Reality Roots

Conjugate Pair

Existence & Form
Time & Space
Frequency & Wavelength

Reality Roots
Shape of Nature
Nature's Symphony from the Vibrating Waves of TimeSpace
Dancing Entities to the Rhythm of Emergence
"Classical" Systems Waltzing on Space Waves Turbulent Jitterbug Particles in "Quantum" Time

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Celestial Bodies








Click links below to view the detail topics of Nature
Shapes of Nature Order Shapes Space Energy

(Page under development)

Groups of Things
Pairs in Unity
Pairs may be understood "One" yet relate in Two (2) ways.
(State of Existence)

Conjugate Pair as "One"
understood to have Unity
(pair that cannot be separated)
a pair by its limits
(Analogy: A line that is One length, but a pair
by the Two ends of line -- Diameter of a Circle)

Conjugate Pair as "One"
understood to have Unity
(pair that cannot be separate)
a pair by limits of opposits
(Analogy: A line that is One length, but a pair
by Two the opposites of its ends -- Radii of a Circle)

Set theory is the branch of mathematical logic that studies Sets.
Sets are collections of objects.
Set Theory is applied most often to objects of mathematics.
However, Set Theory can be used to analysis any group of objects.
Any type of object can be collected into a set.
Just as arithmetic features binary (pair) operations on numbers,
Set Theory features binary operations on sets within Reality.

Pairs of Identities, Three (3), relating inseparably.
"Conjugate Pairs"


defined by
Limits of Three (3) pairs

Each pair as an Identity
One Thing, in its self,
defined by its limits

Set of Three (3)
conjugate pair



(Derivative of Space)
defined by
Limits of Three (3) pairs
Univers is





Structure of Unity -- Process of Bioty
Unity is Oneness -- Pair is Difference -- Bioty is Two (2) Pair Difference
(Potential Energy)












One time Increment


Ordinary Energy
Two (2) Time

Human understanding of Reality, by no small measure,
is a significant challenge of speculative Knowledge.
Human perception of change gives a clue that Reality must,
in some part, be a process. Further, perception tells us that
Structure is another part of Reality. However, there are different
kinds of structures, those that just stand (buildings) and those
that move -- stars, planets, vehicles, animals and humans, etc.

Why 8 electrons form a stable shell Structure of an Atom

The presence of 8 slots in the shell of Atoms has been taught historicallyfor a long time as if it were a purely empirical observation. However, it can actually be derived from the axioms of quantum mechanics and the geometry of 3-dimensional space. Below is an overview of some of the relevant features of the derivation.

If you solve Schrödinger’s equation for an electron in the potential well of a nucleus, you can find every possible bound state that the electron could possibly have. Finding the solution involves making sure that the electron's wave function repeats itself at intervals that allow it to "fit" around the nucleus.

By analogy, consider a wave on a length of string. It has peaks and troughs at a certain period. The goal is to find which periods (wavelengths) will "fit" on the string, such that if you bring the two ends of the string together to form a loop around the nucleus, there will be no discontinuities in the wave. The resulting waves are called harmonics:

A harmonic of a  wave is a component frequency of the signal that is an integer multiple of the    fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental frequency; therefore the sum of harmonics is also periodic at that frequency. Harmonic frequencies are equally spaced by the width of the fundamental frequency and can be found by repeatedly adding that frequency. For example, if the fundamental frequency (first harmonic) is 25 Hz, the frequencies of the next harmonics are: 50 Hz (2nd harmonic), 75 Hz (3rd harmonic), 100 Hz (4th harmonic) etc.

A loop is sufficient to wrap around a point in a 2 dimensional plane, but electrons and atoms exist in 3 dimensions. Hence, we have to make sure the electron's wave function "fits" in both of the two angular degrees of freedom around a point in 3 dimensions. The waves that "fit" around a central point in 3 dimensions are called spherical harmonics:
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace’s equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre Simon de Laplace in 1782. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of   atomic orbital electro configurations, representation of gravitational fields, geoids, and the magnetic fields of planetary bodies and stars, and characterization of the cosmic microwave background radiation. In  3D computer graphics, spherical harmonics play a special role in a wide variety of topics including indirect lighting ( ambient occlusion, global illumination, precomputed radiance  transfer, etc.) and modeling of 3D shapes.

Since there are two angular degrees of freedom around a point in 3 dimensions, it takes two numbers to identify a given solution that fits. As a result, there are a total of three numbers needed to fully identify a wave function solution:
n: quantization in energy
l (lowercase L): quantization in first angular degree of freedom
m: quantization in second angular degree of freedom
Purely from the math of the solution and the rules of geometry, one finds that the following rules must be satisfied in order for the solution to fit:

n must be a positive integer (1, 2, 3, ...)
for a given n, l can be any integer from 0 to n-1 inclusive
for a given l, m can be any integer from -l to l inclusive
In addition, the properties of the electron dictate that:
for a given n, l, and m, there are two possible electron states: one for spin up and one for spin down
Putting all these rules together, we can count the number of "slots" available for electrons to fill up a given shell.

For the lowest energy shell (n=1), l can be any integer from 0 to 0. That is, it can only be 0. In turn, for l=0, m can be only -0 to 0, meaning it can also only be 0. This means there is only one allowed combination with n=1, namely the one where n=1, l=0, and m=0. Since electrons can be spin up or spin down, this means there are 2 total slots available. This is why hydrogen and helium are stable with 2 electrons -- because those 2 electrons fill the n=1 shell.

For the next energy shell (n=2), l can be 0 to 1. For l=0, m can be only 0. For l=1, m can be -1, 0, or 1.

This forms the following set of allowed values:

n=2, l=0, m=0
n=2, l=1, m=-1
n=2, l=1, m=0
n=2, l=1, m=1

Since there are 4 allowed values, the shell has 8 slots for electrons (accounting for spin).




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