1010011010, Defined

Please note that the post time was edited so that this post comes as part 3 of 3 in a large post, and is displayed after its corresponding piece.

First and foremost, the original negative connotation associated with the number 666 comes directly from the Bible. I don't hold any truth in anything contrived from said source. Sorry Mom, but it is, as all other history is, be it true or false, all hearsay, and one with a logical and curious mind such as mine must take it with a grain of salt... All other darkness is prescribed to the number as descendant to the same source of hearsay and, dare I say, ignorance.

I use the number because, growing up, when I first started to doubt that which was fed to me, I started to first research everything that directly went against what was suspiciously being preached to me. I therefore learned a lot about a lot of things, including this "evil" number, and I became inseparably attracted to it.

Secondly, I have a habit, especially in discussing personal philosophies, or subjects that I have very intricate reason to display, to tend to show only a 'tip of the iceberg,' and to present a debased version of whatever meaning I initially intended. Some people get tattoos because they think they are pretty, and that's all you see. I get tattoos that dignify the ink running as deep into my skin as can be allowed, and you have to dig into me to grasp them. Even so, when the average stranger, client, aquaintence, or even friend inquires as to my underlying meaning, I don't care to share much further than a brief jaunt of what it means to me. But it's all here. I'll be who I am, and say what I feel, for those who mind don't matter, and those who matter don't mind. (As further discussed in regards to my arm tattoos.)

As a mathematical figure, 666 is a natural (1, 2, 3 ...), palindromic (ten animals I slam in a net; race car; dammit, I'm mad!; I, madam, I made radio! So I dared! Am I mad? Am I?, etc), repdigit (11, 444, etc; 666 is trips) number.

666 is abundant (the sum of its divisors is greater than two times itself):

The divisors of 666:

SUM(1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666) = 1482

2(666) = 1332

(interestingly, this is the sum of its palindromic/repdigit divisors, 111 + 222 + 333 + 666 = 1332)

The abundance of 666 is:

1482 - 1332

150

666 is triangular (think bowling pins, 1, 3, 6, 10, etc). This is because it is the sum of the first 36 natural numbers:

1 + 2 + 3 + ... + 34 + 35 + 36 = 666

Since 36 itself is both square and triangular, 666 is the sixth number of the form:

n^2(n^2 + 1) / 2

36(37) / 2

1332 / 2

666

and 666 is also the eighth number of the form of doubly triangular numbers:

n(n + 1)(n^2 + n + 2) / 8

8(9)(64 + 10) / 8

72(74) / 8

5328 / 8

666

666 is the magic total of the prime reciprocal magic square using 149.

I'll use an example, as this square would be 148 digits up and down.

Using 7, the repeating digits are as follows:

1 / 7 = .[1][4][2][8][5][7]

2 / 7 = .[2][8][5][7][1][4]

3 / 7 = .[4][2][8][5][7][1]

4 / 7 = .[5][7][1][4][2][8]

5 / 7 = .[7][1][4][2][8][5]

6 / 7 = .[8][5][7][1][4][2]

And the digits of each column, as well as each row, add up to 27, its magic total.

666 is a Smith number (composite number whose digits' sum is equal to the sum of the digits in its prime factorization):

6 + 6 + 6 = 18

2 * 3 * 3 * 37 = 666

2 + 3 + 3 + 3 + 7 = 18

Of the first million natural numbers, about 2.99% are Smith numbers. The more numbers you count to, the less percentage will be Smith numbers.

Using Roman numerals, 666 is represented as DCLXVI, which are in perfect consecutive descending order, once each:

D = 500

C = 100

L = 50

X = 10

V = 5

I = 1

But perhaps the most remarkable and beautiful of all properties of the number 666 is that it occurs as 13th prime in numbers following the Padovan sequence:

This is also the most complicated of the notable properties of 666:

P(0) = P(1) = P(2) = 1

and

P(n) = P(n − 2) + P(n − 3)

The first 13 values of P(n) are:

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21

This is illustrated in the following diagram:

Also closely related is the Fibonacci sequence:

For me personally, the relationship between these two, and the relationship between the complicated, intricate almost unknowable patterns in the formulas, and the simplistic, breathtaking beauty of the spiraling forms that they create, and the fact that they are found in nature, is completely, utterly, fricking, dumbfounding.

Continuing yet deeper, this Fibonacci sequence is defined by the Golden ratio or φ (an irrational constant, included on this chest piece), and the Golden section, expressed as follows:

φ ≈ 1.6180339887

(a + b) / a = a / b = φ

This ratio describes the relationship between lengths of sides, when:

And so, this ratio, being used in the Vitruvian Man, as well as many other works of art, in accordance to geometric proportions found in nature, most notably between human body parts and human faces, especially as illustrated by da Vinci (among many), and, having also been used in works of music, can be found universally in nature, organic and inorganic. These self-similar, recurring patterns, described in simplest term as fractals, build the shapes defining the following examples:

branches in plants

veins in leaves

skeletons of animals

branches of their veins and nerves

proportions of chemical compounds
lightning arcs

geometry of crystals

Imitating the truest reality possible, it is said that obvious occurences of the Golden ratio in nature, and art forms using the Golden ratio as a basis of proportion, are the greatest performances of beauty.

The Golden ratio is also found in connection with 666, in the following expression:

-φ = sin 666^o + cos (6 * 6 * 6^o)

Sine and cosine, describing relationships between triangles and circles, gives yet another reason to incorporate a circle as I so desire. I've often thought of that for a tattoo, and/or the usage of the symbol for pi to recognize the awesomeness of the circle. Now I shall have that, and more.

Finally, the four lengths of line segments in a pentagram (shown in red, orange, green, and blue) are in Golden ratio to one another:

Pi, when written in binary, has several strings where the binary for 666 occur, naturally. The first instance of this is just after the 600th place after the decimal. It reads 1010011010 from the 601st to 610th places consecutively. There are 22 occurrences of this string in the first 32,768 digits.

Math. Nature. Beauty. Tell me how you get Satan out of that.