Matern. Univ. di

Trieste 2 (1970), 125-128. [4.4]

B. Hesselbach 1933 Über zwei Vierecksätze der Kreisgeometrie, Abh. Math. Sem. Univ. Hamburg 9

(1933), 265-271. FM 59 p. 1232. [4.4]

D. Hubert 1899 Grundlagen der Geometrie, Teubner, Leipzig 1899. FM 30 p. 424. [4.4] 1900 Mathematische Probleme, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl 1900

(1900), 253—297. (Modified French translation: Sur les problemes futures des mathematiques. C. R. 2. Congr. Internat. Math. Paris 1900 (1902), 58-114.) FM 31 pp. 68, 905. [2.5]

D. Hubert and S. Cohn-Vossen 1932 Anschauliche Geometrie, Grundlehren Math. Wiss. Vol. 37. Springer, Berlin

1932. (English translation: Geometry and the Imagination, Chelsea, New York 1952.) FM 58 p. 597; MR 13 p. 76. [2.7]

S. Jendrol and E. Jueovic 1971a On the toroidal analogue of Eberhard's theorem, (to appear). [2.4]

1971b On á conjecture by B. Grünbaum, (to appear). [2.4]

H. F. Jensen 1971 An upper bound for the rectilinear crossing number of the complete graph, J.

Combinatorial Theory (to appear). [3.4]

R. Jerrard 1961 Inscribed Squares in plane curves, Trans. Amer. Math. Soe. 98 (1961), 234-241.

MR 22 #11354. [4.2]

F. John 1937 Polar correspondence with respect to á convex region, Duke Math. J. 3 (1937),

355-369. FM 63 p. 669. [4.4]

C. Jordan 1920 Sur la Classification des constellations, C. R. Congr. Internat. Math. Strasbourg

1920, 410-436. FM 48 p. 652. [2.6]

E. Jueovic 1967 Beitrag zur kombinatorischen Inzidenzgeometrie, Acta Math. Acad. Sei. Hung.

18 (1967), 225-259. MR 36 #766. [3.4]

E. Jueovic and M. Trenkler

1971 On 4'Valent graphs imbedded in orientable 2-manifolds, (to appear). [2.4]

W. Jung 1937 Untersuchungen über symmetrische Geradenkomplexe, Thesis, Univ. Berlin 1937.

47 pp. FM 63 p. 599. [2.3]

102 BRANKO GRUNBAUM

W. Jung and E. Melchior 1936 Symmetrische Geradenkomplexe. Ein Beitrag zur Theorie der Konfigurationen,

Deutsche Math. 1 (1936), 239-255. FM 62 p. 735. [2.3]

P. C. Kainen

1968 On á problem ofErdös, J. Combinat. Theory 5 (1968), 374-377. MR 38 #72. [3.4]

S. Kakeya 1916 On the inscribed rectangles of á closed convex curve, Tohoku Math. J. 9 (1916),

163-166. FM 46 p. 1116. [4.2]

S. Kantor 1881 Die Configurationen (3, 3), S.-Ber. Math.-Nat. Kl. Äkad. Wiss. Wien 84 (1881),

1291-1314. FM 13 p. 460. [3.1]

F. Kärteszi 1963 Alcuni problemi della geometria d'incidenza, Confer. Sem. Math. Univ. Bari No.

88 (1963), 14 pp. FM 31 #3926. [2.3]

1964 Intorno á punti allineati di certi reticoli circolari, Rend. Sem. Matern. Messina 9 (1964/65), 1-12. MR 40 #7923. [2.3]

L. M. Kelly and W. 0. J. Moser

1958 On the number of ordinary lines determined by ç points, Canad. J. Math. 10 (1958), 210-219. MR 20 #3494. [2.2; 2.3; 2.6]

L. M. Kelly and R. Rottenberg

1971 Simple points in pseudoline arrangements, (to appear). [3.2; 3.3]

E. Kivikoski 1925 Ein einfacher Beweis des Jordan sehen Kurvensatzes nebst der projektiven Ver

allgemeinerung des Satzes, Soc. Scient. Fenn., Comm. Phys.-Math. 2, 21 (1925), 12 pp. FM 51 p. 459. [3.1]

R.Klee 1938 Über die einfachen Konfigurationen der euklidischen und der projektiven Ebene,

Focken und Oltmanns, Dresden 1938. FM 64 p. 1296. [2.1; 3.1]

D. J. Kleitman

1970 The crossing number of KSn, J. Combinat. Theory 9 (1970), 315-323. [3.4]

M. Kneser 1949 Eibereiche mit geraden Schwerlinien, Math.-Phys. Semesterber. 1 (1949), 97—

98. MR 11 p. 386. [4.2]

T. Kojima 1919 On characteristic properties of the conic and quadric, Tohoku Sei. Reports 8

(1919), 67-78. FM 47 p. 682. [4.4]

K. Kommerell

1941 Die Pascalsche Konfiguration 93 , Deutsche Math. 6 (1941), 16-32. MR 3 p. 179. [2.7]


A. Kosirtski 1957 On involutions and families of compacta, Bull. Acad. Polon. Sei. Cl. III, 5 (1957),

1055-1060. MR21 #2244. [4.1] 1958 On á problem of Steinhaus, Fund. Math. 46 (1958), 47-59. MR 24 #A2379.

