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"What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobachevski to Euclid." Lobachevski was the first to publish non-Euclidean geometry. An unabridged printing, to include all figures, from the translation by Halsted.

The secrets of life and the Universe revealed...
Living with Geometry is about how civilization has been using and abusing Geometry and other related scientific subject areas for our finite needs in the physical, material plane for ages gone by, and how we must now turn to the natural world, learn from it and employ the ancient and trusted sciences of a wondrous yesteryear to carry on in future, in order for the human spirit to be freed from its material confines, and how the physical environment can be saved along with us.
It is about what went missing, what went wrong, and what we need to learn about, in order to deal with the problem. The project became literally, "A Survey for the Truth."

Visualize hypercheckers and hyperchess in the 4th dimension, rotate a Rubik's hypercube, stack tesseracts together to build a hyperpyramid or a hypercross, view a hypertable with hyperchairs, climb hyperstairs, open the hyperdoor to a hyperhouse, watch hypertelevision, read a hyperbook, arrange glomes in a 4D lattice structure to create hypercrystals, stack hyperfruits at a hypersupermarket, record the position of a hyperstar with hypercelestial coordinates, make a hypermap using hypercompass directions, watch a spinning hyperplanet with tilt revolve around a hypersun, see a sample alphanumeric system for writing and arithmetic in 4D space, line up a shot on a hyperbilliards table, enjoy hyperbowling or hypertennis, and contemplate an object's reflection from a hyperplanar or hyperspherical mirror - all on this colorful journey through the fourth dimension. Each page consists of colorful images of four-dimensional objects with a paragraph caption describing the figures at the bottom.

These full-color illustrations on 8"x10" pages are at once aesthetically captivating and instructive. Each page consists of colorful images of tesseracts (4D hypercubes) or glomes (4D hyperspheres) with a paragraph caption describing the figures at the bottom. Color is used effectively to show how to visualize the features of tesseracts and glomes, how to draw tesseracts in perspective, how a tesseract unfolds, how the features of tesseracts and glomes change as they rotate, how to find the intersection of a tesseract or glome with a hyperplane, how hyperspherical coordinates are defined, how to understand hypercompass directions, and how to draw longitudes, hyperlatitudes, and latitudes. Rectangular hyperboxes and a hyperellipsoid are also shown. Subsequent volumes of this series will build upon these fundamental 4D objects to help you imagine features of a 4D world such as a hyperchair, a hypercross, a hyperpyramid, a hyperhouse, crystal structures, and simple hypermachines.

AUTHOR: Chris McMullen earned his Ph.D. in particle physics from Oklahoma State University. Dr. McMullen currently teaches physics at Northwestern State University of Louisiana. His background on the geometry and physics of a possible fourth dimension of space includes a half-dozen research papers on the prospects of discovering large extra dimensions at the Large Hadron Collider.DESCRIPTION: This book takes you on a visual tour of a fourth dimension of space. It is much more visual and conceptual than algebraic, yet it is detailed and technical, with the intention of satisfying the needs of mathematically-minded readers familiar with the fundamentals of algebra, geometry, and graphing. Here is a sample of what you will find: A fascinating tour of the second and lower dimensions, which will help to understand the fourth dimension by analogy.A chapter dedicated toward imagining what it might be like to live in a hypothetical 4D hyperuniverse. This includes details like 4D wheels with axles, a 4D staircase, and a 4D room.Pictures of flat 4D objects called polytopes, like the tesseract, pentachoron, and icositetrachoron. A unique graph of a hecatonicosachoron has 12 of its 120 bounding dodecahedra highlighted to help visualize its complicated structure.In-depth discussion of the hypercube, including numerical patterns, rotations, cross sections, and perspective. Watch a tesseract unfold.Visual intersections of 15 pairs of perpendicular planes and 6 pairs of orthogonal hyperplanes in 4D space.Unique graphs of curved hypersurfaces in 4D space, like the glome, spherinder, cubinder, and hyperparaboloid.PUZZLES: Several puzzles are included to challenge the reader to contemplate the fourth dimension. Answers are included at the back of the book.AUDIENCE: This book is highly visual and very conceptual such that anyone with an appreciation for geometry may understand it, while at the same time including ample detail to also satisfy readers with a strong background in mathematics.

Bertrand Russell was a prolific writer, revolutionizing philosophy and doing extensive work in the study of logic. This, his first book on mathematics, was originally published in 1897 and later rejected by the author himself because it was unable to support Einstein's work in physics. This evolution makes An Essay on the Foundations of Geometry invaluable in understanding the progression of Russell's philosophical thinking. Despite his rejection of it, Essays continues to be a great work in logic and history, providing readers with an explanation for how Euclidean geometry was replaced by more advanced forms of math. British philosopher and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872-1970) won the Nobel Prize for Literature in 1950. Among his many works are Why I Am Not a Christian (1927), Power: A New Social Analysis (1938), and My Philosophical Development (1959).

An Unabridged Printing, To Include All Exercises: Foundations - Elementary Theorems On Order - The Affine Group In The Plane - Euclidean Plane Geometry - Ordinal And Metric Properties Of Conics - Inversion Geometry And Related Topics - Comprehensive Index

"Lobachevsky believed that another form of geometry existed, a non-Euclidean geometry, and this 1840 treatise is his argument on its behalf. Line by line in this classic work he carefully presents a new and revolutionary theory of parallels, one that allows for all of Euclids axioms, except for the last. This 1891 translation includes a bibliography and translator George B. Halsteds essay on elliptic geometry. Russian mathematician NICHOLAS LOBACHEVSKY (17921856) is best remembered as the founder (along with Janos Bolyai) of non-Euclidean geometry. He is also the author of New Foundations of Geometry (18351838) and Pangeometry (1855)."

This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Until now, much of the material has been available only in the periodical literature.

The name non-Euclidean was used by Gauss to describe a system of geometry which differs from Euclid's in its properties of parallelism. Such a system was developed independently by Bolyai in Hungary and Lobatschewsky in Russia, about 120 years ago. Another system, differing more radically from Euclid's, was suggested later by Riemann in Germany and Cayley in England. The subject was unified in 1871 by Klein, who gave the names of parabolic, hyperbolic, and elliptic to the respective systems of Euclid-Bolyai-Lobatschewsky, and Riemann-Cayley. Since then, a vast literature has accumulated. The Fifth edition adds a new chapter, which includes a description of the two families of 'mid-lines' between two given lines, an elementary derivation of the basic formulae of spherical trigonometry and hyperbolic trigonometry, a computation of the Gaussian curvature of the elliptic and hyperbolic planes, and a proof of Schlafli's remarkable formula for the differential of the volume of a tetrahedron.

This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting "gyrolanguage" of the book one attaches the prefix "gyro" to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Moebius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Moebius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.