Author: John Stillwell Publisher: Springer Science & Business Media ISBN: 0387255303 Category : Mathematics Languages : en Pages : 240
Book Description
This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises
Author: John Stillwell Publisher: Springer ISBN: 9781441920638 Category : Mathematics Languages : en Pages : 229
Book Description
This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises
Author: Parker Emmerson Publisher: Parker Emmerson Publishes on Lulu ISBN: 1329787293 Category : Education Languages : en Pages : 893
Book Description
The Cone of Perception describes the algebra of orbifold circle folding into a cone with fixed parameters, i.e. an invariant. This is like a mathematical quest to discover a wealth of forms and equations. I began by deciding I was going to make a scientific discovery and by asking the simple question, "at what angle do we perceive two equal line segments in golden ratio with each other?" Diagramming out this scenario, I slowly realized that one could fold the lines of sight onto each other, and the resulting shape formed a cone. Then, I attempted to describe this action algebraically in a phenomenological manner. The difference between the circumferences of two circles equals an arc length of either circle, and this can be applied to the Pythagorean theorem, the realm of relativistic physics. I also illustrate where paradoxes arise in this train of thinking and in my later works, The Sphere of Realization and The Book of Eternity, ameliorate these paradoxes entirely. One can fold a circle into a cone. When a sector of a circle is collapsed (removed, we may, "fold up," the resulting shape into a cone. Over 500 pages of mathematical formulas and graphs at your fingertips. This is the research of several years piecing together potential visualizations of the perceptual cone phenomenon. Extensive, in depth description of perceptual forms included. However, with all these equations, finding a new solution is not difficult. Great for anyone who needs to come up with a mathematical thesis in algebra, geometry, topology, or philosophy. The Cone of Perception includes many graphs and solutions to the equations of perceiving a circle to be one size and then perceiving a circle of a different size. The Cone of Perception is a work that confronts the perceptually evident purely geometric truth. The quest to discover this wealth of mathematical forms and equations began by deciding I was going to make a scientific discovery and by asking the simple question, "at what angle do we perceive two equal line segments in golden ratio with each other?" Diagraming out this scenario, I slowly realized that one could fold the lines of sight onto each other, and the resulting shape formed a cone. Then, I attempted to describe this action algebraically. The difference in circumferences of two circles equals an arc length, and this can be applied to the Pythagorean theorem and the realm of relativistic physics. I discovered certain fundamental structures within the ideal Platonic forms in the Euclidean and Pythagorean sense that can be used to perform a phenomenological description of perception and our perceived reality which is more accurate to the true nature of the Universe than current physics and beliefs about our physical reality. One can fold a circle into a cone. When a sector of a circle is collapsed (removed), we may "fold up" the resulting shape into a cone. The book relates the system of a circle transforming through a cone to the perceptual theories of Gibson, Koffka, Husserl, and Sense Data theory. It also delves into the mathematics of perceiving a difference in circumferences and presents a computational solution to the velocity variable within the Lorentz transformation. This solution is found only when using the exact speed of light in scientific notation. The auspicious symbols of the umbrella and the conch in Buddhist philosophy are perhaps a hidden message, or a hint to the true nature of reality delivered down through the ages to those who might seek to perceive and inquire. However, the mathematical expression of the, "umbrellic transformation," is one rarely discussed in Buddhist circles that I have encountered if ever, and it is certainly not vocally embodied in the vibrant message promoted and propagated by the majority of the Buddhist community, though many Buddhists do have a respect for the sciences, and math is highly prized in the societies of India and Nepal. We are only beginning to understand what the meaning of the, "phenomenological velocity," solution truly is and how the curvatures that result from the solutions to the v-variable are effecting the perceived phenomena in our reality. The idea that we can solve for something that cancels out with itself, that we can prove it cancels out with itself, yet we can solve in a non-trivial way that there is a complex polynomial equation that fits as a solution is a bit mystifying, however it is real. We ask ourselves, "why do the galaxies spiral?" We ask ourselves, what is the phenomenon of, "dark matter," and we lack answers to these basic questions, but with the new dimension (or metric) that has emerged from within the structure of the circle's folding into a cone, and the new solution to the v-variable within Lorentz coefficient as presented within The Geometric Patterns of Perception (Emmerson, 2009), we have a way forward. Physicists have assumed that mass is a real phenomenon, and have based all their formulations upon this concept. However functional the postulate of mass's, "being," is, it is still an assumption on its face. Just because a theory works, does not mean it's technically correct. Does one actually perceive a mass? Or has one inferred that a concept of mass must exist as the basis of reality, and if so, "on what notion was this inference based?" The Geometric Pattern of Perception Theorems base their functionality of describing the motion of and perceived being of, "objects," in the world through pure algebra and geometry of the transformation of ideal shapes. Through perceiving and describing these transformations phenomenologically, we can extract a plentitude of equations describing transformation and motion, which act as articulation of perceived phenomena of transformation and motion and may suffice for explaining curvature of space time relating with gravity, including the curvature perceived as correlating with dark matter. People speak of Energy to describe the phenomenon of that which is neither created nor destroyed, but really, all that is needed to describe that phenomenon is contained within the phenomenological velocity," equation, also known as V-Curvature, since it's not really even necessary to consider it velocity. We have a wave equation within the fabric of perceived reality, the expressions of which were derived from the most basic, fundamental ideal forms, that never equals zero, meaning it most likely never began, and it certainly will never end (or it can't be created, and it can't be destroyed). From this (loose) definition of Energy, we now have a theoretical "mass-energy," relation, if we still need to cling to the concepts of mass and energy.