[4.1]

K. Koutsky and V. Poläk 1960 Note on the omittable points in complete Systems ofpoints and straight lines in

the plane, [In Czech. Russian and English summaries.] Casopis Pest. Mat. 85 (1960), 60-69. MR 24 #A448. [2.3]

T. Kubota 1939 Ein neuer Aufbau der euklidischen Geometrie in der affinen Ebene, Monatsch.

Math. Phys. 48 (1939), 96-102. FM 65 p. 638. [4.2]

G. Landsberg 1911 Beiträge zur Topologie geschlossener Kurven mit Knotenpunkten und zur Krön-

eckerchen Charakteristikentheorie, Math. Ann. 70 (1911), 563-579. FM 42 p. 509. [3.4]

D. W. Lang

1955 The dual of á well known theorem, Math. Gazette 39 (1955), 314. [2.3]

R. Lauffer 1953a Zur Topologie der Konfiguration von Desargues. I, Math. Nachr. 9 (1953), 235-

240. MR 14 p. 1008. [2.7] 1953b Zur Topologie der Konfiguration von Desargues. II, Math. Nachr. 10 (1953),

179-180. MR15p. 339. [2.7]

B. Leclerc and B. Monjardet

1969 Representation graphique d'un graphe, Math, et Sei. humaines 26 (1969), 5 1 -57. [3.4]

N. J. Lennes

1911 Theorems on the simple finite polygon and polyhedron, Amer. J. Math. 33 (1911), 37-62. FM 42 p. 511. [2.5]

F. Levi 1922 Die gegenseitige Lage von 5 und 6 Punkten in der projektiven Ebene, Ber. Ver.

Sachs. Ges. Wiss. Leipzig. Math. Phys. Kl. 74 (1922), 34-39. FM 48 p. 721. [2.7]

1926 Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade, Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig 78 (1926), 256-267. FM 52 p. 575. [2.4; 3.1; 3.2; 4.3]

1929 Geometrische Konfigurationen, Hirzel, Leipzig 1929. FM 55 p. 351. [2.6; 2.7]

B. Lindström 1971 On the realization of convex polytopes, Euler's formula and Möbius funetions,

Aequat. Math, (to appear). [2.5]

L. Locher-Ernst

1951 Polarentheorie der Eilinien, Eiern. Math. 6 (1951), 1-7. MR 12 p. 436. [4.4]


S. Loyd

1914 Cyclopedia ofpuzzles, Lamb, New York 1914. [2.3]

S. Mac Lane 1937 Á combinatorial condition for planar graphs, Fund. Math. 28 (1937), 22—32.

FM 63 p. 548. [3.4]

K. Maier 1939 Die Desarguessche Konfiguration, Deutsche Math. 4 (1939), 591-641. FM 65

p. 664. [2.7]

A. Marchaud

1948 Sur les ovales, Ann. Soc. Polon. Math. 21 (1948), 324-331. [4.4] 1959 Un theoreme sur les corps convexes, Ann. Sei. Ecole Norm. Sup. (3) 76 (1959),

283-304. [4.4]

M. L. Marx 1969 The Gauss realizability problem, Proc. Amer. Math. Soc. 22 (1969), 610-613.

MR 39 #6297. [3.4]

Yu. V. MatiyaseviS

1971 Diophantine representation ofenumerable predicates, Izvestia Akad. Nauk SSSR 35 (1971), 3-30. [2.5]

K. 0 . May 1954 The use of Condensed graphs in analytic geometry, Amer. Math. Monthly 61

(1954), 31-32. MR 15 p. 460. [4.4]

P. McMullen

1969 Linearly stable polytopes, Canad. J. Math. 21 (1969), 1427-1431. MR 40

#6364. [2.7]

1971 On zonotopes, Trans. Amer. Math. Soc. 159 (1971). [2.1; 2.7]

E. Melchior 1937 Untersuchungen über ein Problem aus der Theorie der Konfigurationen, Sehr,

math. Sem. Inst, angew. Math. Univ. Berlin 3 (1937), 181-206. FM 63 p. 1199. [2.3]

1940 Über Vielseite der projektiven Ebene, Deutsche Math. 5 (1940), 461-475. M R 3 p . 13. [2.1; 2.3; 2.7]

