It is important to read the previous three articles (The Fundamentals of Syndex, The Classification of Number, and An Introduction To

As we saw with

We can summarize this archetype with a simple rhyme: An in-and-out pulsation leads to a spiral rotation.

This pattern shows up all throughout Nature, and even within our symbols for describing it! However, the cyclical behavior only truly becomes apparent within numbers when they are placed along the outside of a circle in particular increments. This principle is referred to as

A

6 is the first Perfect Number. Let's take this number, double it, and then multiply the result by each of the Prime numbers in succession, beginning with the number 2:

6 doubled is 12

12 × 2 = 24,

24 × 3 = 72,

72 × 5 = 360,

360 × 7 = 2520,

etc.

Primes are shown in red and Highly Composite Numbers in blue within the above sequence. While this process may seem somewhat arbitrary at first, notice that we are actually generating some of the most "factor-able" numbers (i.e.: Highly Composite Numbers) by means of the least "factor-able" numbers (i.e.: Primes).

Whenever we multiply a number by another, those two numbers become factors of the "product" or result. For example, 2520 is divisible by both 360 and 7 because it is the result of 360 × 7. This operation is referred to as

If we wanted to produce a number that was divisible by as many consecutive numbers as possible, then we could multiply together every number that came before it. In traditional mathematics, this operation is referred to as a "factorial" and is represented by an exclamation point. For example, say that we wanted a number that was divisible by all of the numbers 1 through 4:

4! = 1 × 2 × 3 × 4 = 24 (which is divisible by 1, 2, 3, 4, and a few multiples of those numbers, like 6, 8, and 12)

This is a simple example. However, this operation quickly produces very large numbers when we use a higher "factorial". If we wanted a number with more consecutive factors, this process can become a bit unwieldy. But notice that every successive number produced within

12 (the first number to have five of the base-ten numerals as factors: 1, 2, 3, 4, and 6),

24 (the first number to have six of the base-ten numerals as factors: 1, 2, 3, 4, 6, and 8),

72 (a number which has seven of the base-ten numerals as factors: 1, 2, 3, 4, 6, 8, and 9),

360 (the first number to have eight of the base-ten numerals as factors: 1, 2, 3, 4, 5, 6, 8, and 9),

2520 (the first number to have all nine of the base-ten numerals as factors: 1, 2, 3, 4, 5, 6, 7, 8, and 9).

Instead of using factorials, we have produced some of the lowest numbers with as many consecutive factors as possible using the fewest number of multiplications as possible. This phenomenon is called

The numbers within

We will give a letter name to each of these

etc.

What is special about these circles of numbers? They make a relationship between the Primes, Squares, and Composites clearly visible. This is because numbers at one part of the continuum can affect others that are seemingly distant from them. For example, 9 is not Prime because 3 comes before it. This is true for every odd number that is a multiple of some Prime. One of the characteristics of all

[There are some concepts out there that produce similar results, such as "Number Spirals" (e.g.: the "Ulam Spiral", Gary Croft's "Prime Spiral Sieve", Scott Nelson's "Daisy Integer Matrix", etc.) and various other "Wheel Factorization"-type algorithms. However, few exhibit symmetry both vertically

Let's look at the first

The

The grid is the same information as the circle, but in tabular form. [It is similar to Jeffrey Ventrella's "Divisor Plot".] The little rows of eight boxes (labeled as columns B through I) are one of the ways in which factors are shown in

Note that columns B through I correspond to our "octave" of numbers 2 through 9. Therefore, sweeping across an entire row immediately shows all of the factors of each number to the left of it, and sweeping down a column shows the multiples of the number associated with that column as a whole. For example, column B is like counting by twos, column C is like counting by threes, and so on.

To put it simply: Columns show multiples, while rows show factors. Because multiplication and division are "inverse operations", multiples and factors are related to one another. Primes, Composites, and Squares (and all of their factors and multiples) are tightly integrated into one cohesive system.

[This is equivalent to Peter Plitchta's "Prime Number Cross".]

The

Notice that boxes B through I of our

The only base-ten factor that is missing from 360 is the number 7...

There is much mysticism revolving around the number 7. It embodies many cycles and symmetries throughout the number continuum as a whole. For example, division by seven always yields a repeating decimal that has the same digits in a different order. Notice that 3, 6, and 9 are the only base-ten numbers that are missing from this sequence:

1 ÷ 7 = 0.142857...

2 ÷ 7 = 0.285714...

3 ÷ 7 = 0.428571...

4 ÷ 7 = 0.571428...

5 ÷ 7 = 0.714285...

6 ÷ 7 = 0.857142...

7 ÷ 7 = 1

8 ÷ 7 = 1.142857...

9 ÷ 7 = 1.285714...

etc.

Why does this work? As a whole number, 142857 is referred to as a "Cyclic Number". This means that multiples of it produce another number which are the same digits just shifted over:

142857 × 1 = 142857

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142

142857 × 7 = 999999

Therefore, this sequence shifted over three digits and added to itself produces 9 repeatedly:

142857 + 857142 = 999999

285714 + 714285 = 999999

428571 + 571428 = 999999

Within various spiritual teachings (such as the "Fourth Way"), these patterns are sometimes referred to as "The Law of Seven" and "The Law of Three". They are usually represented by a symbol called

The six-pointed figure within

We will use

This number wheel contains 9 "spokes" or "rays", and the entire spiral of numbers is taken out to 121 (which is 11 × 11). Squares are contained within squares. The

Now, let's get to

[In Progress...]