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The document explores the concept of incommensurability in mathematics, focusing on the relationship between numbers and their square roots. It introduces the square roots spiral as a visual representation of incommensurable magnitudes. The text then contrasts the square roots spiral with two other well-known spirals: the logarithmic spiral and the Archimedean spiral. It details the construction and properties of each spiral, highlighting similarities and differences among them. Finally, the document concludes by suggesting the potential use of these spirals in defining a metric relationship for a specific geometry called CPS Geometry.
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The text discusses the concept of straight lines in CPS Geometry, a system where points are infinitesimal spheres arranged in a specific pattern. It explores the concept of lines as patterns that extend infinitely in both directions and can be defined by any two points in the space. The text then investigates patterns formed by lines emanating from a central point, analyzing these patterns based on the surrounding layers of points, which are arranged in cuboctahedron structures. The text also considers how these patterns arise from the arrangement of rhombic dodecahedrons, which fill space in CPS Geometry. The text concludes by highlighting the potential for further exploration of these line patterns and their connection to complex analysis concepts such as Möbius transformations.
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The source explains the connection between the Fibonacci sequence and the Golden Ratio, also known as the Golden Section. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers (e.g., 1, 1, 2, 3, 5, 8). The Golden Ratio is an irrational number, approximately 1.618, that appears in various natural and mathematical phenomena. The source shows that the ratio of consecutive terms in the Fibonacci sequence approaches the Golden Ratio as the sequence progresses. It then explores the relationship between the Golden Ratio and the Close Packing of Spheres (CPS) geometry, which is a system for studying the arrangement of spheres in three-dimensional space. The source argues that the CPS geometry provides a geometric interpretation of the Golden Ratio and the Fibonacci sequence, demonstrating that the relationship between these concepts is not just a mathematical curiosity but has a basis in the natural world.
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The text explores the Golden Ratio, also known as the Golden Section, and its significance in classical geometry. It highlights three primary ways the Golden Ratio manifests itself: through Euclid's definition of dividing a line into extreme and mean ratio, in the construction of a regular pentagon, and as a key element in constructing an icosahedron. The text emphasizes the fractal nature of the Golden Ratio, showcasing how its principles can be applied across different scales, from microscopic to macroscopic. It also draws connections between the Golden Ratio, the Theorem of Pythagoras, and complex functions, suggesting potential applications in advanced mathematical fields.
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The source explores the concept of similarity in geometry, arguing that traditional Euclidean geometry’s reliance on the parallel postulate is not the most fundamental approach. Instead, the source proposes a "CPS Geometry" based on the close-packing of spheres, where similarity arises from the inherent patterns and structures within this arrangement. This framework introduces the idea of "quantization" and suggests that the similarity theorem, rather than being a consequence of parallel lines, is a result of the inherent properties of the CPS arrangement.
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The text describes the 13 Archimedean solids in terms of their relationship to the close-packing of spheres (CPS) arrangement. The author explains how these semi-regular polyhedrons, such as the cuboctahedron, truncated tetrahedron, and truncated icosahedron, can be constructed by manipulating Platonic solids within the framework of CPS. The text emphasizes that the CPS arrangement, where points are considered infinitesimal spheres, offers a fundamental understanding of geometrical concepts such as similarity and quantization of space. The text then explores the relationship between the CPS Geometry and other geometric and mathematical systems, including the Cartesian Geometry and Number Theory.
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This excerpt from "12-The Rhombic Dodecahedron in CPS.pdf" explores the presence of the rhombic dodecahedron in the Close Packing of Spheres (CPS) model. It argues that the shape of the rhombic dodecahedron, a space-filling form, emerges from a multitude of spheres arranged in a specific pattern. The text then connects this pattern to the concept of minimum surfaces, exemplified by soap films, demonstrating the emergence of soap film-like surfaces within the CPS framework. This connection is further supported by the observation of angles and geometric conditions, which correspond to Joseph Plateau's laws governing soap film behavior. Finally, the source highlights the potential of the rhombic dodecahedron and its associated patterns for understanding the design of complex structures, particularly in the context of Platonic Structures, which are 3D structures built using simple components.
