The Geometry of Closed Packed Spheres

The document explores the concept of  incommensurability  in mathematics, focusing on the relationship between numbers and their square roots. It introduces the  square root s 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 const...

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 o...

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 th...

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 nat...

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 "quantiz...

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 a...

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, dem...

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 CP...

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 ortho...

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, s...

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, exemplifie...

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 ...

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  p...

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...

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.andO...

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.andO...

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.andO...

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.andO...

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 ,...

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=y This 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...

US Patent: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/9245059 A system and method for determining a location within a physical structure are described. The location within the physical structure can be determined by storing a model of a physical structure comprising a plurality of nodes connected by a plurality of struts, each of the nodes and struts of the model correspondin...

The ration of the most precise measurements of the mass of the proton and the mass of the electron is very closed to 694 multiplied by square root of 7. See: https://www.amazon.com/dp/B09RJNMZPV

The Geometry of Closed Packed Spheres Mission statement: To change minds, to open eyes, to educate and inspire people designing and building better worlds. Beauty makes beautiful things beautiful! A sphere can be completely surrounded by exactly twelve other identical spheres. Close-packing of spheres helps us explore the shape of the physical space. A good design of a 3D structure shall obey the...