Spreads Of Three Dimensional And Five Dimensional Finite Projective Geometries
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Author | : |
Publisher | : |
Total Pages | : |
Release | : 2009 |
Genre | : Finite fields (Algebra) |
ISBN | : 9781109392982 |
This thesis is primarily concerned with specific types of spreads of three-dimensional and five-dimensional projective geometries over finite fields. Spreads are a partition of a projective geometry, and are used to construct translation planes through the Andre/Bruck-Bose construction. This thesis uses the Bruck-Bose model, which is more geometric in nature. The types of spreads examined include the following: spreads of five-dimensional projective geometries for three-dimensional flag-transitive affine planes, polarity-paired spreads of three-dimensional projective geometries, and spreads of five-dimensional projective geometries constructed from a three-dimensional circle geometry. In the introduction to the thesis, a short historical account is given of some aspects of modern incidence geometry. Specifically, a partial history of the theory of projective and affine planes that leads to the study of translation planes. In Chapters Two and Three the definitions of a projective plane and translation plane are given, along with properties of these objects that will be useful in their study. Also the classical (Desarguesian) projective plane and the classical projective geometries are defined. It is these higher-dimensional Desarguesian geometries that are needed for the Bruck-Bose model of translation planes. The Andre/Bruck-Bose construction is explained in Chapter Four. This includes a discussion of the Miquelian inversive plane, which can be used to model a fundamental family of spreads called "regular". In Chapter Five spreads of five-dimensional projective geometries are used to construct odd order three-dimensional flag-transitive affine planes. This involves examining the way that planes in the spread intersect a partition of a five-dimensional geometry. Chapter Six is concerned with polarities of three-dimensional geometries applied to spreads of that geometry, leading to the concept of polarity-paired spreads. The symplectic polarity-paired spreads are used to construct a certain class of ovoids of a specific generalized quadrangle. In Chapter Seven a three-dimensional circle geometry is used to construct spreads of five-dimensional projective geometries. This circle geometry and spreads constructed from a regular spread mirror the concept of the Miquelian inversive plane and its relationship to subregular spreads from a regular spread of a three-dimensional projective geometry. Finally, the possibility of further work is discussed in Chapter Eight.
Author | : Alfred S Posamentier |
Publisher | : World Scientific |
Total Pages | : 441 |
Release | : 2021-11-24 |
Genre | : Mathematics |
ISBN | : 9811237123 |
The book presents a comprehensive overview of various aspects of three-dimensional geometry that can be experienced on a daily basis. By covering the wide range of topics — from the psychology of spatial perception to the principles of 3D modelling and printing, from the invention of perspective by Renaissance artists to the art of Origami, from polyhedral shapes to the theory of knots, from patterns in space to the problem of optimal packing, and from the problems of cartography to the geometry of solar and lunar eclipses — this book provides deep insight into phenomena related to the geometry of space and exposes incredible nuances that can enrich our lives.The book is aimed at the general readership and provides more than 420 color illustrations that support the explanations and replace formal mathematical arguments with clear graphical representations.
Author | : Peter Dembowski |
Publisher | : Springer Science & Business Media |
Total Pages | : 414 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 9783540617860 |
Peter Dembowski was born in Berlin on April 1, 1928. After studying mathematics at the University of Frankfurt of Main, he pursued his graduate studies at Brown Unviersity and the University of Illinois, mainly with R. Baer. Dembowski returned to Frankfurt in 1956. Shortly before his premature death in January 1971, he had been appointed to a chair at the University of Tuebingen. Dembowski taught at the universities of Frankfurt and Tuebingen and - as visiting Professor - in London (Queen Mary College), Rome, and Madison, WI. Dembowski's chief research interest lay in the connections between finite geometries and group theory. His book "Finite Geometries" brought together essentially all that was known at that time about finite geometrical structures, including key results of the author, in a unified and structured perspective. This book became a standard reference as soon as it appeared in 1968. It influenced the expansion of combinatorial geometric research, and left its trace also in neighbouring areas.
Author | : Albrecht Beutelspacher |
Publisher | : Cambridge University Press |
Total Pages | : 272 |
Release | : 1998-01-29 |
Genre | : Mathematics |
ISBN | : 9780521483643 |
Projective geometry is not only a jewel of mathematics, but has also many applications in modern information and communication science. This book presents the foundations of classical projective and affine geometry as well as its important applications in coding theory and cryptography. It also could serve as a first acquaintance with diagram geometry. Written in clear and contemporary language with an entertaining style and around 200 exercises, examples and hints, this book is ideally suited to be used as a textbook for study in the classroom or on its own.
Author | : David John Oakden |
Publisher | : |
Total Pages | : 258 |
Release | : 1973 |
Genre | : Geometry, Algebraic |
ISBN | : |
Author | : Jeffrey R. Weeks |
Publisher | : Marcel Dekker |
Total Pages | : 352 |
Release | : 1985 |
Genre | : Mathematics |
ISBN | : |
The Shape of Space brings topology to the general reader by showing how to visualize manifolds directly ... complements existing textbooks, which often deal only in abstractions, by offering a wealth of concrete examples ... includes the first elementary exposition of William P. Thurston's revolutionary discoveries ... applies topology to cosmology ... gives the first simple pictorial exposition of the Gauss-Bonnet formula ... builds intuition with more than 140 hands-on exercises, all with complete solutions ... and offers over 170 illustrations. An annotated bibliography lists useful references for further study on specific topics.
Author | : Izu Vaisman |
Publisher | : CRC Press |
Total Pages | : 290 |
Release | : 2020-11-26 |
Genre | : Mathematics |
ISBN | : 1000146340 |
This book presents to the reader a modern axiomatic construction of three-dimensional Euclidean geometry in a rigorous and accessible form. It is helpful for high school teachers who are interested in the modernization of the teaching of geometry.
Author | : Catherine Anne Baker |
Publisher | : CRC Press |
Total Pages | : 400 |
Release | : 2020-10-14 |
Genre | : Mathematics |
ISBN | : 1000103250 |
This book is a compilation of the papers presented at the conference in Winnipeg on the subject of finite geometry in 1984. It covers different fields in finite geometry: classical finite geometry, the geometry of finite planes, geometric structures and the theory of translation planes.
Author | : Catherine Anne Baker |
Publisher | : CRC Press |
Total Pages | : 399 |
Release | : 2020-10-15 |
Genre | : Mathematics |
ISBN | : 1000146685 |
This book is a compilation of the papers presented at the conference in Winnipeg on the subject of finite geometry in 1984. It covers different fields in finite geometry: classical finite geometry, the geometry of finite planes, geometric structures and the theory of translation planes.
Author | : Dieter Jungnickel |
Publisher | : Springer Science & Business Media |
Total Pages | : 242 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461313953 |
Designs and Finite Geometries brings together in one place important contributions and up-to-date research results in this important area of mathematics. Designs and Finite Geometries serves as an excellent reference, providing insight into some of the most important research issues in the field.