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This source discusses the five Platonic solids, or perfect bodies: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The author argues that these solids are not mystical, but rather can be explained using the principle of close-packing of spheres in a specific arrangement called the CPS Space. The source presents detailed patterns and structures of the Platonic solids within the CPS arrangement, showcasing how these solids can be assembled from identical spheres. It emphasizes the importance of building physical models to understand these structures and challenges the traditional view of space as being composed of cubes. The source concludes by exploring the relationship between the icosahedron and dodecahedron as dual structures, illustrating their interconnections and the existence of a related structure called the Great Stelled Dodecahedron.
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The text explains the concept of close packing of spheres, a principle that describes how spheres can be arranged in three-dimensional space to achieve the densest possible packing. It highlights the two primary lattice patterns used in this arrangement: the square lattice and the hexagonal lattice. The text then explores the relationships between these patterns, including how they relate to orthogonal and tetragonal planes and axes, and how these relationships are used to define a more complete coordinate system. The text concludes by emphasizing that the close packing of spheres represents a foundational concept for understanding space, matter, and spatial structures.
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The provided text introduces Closed Packed Spheres (CPS) Geometry, an alternative geometric system that challenges traditional Euclidean geometry. Unlike Euclidean geometry, which defines points as dimensionless and structureless, CPS Geometry views points as infinitesimally small, identical spheres arranged in a close-packed pattern. This arrangement allows for the natural emergence of lines, surfaces, and solids as inherent characteristics of the CPS space, replacing the more abstract and ambiguous definitions of Euclidean geometry. The text further suggests that CPS Geometry offers a more harmonious and structured approach to geometric understanding.
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The source criticizes the axiomatic method of Euclidean geometry, arguing that it stifles creativity and prevents discovery by imposing a rigid, bureaucratic system. It proposes instead a "Mental-Experimental Method" that relies on mental visualization and experimentation to understand geometric principles. The author advocates for a more intuitive and experiential approach to geometry, exemplified by their development of CPS Geometry, which emphasizes hands-on exploration and the tangible observation of geometric patterns and relationships.
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The text draws a parallel between the incommensurability of the square root of two and the distribution of prime numbers, arguing that neither can be fully understood or expressed using simple patterns. The author then references Plato's dialectic method, which utilizes a series of hypotheses to reach a higher understanding of knowledge. This method is seen as analogous to the process of reasoning about the distribution of primes, suggesting that it requires a multi-layered approach to fully comprehend its complexity.
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The provided text explores the concept of incommensurability, specifically focusing on the square root of 2. The text outlines two methods for understanding incommensurability: a geometric approach that is intuitive but potentially less rigorous, and an arithmetical approach that uses logic and number theory to provide a more formal proof. The arithmetical approach is illustrated by the proof by contradiction, which demonstrates that the square root of 2 cannot be expressed as a ratio of two integers. The text argues that despite the emphasis on the arithmetical approach in modern mathematics, there is value in exploring the potential for combining geometric and arithmetical methods to gain deeper insights.
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The text discusses the discovery of incommensurable magnitudes, a fundamental concept in mathematics. This discovery, made by the Pythagoreans, demonstrated that not all line segments can be measured using a common unit of length. The text uses the example of a square's diagonal and its side to illustrate this concept. The process of repeatedly trying to find a common unit of length between the diagonal and side of the square reveals that this task is impossible, as the process continues indefinitely. This led to the creation of irrational numbers, a new category of numbers that cannot be expressed as fractions of integers. This discovery had a profound impact on the development of mathematics, expanding the understanding of numbers and their relationship to geometry.
