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PODCAST

The Geometry of Closed Packed Spheres

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 principles, freedom, and constraints imposed by the physical space around us.

science mathematics

All Episodes

00:13:32

18. The Square Roots Spiral

The Geometry of Closed Packed Spheres ·

2024/10/18

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

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00:08:06

17. Lines Patterns in Space

The Geometry of Closed Packed Spheres ·

2024/10/17

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

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00:08:06

16. The Fibonacci Sequence

The Geometry of Closed Packed Spheres ·

2024/10/16

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

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00:08:55

15. The Golden Section

The Geometry of Closed Packed Spheres ·

2024/10/11

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

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00:07:24

14. Similarity Theorem in CPS Geometry

The Geometry of Closed Packed Spheres ·

2024/10/10

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

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00:08:24

13. Archimedean Solids

The Geometry of Closed Packed Spheres ·

2024/10/09

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

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00:11:04

12. The Rhombic Dodecahedron in CPS

The Geometry of Closed Packed Spheres ·

2024/10/07

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

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00:06:23

11. Platonic Solids in CPS Geometry

The Geometry of Closed Packed Spheres ·

2024/10/04

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

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00:10:52

10. Close Packing of Spheres

The Geometry of Closed Packed Spheres ·

2024/10/03

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

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00:09:53

09. CPS Geometry

The Geometry of Closed Packed Spheres ·

2024/10/02

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

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00:05:46

08. The Mental-Experimental Method

The Geometry of Closed Packed Spheres ·

2024/10/01

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

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00:06:22

07. The Dialectic Process

The Geometry of Closed Packed Spheres ·

2024/09/30

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

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00:04:24

06. The Incommensurables – Arithmetical Proof

The Geometry of Closed Packed Spheres ·

2024/09/28

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

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00:03:57

05. The Incommensurables – Geometrical Proof

The Geometry of Closed Packed Spheres ·

2024/09/27

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

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00:07:34

04. The Euclidian Algorithm

The Geometry of Closed Packed Spheres ·

2024/09/26

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

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00:07:55

03. Gauss Estimation An Epistemological Problem

The Geometry of Closed Packed Spheres ·

2024/09/25

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

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00:08:28

03 (Old). Gauss’s Estimation – An Epistemological...

The Geometry of Closed Packed Spheres ·

2024/09/24

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

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00:09:25

02. A Perfect Experiment

The Geometry of Closed Packed Spheres ·

2024/09/23

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

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00:05:24

01. The Scientific Method

The Geometry of Closed Packed Spheres ·

2024/09/22

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

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00:08:45

Preface to CPS Geometry Book

The Geometry of Closed Packed Spheres ·

2024/09/22

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 that points in space, typically viewed as dimensionless, should be considered as infinitesimal spheres

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23 results

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