The Web of Compositability
A mathematical proof on the Nature of Whole Numbers
as a self organized system revolving around
12 positions of compositability.
Heres a PDF version:
Often when we think of numbers we understand it in the format of language acquisition, meaning that we learn it as it is taught to us, rather than coming to terms with the reality of comprehending quantity. Though we ascribe names and symbols to numbers, it is different from language since numbers are the system we use to understand the difference between an object and multiple objects. Its there whether or not we acknowledge it. There is 1 sun in our solar system, no matter how we communicate that info to each other. We have 2 eyes and ears, no matter how that is perceived or described. I say this because numbers are different from language, in the sense that we are comprehending the relationship of quantity and relationship of number systems, rather than fabricating their meaning through the development of culture and custom, as with language.
before we start…
lets define a few things
A number is a mathematical object used to count, label, and measure.*
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the abstract study of topics encompassing quantity, structure, space, change, and other properties; it has no generally accepted definition.*
French mathematician Claire Voisin states “There is creative drive in mathematics, it’s all about movement trying to express itself.”*
*taken from Wikipedia
1a) When we place numbers around a circle with 12 positions, like a clock, and then continue the number sequence, patterns emerge locking evens, or 2x, into a hexagon, multiples of 3, or 3x into a square, and multiples of 4, or 4x into a triangle. Prime numbers emerge in positions 5, 7, 11, and 1. Prime numbers then interacting with themselves and “knock out” primes by creating semi primes (ex: 25, 35, 49, 55, 65, etc.)
1b) The pattern of compositability begins dictated by the first 6 numbers and then reverses back to 12, allowing 4 place positions for prime numbers every 12 numbers.
1c) The pattern of compositability reads like this starting from 1: prime, 2x, 3x, 4x, prime, 6x (2x and 3x), prime, 4x, 3x, 2x, prime, 12x (3x and 4x, 2x and 6x) and then it repeats.
or like this: when prime is 1x
1x 2x 3x 4x 5x(1x) 6x(2x3x)7x (1x) 8x(4x) 9x(3x) 10x(2x) 11x(1x) 12x(3x4x/2x6x)
1x is prime (positions and numbers 5 and 7, 11 and 1)
Any number can be placed into 1 of 12 positions
2. When we keep wrapping numbers sequentially around the 12 positions of the clock we can begin to see the shape systems the multiples of each number form. 13 follows 1, 14 follows 2, 15 follows 3, etc. 24 then sits on top of 12 and the sequence keeps repeating. (ex #1)
3. If we connect the evens by lines, or 2x (all multiples of 2. ex: 4, 6, 8, 10, 12) we get a hexagon. This pattern goes on forever. It is embedded in the system. The next layer follows the same positions 14, 16, 18, 20, 22, and 24. (ex #2)
4. When we connect the multiples of 3 (3, 6, 9, 12), or 3x, with lines we get a square tilted on end. (ex #3)
5. When we connect the multiples of 4 (4, 8, 12, 16), or 4x, we get a perfect triangle. (ex #4)
6. The numbers 2, 3, and 4 create the dominant systems of compositability.
7. The numbers 6, 8, 9, 10, and 12, as well as all multiples of 2, 3, and 4 are embedded in the shape patterns of 2x, 3x, and 4x. (9x follows 3x backwards, 8x follows 4x backwards, 15x follows 3x, etc)
8. The numbers 5, 7, or 5x and 7x “bounce” back and forth across the system, like a star. The numbers that follow these positions follow the same path (17x and 19x, respectively, as well as all other numbers that fall in these positions. (ex #5)
9. The numbers 11 and 13, or systems 11x and 13x, cascade around the system like spirals. (ex #6)
The traditional multiplication table lines up numbers in 2 directions, with their squares reflecting diagonally down the center. Though it repeats the table, it also attempts to tease out the numbers, preventing their interrelationships from appearing.
The Multiplication Spiral
includes 1 instance of every whole number and allows for a deeper understanding of how multiplication patterns or number systems interact. Say like how 9 intersects the system of 4 at 36, which is also a multiple of 12, and the square of 6.
Prime Numbers and the Sieve of Eratosthenes
The traditional and typical way to look at prime numbers is through the linear chart of 10’s, or the Sieve of Eratosthenes. As you can see there is some pattern but it is not real consistent or easy to predict.
As a 12 position spiral, the primes emerge in an easy to see pattern just outside the end points 12 and 6 (5, 7, 11, 13). This is also supported by the equation to find primes: 6x +/-1. What becomes the interesting pattern here is the exceptions to the prime fields, or the semiprimes. 1st is 25, the square of 5, and 35 which is 5×7. The next one is 49, the square of 7. 5 and 7 are the 1st primes, the 1st twin primes, and the largest system of exceptions to the prime field.
Once the simple divisibility rules are known of 2, 3, and 4, numbers can be “glanced at” and then placed into position
2x If a number is even than it is divisible by 2.
4x If a number can be cut in half and is still even, then it is divisible by 4, and can only reside in positions 4, 8, and 12. ( 36/2 = 18 (even) so 36 is divisible by 4). For the sake of space, lets call this feature, the ability to be divided by 4, double-even.
3x If you add the digits of a number together and that sum is divisible by 3, then the original number is too. (354 = 3+5+4 = 8 + 4 =12 = 1+2 +=3. Therefore 354 is divisible by 3. Every third number is. If all digits of numbers are added together to show the properties of divisibility, then all whole numbers can be reduced to one of nine single digit numbers. Also, if every third number is divisible by 3, and some number is not, then either by adding 1 or subtracting 1, it is.
12x If a number can be divided by 3 and 4, then it can also be divided by 12.
With these 2 (sum of the digits and double-even) ways to understand divisibility, it becomes possible to place any number into 1 of 12 positions.
Placing Numbers Into the Web Positions.
If a number is double even and divisible by 3, it is divisible by 12 and in the 12th position.
If a number is even, but odd when halved, and divisible by 3, it is in the 6th position
If a number is odd and divisible by 3 it is in the 3rd or 9th position. If you can add 1 to the number and make it a double even (divisible by 4) then it is in position 3. If you can subtract 1 and make it a double even, then it exists in position 9.
If a number is double even, but not divisible by 3, then it exists in positions 4 or 8. If you can subtract 1 and make it divisible by 3, then it exists in position 4. If you can add 1 and make it divisible by 3, then it resides in position 8.
If a number is even but not double even, and you can add 1 to make it divisible by 3, it exists in position 2. If you can subtract 1 and make it divisible by 3, then it belongs in position 10.
If a number is odd, and you can subtract 1 and make it divisible by 3 and a double even, it belongs in position 1. If you can add 1 and make it divisible by 3 and a double even, then it belongs in position 11.
If a number is odd, you can add 1 and make it divisible by 3, and a single even, then its in position 5. If a number is odd, you can subtract 1 and make it divisible by 3, and a single even, then its in position 7.
Other Interesting Functions of Primes and 12x.
All twin primes added together equal 12x.
All primes squared (besides 2 and 3) equal 12x +1. Below is the pattern of the sum of the digits of each position.
Compositability: The ability of a number to be divided by another number besides itself and 1.
Prime: A number with no divisors.
Semiprime: A number that is the sum of 2 primes.
Twin Primes: when 2 prime numbers are separated by only 1 number (5 and 7, 11 and 13).