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Chapter 4 of the book:“From Riemann Hypothesis to CPS Geometry and Back Volume 1 (https://www.amazon.com/dp/B08JG1DLCV) ”, Canadian Intellectual Property Office Registration Number: 1173734 (http://www.ic.gc.ca/app/opic-cipo/cpyrghts/srch.do?lang=eng&page=1&searchCriteriaBean.textField1=1173734&searchCriteriaBean.column1=COP_REG_NUM&submitButton=Search&searchCriteriaBean.andOr1=and&searchCriteriaBean.textField2=&searchCriteriaBean.column2=TITLE&searchCriteriaBean.andOr2=and&searchCriteriaBean.textField3=&searchCriteriaBean.column3=TITLE&searchCriteriaBean.type=&searchCriteriaBean.dateStart=&searchCriteriaBean.dateEnd=&searchCriteriaBean.sortSpec=&searchCriteriaBean.maxDocCount=200&searchCriteriaBean.docsPerPage=10) , Ottawa, ISBN 9798685065292, 2020.On Google Books:https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=yOn Google Play: https://play.google.com/store/books/details?id=jFQjEQAAQBAJThe provided text introduces the Euclidean Algorithm, a method for finding the greatest common divisor (GCD) of two integers. The algorithm is presented both conceptually, as finding the smallest common unit of measurement, and procedurally, using a series of divisions and remainders. The text then explores the algorithm's application to line segments, highlighting the implicit assumption of a common unit of length. It concludes by acknowledging that the algorithm relies on the existence of a common unit (like the number 1 for integers) and may not be directly applicable to all pairs of line segments.
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Chapter 3 of the book:“From Riemann Hypothesis to CPS Geometry and Back Volume 1 (https://www.amazon.com/dp/B08JG1DLCV) ”, Canadian Intellectual Property Office Registration Number: 1173734 (http://www.ic.gc.ca/app/opic-cipo/cpyrghts/srch.do?lang=eng&page=1&searchCriteriaBean.textField1=1173734&searchCriteriaBean.column1=COP_REG_NUM&submitButton=Search&searchCriteriaBean.andOr1=and&searchCriteriaBean.textField2=&searchCriteriaBean.column2=TITLE&searchCriteriaBean.andOr2=and&searchCriteriaBean.textField3=&searchCriteriaBean.column3=TITLE&searchCriteriaBean.type=&searchCriteriaBean.dateStart=&searchCriteriaBean.dateEnd=&searchCriteriaBean.sortSpec=&searchCriteriaBean.maxDocCount=200&searchCriteriaBean.docsPerPage=10) , Ottawa, ISBN 9798685065292, 2020.On Google Books:https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=yOn Google Play: https://play.google.com/store/books/details?id=jFQjEQAAQBAJ
The text explores the historical development of methods used to estimate the distribution of prime numbers. It begins by highlighting the difficulties faced by mathematicians like Gauss in manually calculating prime numbers, especially for large sets. The text then delves into the idea of using logarithms to approximate the number of primes below a given number, a concept that Gauss himself explored. This leads into a discussion of the Prime Number Theorem, which provides a precise asymptotic formula for the distribution of prime numbers. Finally, the text touches upon the logarithmic integral as a refined approximation for the distribution of primes.
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Chapter 3 of the book:“From Riemann Hypothesis to CPS Geometry and Back Volume 1 (https://www.amazon.com/dp/B08JG1DLCV) ”, Canadian Intellectual Property Office Registration Number: 1173734 (http://www.ic.gc.ca/app/opic-cipo/cpyrghts/srch.do?lang=eng&page=1&searchCriteriaBean.textField1=1173734&searchCriteriaBean.column1=COP_REG_NUM&submitButton=Search&searchCriteriaBean.andOr1=and&searchCriteriaBean.textField2=&searchCriteriaBean.column2=TITLE&searchCriteriaBean.andOr2=and&searchCriteriaBean.textField3=&searchCriteriaBean.column3=TITLE&searchCriteriaBean.type=&searchCriteriaBean.dateStart=&searchCriteriaBean.dateEnd=&searchCriteriaBean.sortSpec=&searchCriteriaBean.maxDocCount=200&searchCriteriaBean.docsPerPage=10) , Ottawa, ISBN 9798685065292, 2020.On Google Books:https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=yOn Google Play: https://play.google.com/store/books/details?id=jFQjEQAAQBAJ
The text discusses Gauss's attempts to find a pattern in the distribution of prime numbers. The author examines Gauss's early experiments with counting primes and explores his eventual development of a formula to approximate the number of primes less than a given number. The text also highlights the limitations of this formula and the ongoing challenge of finding a precise mathematical expression for the distribution of primes. The author then discusses the Prime Number Theorem, which provides a more accurate approximation for the distribution of primes, and the logarithmic integral as an even better approximation. Finally, the text touches upon the implications of this problem for our understanding of mathematics and the potential need for new approaches to address it.
