Fuzzy 'Gaussian Prime' Number Force Fields

 

………. Article summary………….. Gaussian curvature ko(3•D) = k1(2•D) • k2(2•D)

 

• Gaussian Curvature: It’s an intrinsic measure of curvature for surfaces, calculated as the product of two principal curvatures, k1 and k2, in 2•D spaces. The document discusses the concept of Gaussian curvature in 3•D and 2•D spaces, represented by the formula ko(3•D) = k1(2•D) • k2(2•D).
• Prime Line Endpoints: The document extends the concept of a 1•D prime line with endpoints to a 2•D Euclidean field, introducing Gaussian prime points π/4 • GPn. It introduces the idea of a 1•D prime line with endpoints and its extension into a Euclidean 2•D field with Gaussian prime points.

·         Conformal Geometry: Explains how Gaussian curvature is zero for a torus and introduces the Theorema Egregium, stating that curvature is preserved under local isometries.

·         Gaussian Primes: Describes Gaussian primes as intersections on a 2•D surface at integer points and their relation to the fundamental theorem of arithmetic, covering both stable real non-composite non-vacuum points and meta-stable imaginary composite vacuum points.

 

                            Gaussian curvature ko(3•D) = k1(2•D) • k2(2•D)

A 1•D prime line endpoints {{Pn-1, Pn, Pn+1}, Pn+2} and {{P1, P2, P3}, P4} has a Euclidean 2•D field extension Gaussian prime point π/4 • GPn, and finite field extensions successor Gaussian prime points π/4•GPn+1, π/4•GPn+2 ,…… π/4•GPn+m          

   Conformal Geometry: Gaussian curvature ko = k1• k2 = 0 for a Torus

       Someone once Asked......What are Gaussian primes?

Someone once Answered......in 2•D Euclidean surface area, the Cartesian grid points/vertices intersect at Gaussian prime integer GPi or a Gaussian not prime ~GPi composite integer points map 1-1 the stream of complex surface points Pn•e^i•π/2.

 Note: ……GPi's congruent to point Pn mod(P4) = 3 , n = n +1 or to ~GPi composite point (2•n +1) mod(P4) = 1 or 0, n = n +1 that satisfy or contradicts the fundamental theorem of arithmetic...for Gaussian integer points represented in 2•D grid of connected points either a finite non-contradictory prime 'permanent' FIELD points that is a set of {stable real-nonComposite-nonVacuum points…space-like} or 'temporal' RING points that is a set of {metaStable imaginary-composite-vacuum points…time-like}. Both GPi and ~GPi cover completely the entire 2•D Euclidean surface space with ring and field type Gaussian integer points.

            Gaussian primes are a fascinating concept in the realm of complex numbers and number theory. They are the complex numbers that cannot be factored into smaller Gaussian integers, analogous to how prime numbers in the set of natural numbers are those that cannot be divided evenly by any other number except for one and themselves. In the Gaussian integers, a Gaussian prime is a number of the form a + i•b, where 'a' and 'b' are integers, and the number cannot be decomposed into the product of two non-unit Gaussian integers.

The characterization of Gaussian primes depends on their norm, which is defined as the sum of the squares of 'a' and 'b'. A Gaussian integer a + i•b is a Gaussian prime if either 'a' or 'b' is zero and the absolute value of the other is a prime number of the form 4•n + 3, or if both 'a' and 'b' are nonzero and a² + b² is a prime number. This definition ensures that the Gaussian primes maintain a unique factorization within the Gaussian integers, much like the fundamental theorem of arithmetic for natural numbers.

Gaussian primes serve as a cornerstone in the study of complex numbers, providing a parallel to the prime numbers in the natural number system. They also bridge the gap between algebra and geometry, showing how algebraic concepts can have geometric interpretations and vice versa. The study of Gaussian primes continues to be an area of active research, with implications for various fields of mathematics and even theoretical physics.

The exploration of Gaussian primes extends beyond their mathematical definition and into their graphical representation. They can be visualized on the complex plane, where they form an intriguing pattern, often depicted in a manner similar to the Ulam spiral for natural prime numbers. This visual representation can provide insights into the distribution of primes and is a subject of study in recreational mathematics as well as more serious number theory research.

