By Gerhard Kowol
In diesem Buch wird am Beispiel der ebenen reellen und komplexen projektiven Geometrie und der davon abgeleiteten Cayley-Klein-Geometrien versucht aufzuzeigen, dass das Mathematisieren eine weit über das Fachspezifische hinausgehende Bedeutung hat - sowohl in erkenntnistheoretischer Hinsicht als auch in Bezug auf Anwendungen. Ersteres wird durch den anschaulich-synthetischen Zugang, der im Laufe der Darstellung durch den analytischen ergänzt wird, belegt und durch philosophische und mathematikhistorische Erörterungen untermauert; letzteres erstreckt sich auch auf wenig bekannte Anwendungen innerhalb der Botanik, Kristallografie, Mechanik und Psychologie. Des weiteren werden bislang kaum bzw. nicht in Buchform dargestellte Themen behandelt wie: Natürliche Geometrie von J. Hjelmslev, Beweis des Parallelenaxioms nach P. Lorenzen (konstruktive euklidische Geometrie), Imaginärtheorie nach L. Locher-Ernst, Wegkurven und Wegflächen, Koordinatisierung der Cayley-Klein-Ebenen. Das Buch ist soweit wie möglich elementar gehalten; nur eine Vertrautheit mit mathematischer Argumentation sowie Grundkenntnisse der euklidischen Geometrie werden vorausgesetzt.
Read or Download Projektive Geometrie und Cayley—Klein Geometrien der Ebene PDF
Similar geometry books
Geometry of Homogeneous Bounded Domains
S. G. Gindikin, I. I. Pjateckii-Sapiro, E. B. Vinberg: Homogeneous Kähler manifolds. - S. G. Greenfield: Extendibility homes of genuine submanifolds of Cn. - W. Kaup: Holomorphische Abbildungen in Hyperbolische Räume. - A. Koranyi: Holomorphic and harmonic capabilities on bounded symmetric domain names. - J.
Discontinuous Groups of Isometries in the Hyperbolic Plane
This publication by means of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World warfare II - to put in writing the 1st chapters of the e-book (in German). while Fenchel,
who needed to get away from Denmark to Sweden end result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned identified
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) used to be entire in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. notwithstanding, a result of fast improvement of the topic, they felt
that giant alterations needed to be made prior to ebook.
When Nielsen moved to Copenhagen college in 1951 (where he stayed till
1955), he used to be a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The information of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and a whole manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The records additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes precise reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), was once a great deal concerned with a radical revision of the curriculum in al-
gebra and geometry, and targeted his examine within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally positioned a lot attempt into his task as editor of the newly began magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen undertaking the way in which he desired to.
After his retirement from the college, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came upon time to
finish the publication undemanding Geometry in Hyperbolic house, which used to be released by means of
Walter de Gruyter in 1989 presently after his loss of life. concurrently, and with an analogous
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination the various imprecise issues that have been within the unique
manuscript. Fenchel advised me that he reflected removal elements of the introductory
Chapter I within the manuscript, considering this may be lined by means of the e-book pointed out above;
but to make the Fenchel-Nielsen publication self-contained he eventually selected to not do
so. He did choose to omit
27, entitled Thefundamental crew.
As editor, i began in 1990, with the consent of the felony heirs of Fenchel and
Nielsen, to supply a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb activity of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the facts studying of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a basic dialogue of the difference to the fashion
of TEX. In so much respects we determined to persist with Fenchel's intentions. even though, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform in the course of the publication. additionally,
we have indicated the start and finish of an evidence within the ordinary kind of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and pride they agreed to put up the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and speedy response. One
particular challenge with the e-book became out to be the replica of the various
figures that are a vital part of the presentation. Christian Siebeneicher had at
first agreed to carry those in ultimate digital shape, yet by means of 1997 it grew to become transparent that he
would now not manage to locate the time to take action. although, the writer provided an answer
whereby I may still carry exact drawings of the figures (Fenchel didn't depart such
for Chapters IV and V), after which they might set up the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his effective
collaboration in regards to the real creation of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this e-book reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few unique symbols, a listing of notation with brief
explanations and connection with the particular definition within the publication has been integrated. additionally,
a entire index has been extra. In either situations, all references are to sections,
not pages.
We thought of including a whole checklist of references, yet made up our minds opposed to it because of
the overwhelming variety of examine papers during this sector. as an alternative, a miles shorter
list of monographs and different entire debts appropriate to the topic has been
collected.
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into lifestyles.
Statistics on Special Manifolds
This publication is anxious with statistical research at the precise manifolds, the Stiefel manifold and the Grassmann manifold, taken care of as statistical pattern areas which include matrices. the previous is represented by means of the set of m x okay matrices whose columns are at the same time orthogonal k-variate vectors of unit size, and the latter by way of the set of m x m orthogonal projection matrices idempotent of rank okay.
- Complex Manifolds and Hyperbolic Geometry: II Iberoamerican Congress on Geometry, January 4-9, 2001, Cimat, Guanajuato, Mexico
- Computations in Algebraic Geometry with Macaulay 2
- Convexity and Optimization in Rn
- Complex Geometry: An Introduction (Universitext)
Additional info for Projektive Geometrie und Cayley—Klein Geometrien der Ebene
Sample text
U ) = U . Wie beim analogen Fall im vorigen Beweis sei ι die Inversion an einem beliebigen Kreis mit Mittelpunkt κ(U ). Die Kreisverwandtschaft κ ¯ = ι ◦ κ hat ¨ dann U als Fixpunkt, so dass nach dem eben Bewiesenen κ ¯ Ahnlichkeitsabbildung ist. Wegen κ = ι ◦ κ ¯ folgt die Behauptung. 16. Jede Kreisverwandtschaft ist das Produkt von Inversionen. ¨ Beweis. Aufgrund des vorigen Satzes gen¨ ugt es zu zeigen, dass eine beliebige Ahnlichkeitsabbildung τ sich als Produkt von Inversionen darstellen l¨asst.
20. Von den dreiseitigen Figuren ist ein gleichseitiges Dreieck jede mit drei gleichen Seiten, ein gleichschenkeliges jede mit nur zwei gleichen Seiten, ein schiefes jede mit drei ungleichen Seiten. B. gleichseitiges, gleichschenkeliges und schiefes Dreieck. Es werden n¨amlich nicht nur die Grundbegriffe sondern s¨amtliche f¨ ur die im 1. Buch behandelte Theorie ben¨ otigten Begriffe vorangestellt. Bemerkenswerter ist, dass auch die Grundbegriffe definiert werden. Da diese jedoch allererste mathematische Begriffe sind, stellen die diesbez¨ uglichen Beschreibungen die Verbindung her zwischen ihnen und Begriffen, die dem Alltagsleben entnommen sind.
Drittel des 19. 3; zur Namensgebung siehe Anmerkung 109). In ihr gelten die ersten vier euklidischen Postulate, nicht jedoch das f¨ unfte, das Parallelenaxiom, wobei man zu ganz anders gearteten Ergebnissen als in der euklidischen Geometrie gelangt. , nicht mehr die urspr¨ unglichen durch die Außenwelt angeregten. 16 Die Geraden dieser Geometrie sind also – obwohl sie denselben Namen tragen – nicht die in der euklidischen Definition 4 beschriebenen geraden Linien. Mancherorts wird diesem wichtigen Umstand dadurch Rechnung getragen, dass zwischen Geraden in der euklidischen und der hyperbolischen Geometrie unterschieden wird (vgl.