V. V. Menon 1966 Á theorem on partitions of mass-distributions, Pacif. J. Math. 16 (1966), 133—

137. MR 32 #4602. [4.2]

A. F. Möbius 1827 Der barycentrischer Calcul, J. A. Barth, Leipzig 1827. (Ges. Werke, Vol. 1,

pp. 1-388. S. Hirzel, Leipzig 1885.) [2.6]

J. W. Moon

1965 On the distribution of crossings in random complete graphs, SIAM Journal 13 (1965), 506-510. MR 31 #3357. [3.4]


T. More, Jr.

1959 On the construction of Venn diagrams, J. Symb. Logic 24 (1959), 303-304. MR 23 #A3683. [3.4]

L. Moser

1952 Problem 130, Mathematics Mag. (1952). [3.4] 1963 Problem p. 65, Canad. Math. Bull. 6 (1963), 113. Solution, ibid., 14 (1971),

129-130. [2.4] 1964 Problem 77, Canad. Math. Bull. 7 (1964), 137. Solution by B. Grünbaum, ibid.,

477-478. [2.4]

T. S. Motzkin

1951 The lines and planes connecting the points of afinite sei, Trans. Amer. Math. Soc. 70 (1951), 451-464. MR 12 p. 849. [2.3]

1957 Types of dissections, Abstract 112t. Bull. Amer. Math. Soc. 63 (1957), 35-36. [2.7]

1967a Nonmixed connecting lines, Abstract 67T-605. Notices Amer. Math. Soc. 14 (1967), 837. [2.7]

1967b Combinatorial realization of centrally Symmetrie convex polyhedrat Research

Problem 3-8. J. Combinat. Theory 3 (1967), 411. [3.1]

1967c "Signs ofminors", Inequalities, edited by O. Shisha, pp. 225—240. Academic Press, New York 1967. MR 36 #6432. [2.7]

T. S. Motzkin and M. O. Rabin

1971 Nonmixed connecting linesf J. Combinat. Theory (to appear). [2.7]

F. R. Moulton 1902 Á simple non-Desarguesian plane geometry, Trans. Amer. Math. Soc. 3 (1902),

192-195. FM 33 p. 497. [4.4] U. S. R. Murty

1969 "Sylvester matroids,"Recent progress in combinatorics, edited by W. T. Tutte, 283-286. MR 42 #1685. Academic Press, New York 1969. [2.3]

1970 Matroids with Sylvester property, Aequat. Math. 4 (1970), 44-50. [2.3] 1971 How many magic configurations are theret Amer. Math. Monthly 78 (1971).

1000-1002. [2.7]

J. R. Musselman 1931 The planar imprimitive group oforder 216, Amer. J. Math. 53 (1931), 333—342.

FM 57 p. 805. [2.7] 1936 Á Classification of planar sixpoints. Tohoku Math. J. 42 (1936), 114-117.

FM 62 p. 734. [2.7]

J. v. Sz. Nagy 1927 Über ein topologisches Problem von Gauss, Math. Z. 26 (1927), 579-592. FM

53 p. 550. [3.4]

S. Nakajima 1928 Eilinien mit geraden Schwerlinien, Japan. J. Math. 5 (1928), 81-84. FM 54 p.

799. [4.2]


J. Noväk 1959 Anwendungen der Kombinatorik auf das Studium ebener Konfigurationen

(124, 163), (Czech., Russian and German Summaries). Casopis PSst. Mat. 84 (1959), 257-282. MR 24 #A447. [2.7]

C. S. Ogilvy 1950 Square inscnbed in arbitrary simple closed curve, Solution of Problem 4325.

Amer. Math. Monthly 57 (1950), 423-424. [4.2]

E. Piegat

1963 0 srednicach figur wypuktych plaskich, Roczn. Polsk. Towarz. Mat. Ser. 2, 7 (1963), 51-56. [4.2]

H. Rademacher and 0. Toeplitz

1930 Von Zahlen und Figuren, Springer, Berlin 1930. FM 56 p. 62. [3.4]

K. Reidemeister 1924 Eine Kennzeichnung der Kugel nach W. Blaschke, J. Reine Angew. Math. 154

(1924), 8-14. FM 50 p. 488. [4.4] 1932 Knotentheorie, Ergebn. Math. u. Grenzgeb. Vol. 1. Springer, Berlin 1932.

