Exploring Scale Symmetry

Welcome to the world of scale symmetry, the last elementary symmetry and the least explored!Find out how this long-neglected element transforms the traditional geometry of lines and planes into a rich landscape of trees, craggy mountains and rolling oceans.Enjoy a visual exploration through the intricate and elaborate structures of scale-symmetric geometry. See unique fractals, Mandelboxes, and automata and physical behaviors. Take part in the author's forage into the lesser-trodden regions of this landscape, and discover unusual and attractive specimens!You will also be provided with all the tools needed to recreate the structures yourself.Every example is new and developed by the author, and is chosen because it pushes the field of scale-symmetric geometry into a scarcely explored region. The results are complex and intricate but the method of generation is often simple, which allows it to be presented graphically without depending on too much mathematical syntax. If you are interested in the mathematics, science and art of scale symmetry, then read on!This is also a book for programmers and for hobbyists: those of us who like to dabble with procedural imagery and see where it leads.

This textbook is perfect for a math course for non-math majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas. It includes many topics commonly found in sampler courses, like Platonic solids, Euler’s formula, irrational numbers, countable sets, permutations, and a proof of the Pythagorean Theorem. All of these topics serve a single compelling goal: understanding the mathematical patterns underlying the symmetry that we observe in the physical world around us. The exposition is engaging, precise and rigorous. The theorems are visually motivated with intuitive proofs appropriate for the intended audience. Students from all majors will enjoy the many beautiful topics herein, and will come to better appreciate the powerful cumulative nature of mathematics as these topics are woven together into a single fascinating story about the ways in which objects can be symmetric.

Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models--the first examples of the musical spaces of the book's title--depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory. Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.

The Architecture of Music Volume 1.0 is the most complete combined chord, scale, and mode encyclopedia for the guitar and piano ever created, an effort almost two decades in the making. The chord encyclopedia includes more than 300 unique chords, and depicts every possible way to play them on the guitar and piano. The scale and mode encyclopedia contains 5 scales (33 modes) in every key, as well as complete lists of chords included in the chord encyclopedia that you are able to play with every note, in every scale, in every key. The book's two new innovative diagrams, the interval diagram and linear circle of fifths, expose the architecture behind chords, scales, and modes, and explain basic music theory without the use of standard musical notation. The information contained within is meant to be used as a comprehensive reference guide and tool for exploring and analyzing chords, scales, and modes through musical improvisation and composition.

Covering general theoretical concepts and the research to date, this book demonstrates that Bose-Einstein condensation is a truly universal phenomenon.

This is an expanded version of the report by the Electroweak Symmetry Breaking and Beyond the Standard Model Working Group which was contributed to Particle Physics — Perspectives and Opportunities, a report of the Division of Particles and Fields Committee for Long Term Planning. One of the Working Group's primary goals was to study the phenomenology of electroweak symmetry breaking and attempt to quantify the “physics reach” of present and future colliders. Their investigations encompassed the Standard Model — with one doublet of Higgs scalars — and approaches to physics beyond the Standard Model. These include models of low-energy supersymmetry, dynamical electroweak symmetry breaking, and a variety of extensions of the Standard Model with new particles and interactions. The Working Group also considered signals of new physics in precision measurements arising from virtual processes and examined experimental issues associated with the study of electroweak symmetry breaking and the search for new physics at present and future hadron and lepton colliders.This volume represents an important contribution to the efforts being made to advance the frontiers of particle physics.

In The Hidden Symmetry of 43 Octatonic Scales and 43 Tetrachords, Creamer provides an extensive explanation and analysis of his system of octatonic (eight-note) harmonizations and melodic organization as well as a series of exercises, complete musical examples and original compositions utilizing the system. While the book includes the diminished scale and all of the eight-note bebop scales, Creamer goes well beyond their traditional use in a jazz context and introduces a vast new musical language where all eight notes are utilized as scale tones creating thousands of chord combinations, tonal colors and melodic possibilities that can be used by improvisers and composers in any musical context for generating new ideas and expanding traditional harmonic and melodic approaches. Guitarists will also benefit from the inherit symmetrical fingerings of the system (eight-notes-per-two-strings) as well as full tablature for all examples have been provided. Much more than a series of possible mathematical combinations, the book is presented as a complete system, and while a thorough theoretical framework is presented for contextual understanding, the music is first and foremost the focus of the work as articulated in the book’s Foreward by Tuck Andress. Perhaps not since George Russell’s book, Lydian Chromatic Concept of Tonal Organization, has a book had such tremendous potential for modern composers and musicians.

We know very little about the time-evolution of many-particle dynamical systems, the subject of our book. Even the 3-body problem has no explicit solution (we cannot solve the corresponding system of differential equations, and computer simulation indicates hopelessly chaotic behaviour). For example, what can we say about the typical time evolution of a large system starting from a stage far from equilibrium? What happens in a realistic time scale? The reader's first reaction is probably: What about the famous Second Law (of thermodynamics)?Unfortunately, there are plenty of notorious mathematical problems surrounding the Second Law. (1) How to rigorously define entropy? How to convert the well known intuitions (like 'disorder' and 'energy spreading') into precise mathematical definitions? (2) How to express the Second Law in forms of a rigorous mathematical theorem? (3) The Second Law is a 'soft' qualitative statement about entropy increase, but does not say anything about the necessary time to reach equilibrium.The object of this book is to answer questions (1)-(2)-(3). We rigorously prove a Time-Quantitative Second Law that works on a realistic time scale. As a by product, we clarify the Loschmidt-paradox and the related reversibility/irreversibility paradox.