Recently, I was playing around with circles and fractals, trying to visualize how a self-contained system would look like. I started drawing circles within circles and counted the total number of circles after each sub-division. The first pattern is the simplest one: after x iterations, you get x circles. The second pattern is more interesting, like a cell splitting in half after each iteration. So after two iterations, there are 3 circles (the initial big one, plus the two smaller ones). The total number of circles gets larger and larger very quickly. Next, I repeated the same procedure, but with a 3-circle sub-division. Another pattern started to form...
While I am drawing these shapes, I always have in mind what the C's have said about "1,2,3" holding the secrets of all existence.
A: Yes, thank you Arkadiusz!!!! Laura is dancing around in wonderland, meanwhile all of creation, of existence, is contained in 1, 2, 3!!! Look for this when you are trying to find the keys to the hidden secrets of all existence... They dwell within. 11, 22, 33, 1/2, 1/3, 1, 2, 3, 121, 11, 111, 222, 333, and so on! Get it?!?!
Q: When you say that the secrets of all existence dwell within 1 2 3 or variations thereof, what kind of secrets are we talking about here?
A: All.
The total number of circles at any point is represented by a repunit (a number with all 1's) in the base corresponding to the number of sub-divisions!
Want to find out the total number of circles after 4 iterations when you constantly split every circle 6 times?
It's 1111 in base-6, every '1' represents an iteration—for a total of 259 circles!
Isn't it remarkable?
When you write down numbers, you are 'secretly' doing algebra and geometry!
I think that the connection to geometry is important because of the way we perceive reality: patterns. It could be that this connection opens doors even to some physics! Who knows.
I think that the connection to geometry is important because of the way we perceive reality: patterns. It could be that this connection opens doors even to some physics! Who knows.
Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n + 1/2). For example:
I think in this context another way to look at it is not that vibration takes on geometric shapes, but that geometry emerges from vibration. Geometry is about relative proportions and these proportions can in some way be linked to vibrational states. The example of the vibrating states of a string (of a guitar of course) illustrate such a relationship. In the earliest quantum model of the hydrogen atom, the electron "orbits" the proton such that its wave forms a steady wave around the center. It is possible that many phenomena can be viewed as interacting waves/vibrations with matched/unmatched frequencies (proportions) and matched/unmatched phases (hence the apparently stochastic nature of certain phenomena). Just some random thoughts.
This, I feel, might be more on track. I think also, consciousness needs to be (not sure how) included in this, vibration to geometric shape/pattern/structures phenomenon. It brings to mind, the observer causes collapse of the wave function phenomenon in quantum mechanics. As with mkrnhr, just some random thoughts.
I think in this context another way to look at it is not that vibration takes on geometric shapes, but that geometry emerges from vibration. Geometry is about relative proportions and these proportions can in some way be linked to vibrational states. The example of the vibrating states of a string (of a guitar of course) illustrate such a relationship. In the earliest quantum model of the hydrogen atom, the electron "orbits" the proton such that its wave forms a steady wave around the center. It is possible that many phenomena can be viewed as interacting waves/vibrations with matched/unmatched frequencies (proportions) and matched/unmatched phases (hence the apparently stochastic nature of certain phenomena). Just some random thoughts.
In that context, maybe the article referenced in Ark's latest blog entry might be of interest, although it deals more with preserved topology than the geometry that comes from 'vibrating' electric and magnetic field lines (where they "evolve like unbreakable filaments embedded in a fluid flow, stretching and deforming while retaining their identity").
Persistent topological structures in physical systems have become increasingly important over the last years. Electromagnetic fields with knotted field lines play a special role among these, since they can be used to transfer their knottedness to other systems like plasmas and quantum fluids. In...
Q: How does one utilize the energies inherent in prime numbers in this respect? Do they represent frequencies or frequency relationships?
A: Verities.
Q: Is there any formula, or any thing about prime numbers that makes it easier to find them... anything about them that is unique?
A: Pyramidal.
Q: Pyramid relationships would help one find prime numbers?
A: Graph.
Q: A pyramid type graph. Okay, anything else about prime numbers? When you said that they were the 'dwellings of the mystics' I had an idea that a prime number could be a dwelling of a mystic because the individual would express in some manner a frequency that related in some way to a prime number. Is that somewhere along the line...? That mystics can traverse all densities because of frequency?
It's fascinating how just by "geometrically tinkering" with numbers, or reading them in a certain order, one can produce sequences that solve or describe the essence of a seemingly unrelated mathematical problem.
@Natus Videre I’m hoping to soon finish my current composition. Every time you post something like your current post I step back from the piece and try to visualize it geometrically. All elements of the piece are based on prime numbers and Fibonacci sequences. It’s been slow in development as I’ve never thought geometrically when composing. Thanks for the inspiration and keep bringing it on!
Recently, I was investigating the link between harmonic numbers and prime numbers.