FM 58 p. 1202. [3.4]

G. Ringel

1956 Teilungen der Ebene durch Geraden oder topologische Geraden, Math. Z. 64 (1955), 79-102. MR 17 p. 651. [3.1]

1957 Über Geraden in allgemeiner Lage, Eiern. Math. 12 (1957), 75-82. MR 19 p. 763. [2.1]

1964 ltExtremal problems in the theory ofgraphs," Theory ofGraphs and its Applications (Proc. Sympos. Smolenice, 1963), pp. 85—90. Publ. House Czechoslovak Acad. Sei., Prague, 1964. MR 31 #4021. [3.4]

S. Roberts

1889 On the figures formed by the intercepts of á System of straight lines in á plane, and on analogous relations in space of three dimensions, Proc. London Math. Soc. 19 (1889), 405-422. FM 20 p. 592. [2.4]

A. Robinson

1963 Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam 1963. MR 27 #3533. [2.5]

Ç. Á. Robinson 1945 Á problem of regions, Amer. Math. Monthly 52 (1945), 33-34. MR6 p. 215.

[3.4]

R. R. Rottenberg

1971 On finite sets of points in P3, Israel J. Math.10 (1971), 160-171. [2.6]

H. J. Ryser 1963 Combinatorial Mathematics, Carus Math. Monograph No. 14. Wiley, New York

1963. MR27p. 51. [2.7] 1968 An extension of á theorem of de Bruijn and Erdös on combinatorial designs,

J. Algebra 10 (1968), 246-261. MR 37 #5103. [2.2]


T. L. Saaty 1967 Two theorems on the minimum number of intersections for complete graphs, J.

Combinat. Theory 2 (1967), 571-584. MR 35 #2796. [3.4] 1969 Symmetry and the crossing number for complete graphs, J. of Research Nat. Bur.

Stand. 73B (1969), 177-186. MR 41 #6715. [3.4] 1971 On polynomials and crossing numbers ïf complete graphs, J. Combinat. Theory

10(1971), 183-184. [3.4]

H. Salzmann

1955 über den Zusammenhang in topologischen projektiven Ebenen, Math. Z. 61 (1955), 489-494. MR 16 p. 845. [4.4]

1967 Topological planes, Advances in Math. 2 (1967), 1-60. MR 36 #3201. [4.4]

J. J. Schäffer 1968 Symmetrie curves, hexagons, and the girth of spheres in dimension 3, Israel J.

Math. 6 (1968), 202-205. MR 38 #1610. [4.2]

L. Schläfli 1852 Theorie der vielfachen Kontinuität, Posthumously published in: Neue Denschriften

der Schweiz. Ges. Naturwiss. 38 (1901), iv + 239 pp. (= Ges. Math. Abh., Vol. 1, 167-387. Birkhäuser, Basel 1950.) [2.1]

H. Schröter 1889 Ueber die Bildungsweise und geometrische Construction der Configurationen 103,

Göttingen Nachr. 1889, 193-236. FM 21 p. 536. [3.1]

B. Segre 1964 Teoria di Gabis, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat.

Pura Appl. 64 (1964), 1-76. MR 29 #6370. [4.1]

A. Seidenberg

1954 Á new decision method for elementary algebra, Ann. Math. 60 (1954), 365—374. MR 16 p. 209. [2.5]

J.-P. Serre

1966 Problem 5359, Amer. Math. Monthly 73 (1966), 89. [2.3]

M. Sholander 1953 Plane geometries from convex plates, Pacif. J. Math. 3 (1953), 667-671. MR

15 p. 246. [4.4]

G. J. Simmons 1971 Á maximal 2-arrangement of sixteen lines in the projeetive plane, (to appear).

[2.4]

L. A. Skornyakov 1954 Topological projeetive planes, (in Russian), Trudy Moskov. Mat. ObSE. 3 (1954),

347-373. MR 16 p. 60. [4.4]

H. A. Smirnova 1956 The problem of g-circles, (in Russian), Mat. Sbornik, N. S. 38 (81) (1956),

397-399. MR 18 p. 227. [3.4]


Ã. J. Smith 1961 Line families in the plane, University of Wisconsin Thesis, 1961. [4.2]

L. G. Snirelman 1929 On certain geometric properties of closed curves, (in Russian), Sbornik rabot mat.

razd. sekc. estestv. tocnyh nauk Komakademii, Moscow 1929. Reprinted in Uspehi Mat. Nauk 10 (1944), 34-44. MR 7 p. 35. [4.2]

A. Sobczyk 1956 Simple line families, Pacif. J. Math. 6 (1956), 541-552. MR 19 p. 56. [4.1]

G. K. C. von Staudt 1847 Geometrie der Lage, Nürnberg, 1847. [2.1]

S. Stein 1954 Families of curves, Proc. Amer. Math. Soc. 5 (1954), 745-747. MR 16 p. 157.

[4.1]

R. Steinberg 1944 Three point collinearity, Solution of Problem 4065, Amer. Math. Monthly 51

(1944), 169-171. [2.2; 2.3]

J. Steiner 1826 Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew.

Math. 1 (1826), 349-364. [2.1; 3.4]

H. Steinhaus 1955 Quelques applications des principes topologiques á la geometne des corps con-

vexes, Fund. Math. 41 (1955), 284-290. MR 16 p. 849. [4.2] 1957 On chords ofconvex curves, Bull. Acad. Polon. Sei. Cl. III, 5 (1957), 595-597.

