## Ψ.1.Natural repetition

### # 1.5.Elementary powers

At the ninth hour Jesus cried with a loud voice, "Eloi! Eloi, lama sabahkthani?"
Which means, "God, my God, why have you left me?"

#### §1.5.1.Exponentiation is a hypercube

With addition `+` as the empty operation, any number of operators `+..` or `+n;` will be empty too. Without operation there is just separation, where all pluses function as spaces `0*` adding nothing but `0` in effect.
Such meaningless separators that represent the `0` concept can be used to define the ground state of an ordered structure. This helps to show, for example that multiplication `a*b` contains an extra repetition concealed in its left item `a`, simply because this is a number variable.

3*2 = 1+1+1++
1+1+1++ (rectangle)

3*3 = 1+1+1++
1+1+1++
1+1+1++ (square)

ai*... {ai#n} = 1+.... {+..#n #ai} (box)

Single `+` divide the space inside a natural number, separating the units `1` that count that number. Lower constructs can be ordered visually in a higher dimension by multiple plus signs or with comma `,` separators. We think of constructions like these in terms of dimensions, occupying a discrete space (not continuous).
This way a rectangle with more than 2 dimensions can be formed, which is called a hyperrectangle or simply a box, when the size of any dimension may differ from that of other dimensions.

Special boxes are the hypercube with a higher dimensional array where all dimension sizes `ai` are equal, the initial number value `a0 = 1+..` {1#a} included, though the number of dimensions may differ.
And then the dimension box where the number of dimensions equals the size of each subdimension, but where (as specified) a different parameter value `a` is allowed.

Usually exponentiation is pictured as a series of (equal) numbers `a` connected with operators `*` of (the preceding operation of) multiplication. But work this out completely and exponentiation in its ground state is just the symmetric ordering of units `1` in a multidimensional cube or hypercube.
The example below is kept brief by omitting unit level separators from the dual repetition. Of the two power patterns, the first reduces to a row of multiplications, the second directly fills a hypercube with ones.

3**4 = 3*3*3*3 (4-dimensional cube or tesseract)
= 1+.... {+..#4 #3} = 111,.... {,..#3 #3}
= 111,..,..,.. {#3 #3 #3}
= 111,111,111,, 111,111,111,, 111,111,111,,,
111,111,111,, 111,111,111,, 111,111,111,,,
111,111,111,, 111,111,111,, 111,111,111,,, = 81

a**b = a*.. {a#b} (repetition over a row is actually..
= 1+.... {+..#b #a}  ..repetition over dimensions..
= a,.... {,..#b- #a} ..in a hypercube)

Exponentiation expressed as sequences of `1..` separated by empty `,` is equally void an operation as addition. However, its virtue lies not in the power to make things happen, but in the sheer economy of writing numbers in the exponential format `a^b` when its ground state of repeated or added groups of ones had caused a lot of overhead.
This chapter meant to remind you of the true nature of the exponential function – it creates hypercubes!

#### §1.5.2.Googol world

number US-name SI-prefix
10^3 thousand k kilo
10^6 million M mega
299792458 m light-second
10^9 billion G giga
13.75 Ga age of our universe
10^12 trillion T tera
9460730472581 km light-year
10^18 quintillion E exa
10^21 sextillion Z zetta
10^24 septillion Y yotta
10^27 octillion H hella *
6.187151E34 length units/meter
1.854860E43 time units/second

In decimal notation we use 10^n powers of ten 10... {0#n} to express numbers, and round off larger numbers to 10^(3×n) powers of a thousand 1000,... {000#m} and convert these to names. But the words differ, past a million 1000,000 people have to make a choice between the short scale American system or the long scale old European.
Our advice is to follow the Americans and continue with a billion and a trillion, but after that to give up on names and use calculator speak instead! It's real easy to parrot E fifteen 1E15 (a quadrillion) or two E twenty 2E20 (200 quintillion) from the calculator screen and also handy, given that we multiply by adding up exponents.

So the folks on planet Earth express large numbers with individual names that have some translation to a power of ten. What's important then is to reach an international consensus on a system of construction, which is transparent and practical in its application.
The table on the right lists the current conventions for integer magnitudes and throws in some astronomical sizes and Planck units. A computer geek would prefer to write a number in its binary form and then count the required bits or file size in bytes, where 1 KB = 8×2^10 bit

Because this is a book about Big numbers we owe you the googol and the googolplex.
The two numbers became part of popular maths culture when the founders of the world's main search engine adopted the name Google in 1997 to reflect their mission to organize a seemingly infinite amount of information on the web. The Google index stores nearly 100PB of web page data as of June 2010. Its company building and campus is called the Googleplex.
Both names were originally put forward by a 9 year old Jewish boy <Milton Sirotta> in 1920 (with the traditional dessert Gogl-Mogl as a likely inspiration) to be used for the new Big numbers of his uncle, the mathematician Edward Kasner, who popularized the family googols with an article (1938) and in a book (1940).

We've expanded these two numbers to a system dubbed the `googolmultiplex` which may become the height of fashion of tomorrow (every Big number aficionado his own claim to fame nay?-).
We'll use Knuth's arrows ^^.. with an occassional \ backslash, which naturally continues from the lines before – in the next chapter these new signs will be explained.

googol = 1111111111**1111111111**11 = 10^100 = 333 bit
googolplex = 10^googol = 10^10^100 ~ 3.32E100 bit

googolduplex = 10^googolplex = 10^10^10^100
googoltriplex = 10^googolduplex = 10^10^10^10^10^2
googol-4-plex = 10^googol-3-plex = 10^^6\^2
googolmultiplex : googol-n-plex = 10^^n\^googol

googolgoogolplex = 10^^googol ~ 10^^(2+10^100)\^2
googolgoogolduplex = 10^^googolgoogolplex = 10^^10^^googol
googolgoogoltriplex = 10^^10^^10^^googol
googolgoogol-n-plex = 10^^^n\^^googol

googol-3-flex = googolgoogolgoogolplex = 10^^^googol
googol-3-flex-n-plex = 10^^^^n\^^^googol
googolmultiflex : googol-m-flex = 10^...googol {^#m}
googoldiflux = googolgoogolflex = 10^...googol {^#googol}

googolflux = googolflex = googolplex = 10^10^100 = gog100
gog(n) = 10^10^n
ma(m)gog(n) = m^m^n
gog = 10^10 ~ maêgog(3.14)

We end with a teaser. All of human experience can be positioned in the data range between googol and googolplex, approximately between `gog5` and `gog27` in double-exponential plexcentages and more accurately in the range between `maêgog13` and `maêgog64` with natural double-exponents. What is your plexenta and range of mind? Consider this.

##### Humanity's hard disk

Your brain is a smart computer that mostly interprets information from the senses. With the visual cortex receiving 20 Mbit/s from both eyes, it processes about 80 KB in a 30 Hz blink, enough bits to put 2000 googols in a row.
The data stream of all the senses combined plus thoughts and emotions, shouldn't be more than 50 Mbit/s for the average waking state. So a long life will fit on 500 TB of disk space (without compression), which is affordable.
With an estimated total of 108 billion people <Grünwald 2008> to date, the story of mankind takes no more than 1E28 bit to express a mathematically random and physically fuzzy number around 10^10^27 – far smaller than a googolplex.