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Chapter 2 of the book:“From Riemann Hypothesis to CPS Geometry and Back Volume 1 (https://www.amazon.com/dp/B08JG1DLCV) ”, Canadian Intellectual Property Office Registration Number: 1173734 (http://www.ic.gc.ca/app/opic-cipo/cpyrghts/srch.do?lang=eng&page=1&searchCriteriaBean.textField1=1173734&searchCriteriaBean.column1=COP_REG_NUM&submitButton=Search&searchCriteriaBean.andOr1=and&searchCriteriaBean.textField2=&searchCriteriaBean.column2=TITLE&searchCriteriaBean.andOr2=and&searchCriteriaBean.textField3=&searchCriteriaBean.column3=TITLE&searchCriteriaBean.type=&searchCriteriaBean.dateStart=&searchCriteriaBean.dateEnd=&searchCriteriaBean.sortSpec=&searchCriteriaBean.maxDocCount=200&searchCriteriaBean.docsPerPage=10) , Ottawa, ISBN 9798685065292, 2020.On Google Books:https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=yOn Google Play: https://play.google.com/store/books/details?id=jFQjEQAAQBAJSummary
The text explores the concept of "mental experiments" in mathematics, arguing that mathematics, like the physical world, can be investigated through a process of experimentation and measurement. The author uses the problem of determining the distribution of prime numbers as an example, highlighting the need to define fundamental concepts like "multiplication" and "addition" before performing such mental experiments. The author concludes that these mental experiments can be performed with perfect accuracy, resulting in measurements that exactly match the "actual" mathematical reality.
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Chapter 1 of the book:
“From Riemann Hypothesis to CPS Geometry and Back Volume 1”, Canadian Intellectual Property Office Registration Number: 1173734, Ottawa, ISBN 9798685065292, 2020.
On Google Books:
https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=y
On Google Play: https://play.google.com/store/books/details?id=jFQjEQAAQBAJ
The podcast explores the scientific method, focusing on its use in discovering and proving laws of nature. It emphasizes the importance of reproducible experiments, controlled environments, and accurate measurements. The author posits that scientists strive to identify underlying mathematical patterns within experimental data, even acknowledging the inevitable presence of errors. Ultimately, the goal is to communicate new discoveries through mathematical formulas and scientific publications.
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Book:
Amazon.com: From Riemann Hypothesis to CPS Geometry and Back: Volume 1 eBook : Trif, Nick: Kindle Store
On Google Books: https://books.google.ca/books/about?id=jFQjEQAAQBAJ&redir_esc=yThis text introduces the concept of "Closed Packed Singularity Geometry" (CPS Geometry), a new geometric framework that challenges traditional Euclidian geometry. The author, Nick Trif, proposes that points in space, typically viewed as dimensionless, should be considered as infinitesimal spheres. CPS Geometry posits that these spheres, arranged in a close-packing pattern, form the basis of space and can be used to understand various mathematical and scientific concepts. The text outlines the fundamental axioms of CPS Geometry, emphasizing its connection to number theory and arguing that it cannot be fully understood using the axiomatic approach of Euclidian geometry.
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