MR 19 p. 573. [4.2]

E. Steinitz

1906 Über die Eulersche Polyederrelationen, Arch. Math. Phys. (3) 11 (1906), 86-88. FM 37 p. 500. [2.2]

1922 Polyeder und Raumeinteilungen, Enzykl. math. Wiss. Vol. 3 (Geometrie), Part 3AB12, pp. 1-139. (1922). [3.2; 3.4]

1923 Über die Maximalzahl der Doppelpunkte bei ebenen Polygonen von gerader Seitenzahl, Math. Z. 17 (1923), 116-129. FM 49 p. 410. [3.4]

E. Steinitz and H. Rademacher 1934 Vorlesungen über die Theorie der Polyeder, Springer, Berlin 1934. FM 60 p.

497. [2.1; 3.2]

E. Stenfors 1923 Der Jordansche Kurvensatz in der projektiven Ebene und die v. Staudischen

Schnittpunktsätze, Soc. Scient. Fenn., Comm. Phys.-Math. Vol. 2, No. 5 (1923), 6pp. FM 49 p. 404. [3.1]

A. H. Stone and J. W. Tukey 1942 Generalized "sandwich" theorems, Duke Math. J. 9 (1942), 356-359. MR 4

p. 75. [4.2]


K. Strambach

1970 Sphärische Kreisebenen, Math. Z. 113 (1970), 266-292. MR 41 #7510. [4.4]

W. Süss, U. Viet and K. H. Berger 1960 "Konvexe Figuren/* Grundzüge der Mathematik, edited by H. Behnke, F. Bach

mann, K. Fladt and W. Süss, pp. 510-529. Vanderhoeck and Ruprecht, Göttingen 1960. [4.2]

J. J. Sylvester

1867 Problem 2473, Mathematical questions and Solutions from the Educational Times, 8(1867), 104-107. [2.1; 2.3]

1886 Problem 2572, Mathematical questions and Solutions from the Educational Times, 45 (1886), 127-128. (Reprinted in Vol. 59 (1893), 133-134.) [2.3]

1893 Problem 11851, Mathematical questions and Solutions from the Educational Times, 59 (1893), 98-99. [2.3; 4.3]

P. G. Tait 1877 Some elementary properties ofclosed plane curves, Messenger Math. (2) 6

(1877), 132-133. (See also: Scientific Papers, Vol. 1., Cambridge Univ. Press 1898, pp. 273-347.) FM 9 p. 393. [3.4]

A. Tarski 1951 Á decision method for elementary algebra and geometry, Univ. of California

Press, Berkeley 1951. MR 10 p. 499. [2.5]

C. J. Titus 1960 Á theory of normal curves and some applications, Pacif. J. Math. 10 (1960),

1083-1096. MR 22 #5014. [3.4]

1961 The combinatorial topology of analytic functions on the boundary of á disc, Acta Math. 106 (1961), 45-64. MR 23 #A339. [3.4]

O. Toeplitz

1911 Verh. Schweiz. Naturforsch. Gesellschaft Solothurn, 1911, p. 497. [4.2]

L. Toniheim

1950 On n-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950), 457-467. MR 12 p. 395. [4.4]

G. Trevisan

1949 Una condizione di allineamento per gli insiemi finiti di punti del piano euclideo, Rend. Seminar. Mat. Univ. Padova 18 (1949), 258-261. MR 11 p. 383. [2.3]

L. R. Treybig 1968 Á characterization of the double point structure of the protection of á poly

gonal knot in regulär position, Trans. Amer. Math. Soc. 130 (1968), 223—247. MR 36 #878. [3.4]

W. T. Tutte 1970 Toward á theory of crossing numbers, J. Combinat. Theory 8 (1970), 45—53.

MR 41 #6720. [3.4]

K. Urbanik 1955 Solution du probleme pose par P, Turin, Colloq. Math. 3 (1955), 200-201. [3.4]


Ï . Veblen and J. W. Young 1918 Projective geometry. II, Ginn, Boston 1918. (Reprint: Blaisdell, New York 1965).