In the context of geometry, the concept of Gaussian curvature is related to the intrinsic curvature of a surface at a point. For a torus, the Gaussian curvature is zero, which is a product of the principal curvature directions k1 and k2 equal k1 •k2 = 0 surface points. This set of zero curvature points indicates that a torus is a flat surface in the context of conformal geometry. The relationship between Gaussian primes and geometry can be seen in the way these primes are distributed on the complex plane, forming a lattice that mirrors the integer lattice on the Euclidean plane.

Gaussian curvature is a fundamental concept in differential geometry, the mathematical study of curves and surfaces. It’s an intrinsic measure of curvature, meaning it only depends on distances measured within or along the surface, not on how the surface is embedded in space. The Gaussian curvature at a product extension point on a surface equal the product point equal product point principal curvatures, k1 • k2. The principal curvatures are the maximum and minimum values of curvature for 2 different normal patch where the product k1 • k2 = 0 mod(ℼ/2) and 1/( k1 • k2) ≠  0 mod(ℼ/2) seeing k1 and k2 translating like 2 perpendicular line direction extension endpoints, endpoints joined by a finite 2^Pn! - n line difference direction.

The effect of Gaussian curvature on a surface can be understood as follows:

·         Positive Gaussian Curvature (k1 • k2 > 0): If both principal curvatures are of the same sign, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome-like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign. Examples of surfaces with positive Gaussian curvature include a sphere.

·         Negative Gaussian Curvature (k1 • k2 < 0): If the principal curvatures have different signs, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle-shaped. Examples of surfaces with negative Gaussian curvature include a hyperboloid.

·         Zero Gaussian Curvature (k1 • k2 = 0): If one of the principal curvatures is zero, the Gaussian curvature is zero and the surface is said to have a parabolic point. Examples of surfaces with zero Gaussian curvature include a plane and a cylinder.

Gaussian curvature is a key factor in determining the local shape and behavior of a surface. It can help us understand whether a surface is stretched or compressed relative to a flat sheet. The significance of Gaussian curvature lies in its ability to describe the intrinsic geometry of surfaces. Here are the key points:

• Intrinsic Measure: Gaussian curvature is an ‘intrinsic measure’ of curvature, meaning it depends only on distances measured along the surface, not on how the surface is embedded in space.
• Characterizes Surface: The sign of Gaussian curvature helps characterize the surface. Positive curvature indicates dome-like elliptic points, zero curvature corresponds to flat or cylindrical parabolic points, and negative curvature is associated with saddle-shaped hyperbolic points¹.
• Theorema Egregium: This theorem, established by Gauss, states that Gaussian curvature is an intrinsic property and is preserved under local isometries, which are deformations that preserve distances.
• Relation to Geometry: Gaussian curvature relates to the principal curvatures of a surface at a point and is the product of these two curvatures. It controls the surface area of spheres around a point and how much the circumference of a small circle deviates from its expected value in flat space.
  

Gaussian curvature is a fundamental concept in differential geometry with several real-world applications. Here are some key areas where it is utilized:

·         Geographic Mapping: Gaussian curvature helps cartographers create more accurate representations of Earth’s surface on maps, accounting for distortions due to the Earth’s curvature.

·         Material Science: It is used to understand the properties of materials at a microscopic level, particularly how they bend and fold, which is crucial in designing materials with specific mechanical properties.

·         Computer Graphics: In computer graphics, Gaussian curvature is used to simulate realistic surfaces and textures on 3•D models, enhancing the visual quality of animations and video games.

·         General Relativity: The concept is also important in Einstein’s theory of general relativity, where it helps describe the curvature of spacetime due to gravitational fields.

These applications show how Gaussian curvature is not just a theoretical construct but a practical tool in various scientific and engineering disciplines.

In Conclusion, this article covers ‘Gaussian curvature’, ‘ prime line endpoints ‘, ‘the Theorema Egregium‘,  and ‘Gaussian primes’, all have significant connections to key concepts in differential geometry. These connections highlight the deep interplay between geometry, topology, number theory, and contrasts stable real non-composite non-vacuum points 2•D (Gaussian prime field points), with meta-stable imaginary composite vacuum points 2•D (~Gaussian prime field and ring points), that share the same 2•D Euclidean geometry with the topology of a torus surface points all spinning with an Euler characteristic χ = Vertice – Edge + Face  =  0 mod (Pn) = 2   topological spin.

 

Abdon EC Bishop (Ceab Abce)