Author: Parker Emmerson Publisher: Lulu.com ISBN: 055770846X Category : Languages : en Pages : 732
Book Description
The Cone of Perception is a work that confronts the perceptually evident purely geometric truth. The difference in circumferences of two circles equals an arc length, and this can be applied to the Pythagorean theorem and the realm of relativistic physics. Over 500 pages of mathematical formulas and graphs at your fingertips. This is the research of several years piecing together potential visualizations of the perceptual cone phenomenon. Extensive, in depth description of perceptual forms. However, with all these equations, finding a new solution is not difficult. Great for anyone who needs to come up with a mathematical thesis in algebra, geometry, topology, or philosophy.
Author: Jeffrey R. Weeks Publisher: CRC Press ISBN: 1351668552 Category : Mathematics Languages : en Pages : 308
Book Description
The Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spaces—stretching students’ minds as they learn to visualize new possibilities for the shape of our universe. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space. Features of the Third Edition: Full-color figures throughout "Picture proofs" have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussion of cosmological applications Intuitive examples missing from many college and graduate school curricula About the Author: Jeffrey R. Weeks is a freelance geometer living in Canton, New York. With support from the U.S. National Science Foundation, the MacArthur Foundation and several science museums, his work spans pure mathematics, applications in cosmology and—closest to his heart—exposition for the general public.
Author: Vladimir Rovenski Publisher: Springer Science & Business Media ISBN: 0387712771 Category : Mathematics Languages : en Pages : 463
Book Description
This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors and transformations using matrices. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines.
Author: Jeffrey Hoffstein Publisher: Springer Science & Business Media ISBN: 0387779949 Category : Mathematics Languages : en Pages : 524
Book Description
An Introduction to Mathematical Cryptography provides an introduction to public key cryptography and underlying mathematics that is required for the subject. Each of the eight chapters expands on a specific area of mathematical cryptography and provides an extensive list of exercises. It is a suitable text for advanced students in pure and applied mathematics and computer science, or the book may be used as a self-study. This book also provides a self-contained treatment of mathematical cryptography for the reader with limited mathematical background.
Author: William Stein Publisher: Springer Science & Business Media ISBN: 0387855254 Category : Mathematics Languages : en Pages : 173
Book Description
This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.
Author: Matthias Beck Publisher: Springer Science & Business Media ISBN: 0387291393 Category : Mathematics Languages : en Pages : 242
Book Description
This textbook illuminates the field of discrete mathematics with examples, theory, and applications of the discrete volume of a polytope. The authors have weaved a unifying thread through basic yet deep ideas in discrete geometry, combinatorics, and number theory. We encounter here a friendly invitation to the field of "counting integer points in polytopes", and its various connections to elementary finite Fourier analysis, generating functions, the Frobenius coin-exchange problem, solid angles, magic squares, Dedekind sums, computational geometry, and more. With 250 exercises and open problems, the reader feels like an active participant.
Author: Omar Hijab Publisher: Springer Science & Business Media ISBN: 0387693165 Category : Mathematics Languages : en Pages : 342
Book Description
Intended for an honors calculus course or for an introduction to analysis, this is an ideal text for undergraduate majors since it covers rigorous analysis, computational dexterity, and a breadth of applications. The book contains many remarkable features: * complete avoidance of /epsilon-/delta arguments by using sequences instead * definition of the integral as the area under the graph, while area is defined for every subset of the plane * complete avoidance of complex numbers * heavy emphasis on computational problems * applications from many parts of analysis, e.g. convex conjugates, Cantor set, continued fractions, Bessel functions, the zeta functions, and many more * 344 problems with solutions in the back of the book.
Author: John Stillwell
Author: John Stillwell
Author: Parker Emmerson
Author: Parker Emmerson
Author: Jeffrey R. Weeks
Author: Vladimir Rovenski
Author: Jeffrey Hoffstein
Author: William Stein
Author: Matthias Beck
Author: Omar Hijab