FM47p . 582. [2.1; 2.2]

U. Viet 1956 Umkehrung eines Satzes von Ê Brunn über Mittelpunktseibereiche, Math.-Phys.

Semesterber. 5 (1956), 141-142. MR 18 p. 667. [4.2]

I. Vincze 1952 Über die Schwerlinie einer geschlossenen, konvexen Kurve, (In Hungarian; sum-

maries in Russian and German), C. R. Premier Congr. Math. Hungrois, 1950, 679-687. Akademiai Kiado, Budapest 1952. MR 15 p. 247. [4.4]

B. L. van der Waerden and L. J. Smid 1934 Eine Axiomatik der Kreisgeometrie und der Laguerregeometrie, Math. Ann. 110

(1934), 753-776. FM 61 p. 600. [4.4]

H. S. White 1932 The plane figure ofseven real lines, Bull. Amer. Math. Soc. 38 (1932), 59-65.

FM 58 p. 675. [2.1]

H. Whitney

1937 On regulär closed curves in the plane, Compositio Math. 4 (1937), 276-284. FM 63 p. 647. [3.4]

C. Wiener

1864 Über Vielecke und Vielflache, Leipzig 1864. [2.1; 3.4]

V. C. Williams 1968 Á proofofSylvester's theorem on collinear points, Amer. Math. Monthly 75

(1968), 980-982. [2.3]

P. de Witte

1966a Combinatorial properties offinite linear Spaces, Bull. Soc. Math. Belg. 18 (1966), 133-141. MR 34 #1913. [2.2]

1966b Á new property of non-tnvial finite linear Spaces, Bull. Soc. Math. Belg. 18 (1966), 430-438. MR 35 #5345. [2.2]

D. R. Woodall 1971 "Thrackles and deadlock," Combinatorial Mathematics and its Applications,

edited by D. J. A. Welsh, pp. 335-347. Academic Press, London 1971. [3.4]

A. M. Yaglom and I. M. Yaglom 1954 Non-elementary problems in elementary presentation, (in Russian), Biblioteka

Mat. KruXka, Moscow 1954. MR 17 p. 18. [2.3]

M. Zacharias

1941 Untersuchungen über ebene Konfigurationen (124, 163), Deutsche Mathem. 6 (1941), 147-170. MR 8 p. 219. [2.7]

1950 Über die harmonisch gekoppelten Hesseschen Konfigurationen (124, 163) und gewisse in ihnen enthaltene Konfigurationen (153), Math. Nachr. 3 (1950), 243-256. MR12p. 523. [2.7]


J. Zaks 1971 On realizing Symmetrie 3-polytopesf Israel J. Math. 10 (1971), 244-251. [3.1]

T. Zamfirescu 1967a Sur les familles continues de courbes. I, Atti Accad. Naz. Lincei Rend. Cl. Sei.

Fis. Mat. Natur. (8) 42 (1967), 771-774. MR 38 #3843. [4.1] 1967b Sur les familles continues de courbes. II, Atti Acead. Naz. Lincei Rend. Cl. Sei.

Fis. Mat. Natur. (8) 43 (1967), 13-17. MR 38 #3843. [4.1] 1967c Theoreme dual concemant les familles continues de courbes, Bull. Acad. Roy.

Belg. (5) 53 (1967), 1385-1391. MR 39 #7505. [4.1] 1968a Sur les familles continues de courbes. III, Atti Accad. Naz. Lincei, Rend. Cl. Sei.

Fis. Mat. Natur. (8) 44 (1968), 639-642. MR 40 #2034a. [4.1] 1968b Sur les familles continues de courbes. IV, Atti Accad. Naz. Lincei Rend. CL

Sei. Fis. Mat. Natur. (8) 44 (1968), 753-758. MR 40 #2034b. [4.1] 1969a On planar continuous families of curves, Canad. J. Math. 21 (1969), 513-530.

MR 39 #7506. [4.1] 1969b Les courbes fermees doubles sans points triple associees á une famille continue,

Israel J. Math. 7 (1969), 69-89. MR 39 #2035. [4.1] K. Zarankiewicz

1954 On á problem ofP. Turän conceming graphs, Fund. Math. 41 (1954), 137—145. MR16p. 156. [3,4]

1959 Bisection of plane convex sets by linest (in Polish), Wiadom. Mat. (2) 2 (1959), 228-234. MR 22 #7055. [4.2]

H. Zeitler 1970 Modelle der euklidischen und nichteuklidischen Geometrie, Praxis Math. 12

(1970), 33-38. [4.4] K. Zindler

1889 Zur Theorie der Netze und Configurationen, S.-B. Math.-Nat.Xl. Akad. Wiss. Wien 98 Part IIa (1889), 499-519. FM 21 p. 535. [2.6]

1921 Über konvexe Gebilde. II, Monatsh. Math. Phys. 31 (1921), 25-57. FM 48 p. 833. [4.2]

1922 Über konvexe Gebilde. III, Monatsh. Math. Phys. 32 (1922), 107-138. FM 48 p. 833. [4.2]

A. C. Zitronenbaum 1959 Bisecting an area and its boundary, Note 2845, Math. Gazette 43 (1959), 130-

131. [4.2]


Notes added in proof (March 1972)

1. (Page 3) Since the completion of the manuscript certain results have come to my

attention which deserve being mentioned together with those discussed in the text. The fol-

lowing notes deal with such results and references.