Harmonic Series Formula
I stumbled upon a question on Mathematics Stack Exchange where someone claimed to be able to find prime numbers with very little calculations using the harmonic series.
I created a table to find primes What I did was understand that if you divide every number in half we get a unique number. e.g. 5 is 5/2, and 3 is 3/2, so by including more columns we create a full...
math.stackexchange.com
Here I reproduced a part of the table provided in the question.
n \ i
1
2
3
4
5
6
7
1
1/1
3/2
11/6
25/12
137/60
49/20
363/140
2
2/1
3/1
11/3
25/6
137/30
49/10
363/70
3
3/1
9/2
11/2
25/4
137/20
147/20
1089/140
4
4/1
6/1
22/3
25/3
137/15
49/5
363/35
5
5/1
15/2
55/6
125/12
137/12
49/4
363/28
6
6/1
9/1
11/1
25/2
137/10
147/10
1089/70
7
7/1
21/2
77/6
175/12
959/60
343/20
363/20
The rows represent numbers (n) and the columns represent the number of iterations (i) up to which the harmonic series was evaluated. The values of the grid correspond to H(n,i). For example, H(3,2) means we take the number 3, and we multiply it with the first two terms of the harmonic series: H(3,2) = 3*(1/1 + 1/2) = 9/2.
Now, take a look at the denominators of the values in the table. Can you spot a pattern that could identify prime numbers?
When n=1:
H(1,1) = 1*(1/1) = 1/1
H(1,2) = 1*(1/1 + 1/2) = 3/2
H(1,3) = 1*(1/1 + 1/2 + 1/3) = 11/6
...and so on.
Now, say you want to know if the number 3 is prime.
H(3,1) = 3*(1/1) = 3/1
H(3,2) = 3*(1/1 + 1/2) = 9/2
H(3,3) = 3*(1/1 + 1/2 + 1/3) = 3*(11/6) = 11/2
Observe the denominators of H(1,1) and H(3,1). What do you notice?
They are the same! Both are equal to 1.
Do the same thing for H(1,2) and H(3,2). What do you see?
Again, they are the same! Both are equal to 2.
A number 'n' is prime if it shares the same denominators as the number 1 for n-1 iterations, and breaks the trend on the nth iteration!
Since a prime number is only divisible by 1 and itself, when the iteration number (i) is equal to the prime number (n), the prime number reduces the denominator because the latter is a multiple of the prime number. In the first two iterations with n=3, the denominator did not change. But on the 3rd iteration, the number 3 got 'diluted' with the 3rd harmonic number because 3 * 1/3 = 1, thus confirming that the number 3 is prime.
Similarly, consider the number 7. Its denominators (1,2,6,12,60,20) are respectively identical to those of the number 1. Once the 7th iteration is reached, the denominator finally diverges (20 vs 140). And that's how we know that the number 7 is prime!
How do we know the number 6 is not prime?
Well, its denominator diverged from the 'benchmark' denominator (created with n=1) on the second iteration! When the number 6 was multiplied by 1/2 in H(6,2) = 6*(1/1 + 1/2) = 6 + 6/2 = 6 + 3/1 = 9/1, the fraction was reduced because 2 divides 6! The number 6 did not 'hold' its denominators for 5 iterations!
This way of finding primes is peculiar in the sense that primality is deduced from the ability of a number to not be altered by any number other than itself. Of course, a prime number can interact with one of its multiples, i.e. 7 *1/14 = 1/2 and thus there could be an effect on the denominator, but we don't need to go all the way to the 14th iteration to ascertain that the number 7 is prime—the 7th iteration is enough!
What if natural numbers were 'vibrational mixtures?'
For example, consider the number 6. Its divisors are 1, 2, and 3. It's a composite number. The number 6 holds the 'vibrational' signature of 1, 2, and 3 within itself. Here is a way to visualize the 'mixtures.'
Each row represents the prime factorization of a number. Each circle represents a node to which a string can be attached. Prime numbers, due to the fact that they are only divisible by 1 and themselves, have only one string. The magnitude of each prime factor is reflected in the length of a string between two nodes. For example, the number 2 is contained 3 times in the number 6, which means that 3 "jumps" are necessary to reach the last node. Similarly, the number 3 takes 2 jumps (of length 3) to reach the last node.
The following is a visualization of the numbers from 2 to 100.
Observations:
Rows that are almost empty represent prime numbers.
Twin primes (primes that have a separation of 2, like 11 and 13) create a 'road line' between them.
All multiples are fractals composed of 'building blocks', i.e. prime numbers!
Number 2: O-O
Number 3: O--O
Number 6: O-OOO-O
Number 12: O-OOO-O-OOO-O
The numbers 2 and 3 are contained in the number 6. The number 6 is contained in the number 12. Therefore, the numbers 2 and 3 are also contained in the number 12.
@Natus Videre The left hand column are all the pairs that can be utilized as in my previous post attachment. I’m curious what your perspective is on this!
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