Download or read book Multiplanes and Multispheres written by John Browne and published by John M Browne. This book was released on 2020-06-27 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Diccionario polit cnico de las lenguas espa ola e inglesa Espa ol ingl s written by and published by . This book was released on 1965 with total page 1480 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A New Branch of Mathematics written by Hermann Grassmann and published by Open Court Publishing Company. This book was released on 1995 with total page 555 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a translation of the work of Grassmann, a 19th-century mathematician, who is acknowledged as one of the founders of vector analysis. His ideas have recently been applied in elementary particle physics and computer sciences, as well as in analytic geometry.
Download or read book Hermann Gra mann written by Hans-Joachim Petsche and published by Springer Science & Business Media. This book was released on 2009-12-30 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hermann Günther Graßmann was one of the most remarkable personalities in 19th-century science. A "small-town genius", he developed a groundbreaking n-dimensional algebra of space and contributed to a revolution in the understanding of mathematics. His work fascinated great mathematicians such as W. R. Hamilton, J. W. Gibbs and A. N. Whitehead. This intellectual biography traces Graßmann’s steps towards scientific brilliance by untangling a complicated web of influences: the force of unsolved problems in mathematics, Friedrich Schleiermacher’s Dialectic, German Romanticism and life in 19th-century Prussia. The book also introduces the reader to the details of Graßmann’s mathematical work without neglecting his achievements in Sanskrit philology and physics. And, for the first time, it makes many original sources accessible to the English-language reader.
Download or read book Hermann G nther Gra mann 1809 1877 Visionary Mathematician Scientist and Neohumanist Scholar written by Gert Schubring and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume specialists in mathematics, physics, and linguistics present the first comprehensive analysis of the ideas and influence of Hermann G. Graßmann (1809-1877), the remarkable universalist whose work recast the foundations of these disciplines and shaped the course of their modern development.
Download or read book Automated Deduction in Geometry written by Dongming Wang and published by Springer Science & Business Media. This book was released on 1998-03-18 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed and revised post-workshop proceedings of the International Workshop on Automated Deduction in Geometry, held in Toulouse, France, in September 1996. The revised extended papers accepted for inclusion in the volume were selected on the basis of double reviewing. Among the topics covered are automated geometric reasoning and the deduction applied to Dixon resultants, Gröbner bases, characteristic sets, computational geometry, algebraic geometry, and planet motion; furthermore the system REDLOG is demonstrated and the verification of geometric statements as well as the automated production of proof in Euclidean Geometry are present.
Download or read book The Search for Mathematical Roots 1870 1940 written by I. Grattan-Guinness and published by Princeton University Press. This book was released on 2011-02-11 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.
Download or read book Grassmann Algebra Volume 1 Foundations written by John Browne and published by John M Browne. This book was released on 2012-10-25 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book A History of Vector Analysis written by Michael J. Crowe and published by Courier Corporation. This book was released on 1994-01-01 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers to the final acceptance around 1910 of the modern system of vector analysis.
Download or read book Castilla s Spanish and English Technical Dictionary Spanish English written by Castilla Ediciones S.A. and published by . This book was released on 1958 with total page 1156 pages. Available in PDF, EPUB and Kindle. Book excerpt: Only the fields of engineering technology are included, and the physical, chemical, and biological sciences are excluded, except for those words which are of importance to engineers and technologists. Commercial timbers are included with both their scientific and common names. Important commercial and legal terms are included.
Download or read book Die Ausdchnungslehre Von 1844 Oder Die Lineale Ausdehnungslehre Ein Neuer Zweig Der Mathematik Da written by Hermann Grassmann and published by Legare Street Press. This book was released on 2022-10-27 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