2. (Page 17) See also V. Klee (The use ofresearch problems in high school geometry,

Educational Studies in Math. 3 (1971), 482-489).

3. (Page 21) The problem of determining t3(n) was posed also by Alauda (Question

1664, Intermed. Math. 6 (1899), 245 and 18 (1911), 196-197); the values he indicates for

i3(9) and i3(10) are too small.

4. (Page 21) For a relation between n, f2 and the numbers f;. see E. B. EUiott, Question 6362, Math. Questions from the Educ. Times, 34 (1881), p. 120 (solutions by W.B. Grove, E. Rutter and others).

5. (Page 37) Certain related but easy questions were considered by E. Lucas (Recrea-

tions Mathimatiques, vol. 4. Gauthier-Villars, Paris 1894, pp. 155-194; FM25, p. 336) and in a problem posed by E.-N. Barisien (Question 3901, Intermed. Math. 18 (1911), 171) and solved by Welsch (ibid. 19 (1912), 18-21). Much harder are the problems concerning the different "aspects" of finite sets of points; that is the question what permutations of the given points correspond to the circular Orders in which the points are visible from other points of the (Euclidean) plane. There appear to have been no advances in this question since the results and discussions by C.-A. Laisant (Regions du plan et de Vespace. Assoc. Franf. Avanc. Sei. 1881, pp. 71—76, and Remarques sur la thaorie des regions et des as-

pects, Bull. Soc. Math. France 10 (1881-82), 52-55; FM 14, p. 151), R. Perrin (Sur le

probl&me des aspects, Bull Soc. Math. France 10 (1881-82)» 103-127; FM 14, 152, and Question 27, Intermed. Math. 1 (1894), 7-8) and A. Sainte-Lague (Giomitrie de Situation

etjewe, Memorial Sei. Math. vol. 41. Gauthier-Villars, Paris 1929, pp. 3-6 ; FM 55, p. 974).

6. (Page 37) Two arrangements were inadvertently omitted from this list. They may be obtained by adding to the marked points on the last arrangement of Figure 2.24 one of the vertices, or two "neighboring" vertices, of the "inner pentagon".

7. (Page 37) Let Cx and C2 be two cell complex decompositions of the projeetive plane P. We shall say that Ct and C2 are pieeewise projeetive provided there exists a homeomorphism è of Ñ onto itself, such that for each cell C of Cl the set 0(C) is a cell of C2» and the restriction of è to C eoineides with the restriction to C of a projeetive transformation. Á remarkable result of E. Steinitz (Über ein merkwürdiges Polyeder

von einseitiger Gesamtoberfläche, J. Reine Angew. Math. 130 (1905), 281-307; FM 37, p. 500) may be reformulated as follows: If C is á cell complex decomposition of Ñ which

is pieeewise projeetive with the simple arrangement offour lines, then C itself is á simple

arrangement offour lines. It may be conjeetured that every cell complex decomposition of Ñ pieeewise projeetive with a simple arrangement of lines is itself a simple arrangement of lines.


8. (Page 45) The ten non-isomorphic configurations 103 of pseudolines form the

basis of the attractive game Configurations: Number Puzzles and Patterns for all Ages by HL L.

Dorwart (WFF 'N PROOF, New Haven, 1967).

9. (Page 60) There also exist simplicial arrangements of 12 curves with f2 = 82» and

of 13 curves with f2 = 84.

10. (Page 68) In Figure 3.35 one could add the symbol for a simplicial arrangement

of 13 curves with i2 = 21.

11. (Page 68) W. Meyer (On ordinary points in arrangements, to appear) has proved

that t2 > n/2 for every digon-free arrangement of ç curves.

12. (Page 68) The question of finding upper bounds for ù(€) in ApoJlonian arrangements of ç circles has been proposed by G. de Rocquigny (Question 1179, Intermed. Math. 4 (1897), p. 267 and 15 (1908), p. 169).

13. (Page 71) F. Dumont (Question 1389, Intermed, Math. 5 (1898), p. 246) asked about the maximal number of regions into which the plane may be decomposed by ç co-nics; this obviously corresponds to a special case of k = 2. The answer to Dumont's question given by P. Hendle (ibid. 6 (1899), p. 137) allows pairs of straight lines; if only ellipses are allowed the answer becomes 2(n2 - ç + 1), and is valid for all arrangements of simple closed curves in the Euclidean plane, each pair of which intersect in at most 4 points.

14. (Page 72) Á slightly different code, in some respects simpler than Gauss' code, was described by P. G. Tait (Listing's Topologie, Philos. Mag. (5) 17 (1884), 30-46 = Sei. Papers, vol. 2, pp. 85-98). Proceeding along C we assign the symbols a, b, cf df etc. to the first, third, fifth, seventh, etc. vertex, tili all vertices have names; then we go around C once more, reading off the symbols we find on the second, fourth, sixth, etc. vertex. For example, in the seifintersection pattern of Figure 3.37, the letters á, bf cf d, e, f g could stand for the numbers 1, 3, 5, 2, 4, 7, 6; the Tait code would be dea'gfb c.

