It is important to read the previous two articles, The Fundamentals of Syndex and The Classification of Number, to fully grasp the following. While it may seem complicated, please have patience for it. Feel free to skim sections that do not make sense the first time around, and come back to them again after finishing the article. Try to play around with the patterns described so that they can become more familiar to you. Having a sheet of paper, a pencil, and a simple calculator handy might help.

Most of the time when dealing with number we use what is called "base-ten". This just means that we use ten symbols to represent every quantity:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

We cycle through the same ten symbols over and over again, and every time that we reach a group of ten (such as 10, 100, 1000, etc.), we attach another digit to the right. Therefore, the order of the symbols determines the amount that it represents.

In

2 through 9 makes eight distinct base numerals, an "octave" of sorts. (The prefix "oct-" means "eight".) This concept will come up again and again as we continue...

Within base-ten, the operation of

It might be obvious how 11 is related to mirroring (because it is

Notice how the numbers are moving in two directions simultaneously (i.e.: the tens place seems to count up, while the ones place seems to count down). They also contain an octave (of 4 forward and 4 reverse) that seems to turn around at 49.5. The term

These patterns persist into higher digit numbers when we combine 9 and 11. For example, to see the mirrorings in 3 and 4-digit numbers we can take multiples of 99 (which is 9 × 11) and multiples of 1089 (which is 99 × 11):

This continues up to any number of digits. For example, 5-digit mirrorings are produced with multiples of 10890 (which is 990 × 11), 6-digit mirrorings are produced with multiples of 108900 (which is 9900 × 11), and so on. Robert Marshall sometimes depicts this mirroring quality embodied by 9 and 11 with a crab (probably due to the symmetrical nature of its body shape and how many species of crab walk sideways):

[This little guy is derived from the art piece "Crab" by M.C. Escher.]

A diagram called the

While these patterns are beautiful to look at, they also have a deep significance to mathematics as a whole, for they hint at an underlying dynamic permeating the entirety of number itself. This dynamic is called

The reflective patterns of 9 and 11 are a perfect example of two-way, wave-like motions throughout the base-ten system (an "

This might seem complicated, but all that we need to remember is that

In general, you might have already noticed that 9 is a Square (specifically 3

• 11 is known as the

• 9 is known as the

We cannot emphasize this enough: In order to get a full comprehension of number, we need to consider how both of these are continuously appearing throughout

The following diagram is called

It may seem like a lot is going on within this diagram, but we will take it one step at a time. Here are some of the key features of it:

• The numbers within the right-hand column are Squares derived from the numbers within the left-hand column. In other words, multiply the number on the left by itself to get the number on the right.

• The small arcs on either side of the 1-digit numbers within the left-hand column are showing which ones add up to 9 and 11, while the black dots on either side of the entire left-hand column are marking multiples of 9 and 11. Notice that multiples of 9 and 11 coincide or "sync up" at 99. The

• The arcs made with dashed lines are highlighting significant symmetries within the numbers of

• Both 11 and 101 are

•

144 441

and

169 961

The numbers in the first pair are made by the

The numbers in the second pair are made by the

These pairings lead to some amazing interactions between

Other "pretzel-like" patterns can be uncovered with a little attention to how 9 and 11 affect other numbers:

• The only 2-digit

16 is preceded by 15, a

16 is succeeded by 17, a

Finally, 16 + 61 = 77, which is 7 × 11. There is that

Now, let's get a better look at Primes specifically...The following diagram is called

Again, it probably seems more complicated than it actually is. It is okay if the numbers are too small to read. We just want to show the general pattern that they make and we will repeat the numbers that we are referring to below. Here are a couple of the key features of this diagram:

• The numbers within the right-hand column are the Primes, while the numbers in the left-hand column are their place within the sequence. For example, if we were looking for the 149th Prime, then all we need to do is look for 149 in the left-hand column, and then find the number directly across from it within the right-hand column, which is 859. Therefore, the 149th Prime is 859.

• All

11 [the 5th Prime]

101 [the 26th Prime]

131 [the 32nd Prime]

151 [the 36th Prime]

181 [the 42nd Prime]

191 [the 43rd Prime]

313 [the 65th Prime]

353 [the 71st Prime]

373 [the 74th Prime]

383 [the 76th Prime]

727 [the 129th Prime]

757 [the 134th Prime]

787 [the 138th Prime]

797 [the 139th Prime]

919 [the 157th Prime]

929 [the 158th Prime]

• The arcs made with dashed lines are showing

13 31

17 71

37 73

79 97

There are four in total.

107 701

113 311

149 941

157 751

167 761

179 971

199 991

337 733

347 743

359 953

389 983

709 907

739 937

769 967

There are fourteen in total.

For now, we are only going to look at the 2-digit

All of them are separated by 9 and 11! Let's look more closely at how they create a symmetry within the stretch of numbers between 1 and 99:

The arcs connect each of the numbers within the 2-digit

13 + 31 = 44 and 44 ÷ 2 = 22

17 + 71 = 88 and 88 ÷ 2 = 44

37 + 73 = 110 and 110 ÷ 2 = 55

79 + 97 = 176 and 176 ÷ 2 = 88

Notice that the arrows point to 22, 44, 55, and 88.

You might be wondering where the other

39 93

While neither of these numbers is actually Prime, they are very nearly so as 39 = 3 × 13 and 93 = 3 × 31.

We have been continuously using the operation of

Becomes

143 + 341 = 484

Becomes

587 + 785 = 1372

1372 + 2731 = 4103

4103 + 3014 = 7117

If there is a trail of one or more zeros at the end of a number, we can just ignore them when we flip it. For example:

330 + 033 = 363

This operation is called finding a

Once we have a

In this example, we are adding and subtracting 99 from 555. We use 99 because 555, our

This operation will be referred to as

There is a wave pattern that permeates all of number (

•

•

•

This sorts the Primes, Squares, and Composites into the twelve

While all of this may seem trivial, it is actually

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