Grassmann Algebra Volume 1 Foundations

Author: John Browne
Editor: John M Browne
ISBN: 1479197637
File Size: 63,96 MB
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Grassmann Algebra Volume 1: Foundations Exploring extended vector algebra with Mathematica Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors, trivectors, … multivectors. The extensive exterior product also has a regressive dual: the regressive product. The pair behaves a little like the Boolean duals of union and intersection. By interpreting one of the elements of the vector space as an origin point, points can be defined, and the exterior product can extend points into higher-grade located entities from which lines, planes and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involving these entities and the dual products. By introducing the (orthogonal) complement operation, the scalar product of vectors may be extended to the interior product of multivectors, which in this more general case may no longer result in a scalar. The notion of the magnitude of vectors is extended to the magnitude of multivectors: for example, the magnitude of the exterior product of two vectors (a bivector) is the area of the parallelogram formed by them. To develop these foundational concepts, we need only consider entities which are the sums of elements of the same grade. This is the focus of this volume. But the entities of Grassmann algebra need not be of the same grade, and the possible product types need not be constricted to just the exterior, regressive and interior products. For example quaternion algebra is simply the Grassmann algebra of scalars and bivectors under a new product operation. Clifford, geometric and higher order hypercomplex algebras, for example the octonions, may be defined similarly. If to these we introduce Clifford's invention of a scalar which squares to zero, we can define entities (for example dual quaternions) with which we can perform elaborate transformations. Exploration of these entities, operations and algebras will be the focus of the volume to follow this. There is something fascinating about the beauty with which the mathematical structures that Hermann Grassmann discovered describe the physical world, and something also fascinating about how these beautiful structures have been largely lost to the mainstreams of mathematics and science. He wrote his seminal Ausdehnungslehre (Die Ausdehnungslehre. Vollständig und in strenger Form) in 1862. But it was not until the latter part of his life that he received any significant recognition for it, most notably by Gibbs and Clifford. In recent times David Hestenes' Geometric Algebra must be given the credit for much of the emerging awareness of Grass­mann's innovation. In the hope that the book be accessible to scientists and engineers, students and professionals alike, the text attempts to avoid any terminology which does not make an essential contribution to an understanding of the basic concepts. Some familiarity with basic linear algebra may however be useful. The book is written using Mathematica, a powerful system for doing mathematics on a computer. This enables the theory to be cross-checked with computational explorations. However, a knowledge of Mathematica is not essential for an appreciation of Grassmann's beautiful ideas.

Grassmann Algebra

Author: John Browne
Editor:
ISBN:
File Size: 10,74 MB
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This book is an abridged draft edition. It is being published now in this form due to circumstances beyond the author's control. There will not be another edition. You will miss some things you might have expected: page numbers in the table of contents, a comprehensive bibliography and references, an index, a well-reviewed text .. . This book (Volume 2) follows on from Volume 1 so you may also need Volume 1 for reference. The book shows how the hypercomplex and associative algebras are hidden in Grassmann algebra. It shows how the quaternions and octonions and their split variants, and the geometric and Clifford algebras, are simply Grassmann algebra. It does this by extending its two familiar product operations, the exterior and interior products to define a suite of products together called the generalized Grassmann product. It shows how hypercomplex, geometric and Clifford products may then be defined as linear combinations of generalized Grassmann products in which the scalar coefficients are restricted to unity or negative unity only. This binary variability is sufficient to endow properties to a product operation, for example associativity. The book concludes by finding four associative product operations, two of which being the geometric and Clifford products. But these work on Grassmann entities only, so they are operations of the Grassmann algebra. In sum: this book shows how some important linear algebras such as the hypercomplex, geometric and Clifford algebras can be constructed entirely within Grassmann algebra by defining specialized product operations using only the exterior and interior products.

Multiplanes And Multispheres

Author: John Browne
Editor: John M Browne
ISBN:
File Size: 47,72 MB
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This book is a collection of notes exploring multiplanes and multispheres using Grassmann algebra with Mathematica. A multiplane is an m-dimensional generalization of the notions of point, line, plane and hyperplane. A multisphere is an m-dimensional generalization of the notions of point-pair, circle, sphere and hypersphere. Grassmann algebra is a generalization of the notions of scalars, vectors and vector spaces. Mathematica is a system for doing mathematics on a computer.Grassmann algebra has now emerged as one of the more important tools for exploring multidimensional geometry and mathematical physics. It not only generalizes the classic vector algebra to enable construction of (unlocated) bivectors, trivectors and multivectors, it is also an algebra par excellence for working with located entities such as points, lines, planes and multiplanes.But multiplanes are not alone in their space. To every multiplane corresponds a docked multisphere and vice versa. (A docked multisphere passes through the origin.) Corresponding points on a multiplane-multisphere pair are inverses. And because we can easily dock a multisphere by adding a displacement vector to its points, we can work with multispheres by operating on their corresponding multiplanes. For example: we can intersect two multispheres, or a multisphere and a multiplane; construct the best-fit multisphere to a system of points; compute the complex of circles for a Clifford circle theorem, or generate the in-multisphere of a simplex.

Foundations Of Game Engine Development Volume 1

Author: Eric Lengyel
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ISBN: 9780985811747
File Size: 36,61 MB
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"The first volume of Foundations of Game Engine Development discusses the mathematics needed by engineers who work on games or other types of virtual simulations. The book begins with conventional treatments of topics such as linear algebra, transforms, and geometry. Then, it introduces Grassmann algebra and geometric algebra to provide a much deeper understanding of the subject matter and highlight the places where traditional arithmetic with vectors, matrices, quaternions, etc."--Provided by Publisher.

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Author: F.A. Berezin
Editor: Springer Science & Business Media
ISBN: 9401719632
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Author: William Vallance Douglas Hodge
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Author: Shimshon A. Amitsur
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1986 Ieee International Symposium On Circuits And Systems Le Baron Hotel San Jose California May 5 7 1986

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Algebra Volume 3

Author: P. M. Cohn
Editor: Wiley
ISBN: 9780471928409
File Size: 30,63 MB
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 edition. Excerpt: ...bank of the flood oer which there is no return. 426--439. The inhabitants of the first regions of hell. Straightway cries are heard, and wailing loud, and the spirits of infants weeping in the entrance of the door, whom, without sharing the sweets of life, and torn from the breast, the day of doom swept away, and plunged into untimely death. Next to these are they that were condemned to die on a false charge: and these abodes are by no means assigned without allotment, without a judge; Minos rules the scrutiny, and shakes the urn; he convokes the conclave of the silent dead, and learns their lives, and the charges brought against them. The regions that come next in order are filled by a sad company, who, without guilt, have been the authors of their own death by violence, and sick of the light of day have flung away their lives. How ready they would be now, to endure beneath the height of the sky both penury and hard struggles The law of Heaven forbids, and the unlovely marsh with its joyless flood binds them in, and Styx hems them round with nine circles of its stream. 440--476. The Mourning Fields. jEneas sees Dido, and tries in vain to soothe her. Not far from hence the Mourning Fields are shewn, spreading on every side; such is the name by which they call them. Here they whom cruel love has wasted away with pining pain, are concealed in secluded paths, and covered all round by a myrtle grove; not even in death do their woes desert them. In these regions he descries Phaedra, and Procris, and Eri-phyle full of grief, pointing to the wounds dealt by her cruel son, and Evadne, and Pasiphae; in company with them Laodamia goes along, and Caeneus, once a youth, a woman now, and again transformed by fate into her original shape. Among them...

Government Reports Annual Index

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