15. (Page 73) In this context see also A. Sainte-Lague (Les reseaux (ou graphes),

Memorial Sei. Math. vol. 18. Gauthier-Villars, Paris 1926, pp. 41-42; FM 52, p. 576) and the works of E. Kronecker, H. Weber, and others quoted there.

16. (Page 73) Á much earlier Solution of the characterization problem was given, together with many other results, in an apparently forgotten paper by M. Dehn (Über kom

binatorische Topologiet Acta Math. 67 (1936), 123-168; FM 62, p. 656-658).

17. (Page 73) Even if multiple-intersection vertices are allowed, the map determined by each selfintersection pattern is 2-colorable. S. de la Campa (Question 1451, Intermed. Math. 6 (1899), 29—30) posed the problem of determining the possible partitions of the f2 cells on the selfintersection pattern into the two color-classes for preassigned numbers of vertices of various multiplicities (t2 ·). The problem is still open, as is also the question what sequences (t2, t4, t6, · * · , t2k) correspond to selfintersection patterns.

18. (Page 74) Another paper dealing with í^(Ê ç) is R. K. Guy, Sequences associ· ated with á problem of Turin and other problems. Proc. Balatonfüred Combinatorics Conf.


1969, Vol. 4 (1970), 553-569. For a variant of the rectilinear crossing numbers see Ì . Å.

Watkins, Á special crossing number for bipartite graphs: Á research problem. Internat. Conf.

on Combinatorial Math. 1970, Ann. New York Acad. Sei. 175 (1970), 405-410; MR 42

#120.

19. (Page 80) H. Debrunner (Orthogonale Dreibeine in richtungsvollständigen, stetigen

Geradenscharen des R3, Comment. Math. Helv. 37 (1962/63), 36-43; MR 26 #6833) has

proved the following conjeeture of H. Hadwiger (Ungelöste Probleme Nr. 39, Elem. Math.

16 (1961), 30-31; Nachtrag, ibid. 18 (1963), 85): Every spread of lines in E3 contains

three mutually orthogonal lines with á common point. Debrunner also gives an example of

a spread of lines in E3 which contains only one triple of such lines.

20. (Page 83) The first mention of the planar "sandwich theorem" appears to be in F. Levi's paper Die Drittelungskurve (Math. Z. 31 (1930), 339-345; FM 55, p. 434) in which the result is attributed to an oral communication from L. Neder. The main mass-partition theorem of Levi's paper was independently rediscovered and generalized by K. Kura-towski and H. Steinhaus (Une application geometrique du thaoreme de ßrouwer sur les

points invariants, Bull. Acad. Polon. Sei., Cl. 3, 1 (1953), 83-86; MR 15, p. 336) and by

K. Borsuk (An application of the theorem on antipodes to the measure theory, Bull. Acad. Polon. Sei., Cl. 3, 1 (1953), 87-90; MR 15, p. 204).

21. (Page 83) It is not known whether in any of the spreads of "remarkable chords" mentioned on page 81, or in the spread of mideurves, the exceptional curve allowed by Theorem 4.2 actually exists.

22. (Page 84) Mideurves may be generalized to convex bodies Ê in En. For each direction u, the set of centroids of the sets Ê Ð Ç, where Ç varies over all hyperplanes with normal u, forms the mideurve corresponding to u. (For some properties of such curves see, for example, W. Blaschke, Über affine Geometrie IX: Verschiedene Bemerkungen

und Aufgaben. Ber. Verh. Sachs. Ges. Wiss. Leipzig. Math.-Phys. Kl. 69 (1917), 412-420; T. Bonnesen and W. Fenchel, Theorie der konvexen Körper; Erbegnisse der Math. vol. 3, Springer, Berlin 1934; reprint Chelsea, New York 1948; pp. 10-13, and the references given there.) Zindler [1922] proved that F2(L) Ö 0 for the spread L of mideurves of each Ê C E3, and Steinhaus [1955] strengthened this to the assertion that either F^(L) ^ 0 o r c a rd F3 = N. Theorem 4.13 implies F3(L) Ö ö for the spread L of mideurves of every convex body Ê C En.

23. (Page 84) F. Levi (Die Drittelungskurve, Math. Z. 31 (1930), 339-345; FM 55, p. 434) proves the existence of three area bisectors having a common point and enclosing equal angles; his continuity proof applies also to any other preassigned angles between the bisectors.

conferenc - american mathematical society · 2019. 2. 12. · 3.1 pseudolines and non-stretchabl e...

Documents