3.1.4. Poly-Cell and Hyper-E Notation
If BEAF is the most well-known of the less-known large number notations, then in second place probably falls Sbiis Saibian's Extensible-E (ExE) System. It's a large number notation much to the likes of BEAF, divided into subsystems each more advanced than the previous, and with a vast amount of googolisms to its name. Just like we've spent the past few articles learning about Bowers' linear array notation (the first subsystem of BEAF), today we'll learn about Hyper-E notation (E#), the first subsystem of ExE. And just as we spent a good deal of time looking at the backstory of Bowers' legendary notation, we'll go into heavy detail on the origin of Saibian's system, which started as the adventures of an imaginative little second-grader.
Although Sbiis Saibian is a prominent figure in googology who I have brought up again and again throughout this site, little is known about who he is as a person, even less than we know about Jonathan Bowers. The name "Sbiis Saibian" is a pseudonym (he pronounces it /sbeez sigh-bee-uhn/), and his real name is not generally known, nor is what he looks like - so far nobody in the googology community has been able to sleuth out either of these details. An avatar he commonly uses is shown to the right. His fictionpress.com profile tells us he was born 1983 (and that he is male), and he is an American college student who works as a math tutor at the college he attends. He first got into large numbers when he was only in second grade, and you're right about to learn the full story of how he did.
First we'll learn about Sbiis Saibian's poly-cell notation (his childhood googology), then Hyper-E notation (the modern version of that notation), then the vast array of googolisms he defined using Hyper-E notation. Let's begin.
Part 1: Poly-Cell Notation (Sbiis Saibian's childhood googology)
The Backstory: Googolgongs and the Path to Infinity
It all started when Sbiis Saibian was a second grader, which I estimate was around 1991. Saibian says that he was fascinated with mathematics for as long as he can remember, and a concept that he found particularly invigorating was the notion of infinity. As we know, most people think of infinity as the number above all numbers, but since it's not a number in the way one or a googol or Graham's number is, if you really stop and think about infinity it isn't too hard for the layman to confuse the heck out of himself. Young Sbiis Saibian in particular was unable to accept the idea that infinity was unreachable no matter how big a number you devise, which was half of the motivation for his large number quest.
The second half of Saibian's motivation started when he became interested in really big finite numbers. In 2nd grade math class, the largest numbers his class worked with were in the hundred thousands, but from his father he learned about a million, a number that to his resent was almost kept a secret from his math class, not mentioned once in the second grade math textbooks. He then learned about the higher -illions from his father's dictionary, memorizing all the -illions from the everyday billion up to the obscure vigintillion, topped by the king of the dictionary -illions, the centillion, which if you recall, is 1 followed by 303 zeroes. The centillion became Saibian's favorite number, and he says that it was "kind of his googol", since he never learned about the googol as a kid.
Now here's where little Sbiis's quest really takes off. One day, he was waiting with his best friend after school for her father to pick them up and walk them home. When his friend's father arrived, for whatever reason, Saibian talked about all the large numbers he learned about. His best friend's father responded by telling him about a phenomenally large number scientists came up with called a googolgong, which was something like 1 followed by 100,000 zeroes. Saibian was instantly floored, and in his mind popped a huge gong that would ring for a googolgong years when struck. He decided that if "scientists" could come up with such a huge number, he'll try to top them and come up with something even crazier!
Although he didn't know it at the time, his friend's father was probably incorrectly explaining the googolplex. Somehow the man managed to mess up the name of the number, the definition, AND the origin story - that's three botch-ups in one! There never was such a number as a googolgong, only the ever-famous googol and googolplex, which are 1 followed by 100 zeros and 1 followed by a googol zeroes respectively. And it's pretty funny to imagine that "scientists" would be the ones to come up with such a large number for the sake of it - that's just not something that people who study science would do. Instead, a single mathematician and his nephew came up with the googol and googolplex, not a team of mad scientists researching whatever.
Despite how botched up the googolgong was, Saibian was now inspired to generate the largest numbers he could, and write a book on large numbers titled "From One to Infinity" that would show readers through larger and larger numbers, starting with the basic numbers, going through the mechanics of scientific notation, and devising bigger and bigger numbers from there, until the intrepid second grader manages to fabricate a value so incredibly monstrous that it is exactly equal to the elusive mind-screwing "number", infinity!
Young Sbiis wrote this book on a stack of his father's copy paper. He didn't necessarily succeed in the goal of reaching infinity, but as we all know you can't actually do that; nonetheless, he did manage to come up with numbers that are way bigger than stuff most people know about. As you'll learn, he even managed to think up numbers that surpass Graham's number. Now think about that: Graham's number is commonly thought to be an "OH-MY-GOD-IT'S-SO-HUGE-NOBODY-CAN-BEAT-IT" number, and yet a second grader was able to go way beyond it with his imagination. A second grader surpassing a number that is believed by many to be the world's biggest number. Does this give you an idea of just where Graham's number falls in the scope of large numbers, if an imaginative second-grader can beat it? Although Sbiis Saibian lost the papers on which he conjured all these numbers to who-knows-where, he still remembers the contents pretty well, and those contents happen to be the basis of the very notation that is the subject of this article.
Going Beyond Scientific Notation
Almost everyone is acquainted with scientific notation, a mathematical concept that, unlike most things you learn about in this site, is part of any typical grade school curriculum, and is rehashed again and again in science classes. Scientific notation lets you write down large (or small) numbers in the form a*10x, where 1 ≤ a < 10 and x is any integer - for example, 3,500,000 written in scientific notation is 3.5*106. Young Sbiis Saibian, with his interest in large numbers, was fascinated with scientific notation since it lets you write really big numbers compactly. For example, a decillion written out in full is:
but written in scientific notation it is:
which is much more compact. Even Sbiis Saibian's two favorite large numbers, the centillion and the googolgong, can be written as 1*10303 and 1*10100,000 respectively, or more compactly, 10303 and 10100,000.
And so Saibian started to experiment with scientific notation during class time, and figured out how to make numbers that would leave centillions and googolgongs in the dust. He came up with a number equal to 1 followed by a centillion zeroes, which would be written out as 1010303. He named this number a "centillionillion". In fact, this is the only name for a number Sbiis Saibian came up with as a kid. Note that "centillionillion" is not the most technically accurate name for 10 to the power of a centillion: a better name for it would be "centillionplex", and "centillionillion" should mean the centillionth -illion, equal to 103*10303+3. Sbiis Saibian now calls 10 to the centillionth power an "ecetonplex" - the name is a modification of the slightly awkward name "centillionplex", replacing "centillion" with "eceton", an alternate root name for that number.
He found out that you could go further than a "centillionillion" with 1 followed by a googolgong zeroes (10^10^100,000), or 1 followed by a centillionillion zeroes (10^10^10^303), or continuing with numbers like 10^10^10^100,000, 10^10^10^10^303, 10^10^10^10^100,000, etc. It's not long until we'll get to unwieldy numbers like:
But young Sbiis came up with a notation to fix that. He called it "scientific notation notation"; he admits that he wasn't good with names, but didn't really care about names for his googological creations at the time - a bit ironic, since he nowadays tries his hardest to come up with good-sounding names. He notes that better names for this notation would be "stack notation" or "super scientific notation", and we'll call his notation "stack notation" (same name Sbiis Saibian himself uses in articles to refer to that notation).
So how does stack notation work? It's really simple. It takes 3 numbers, the "base", "replicator", and "determinant", and puts the base in a square, the replicator in a rectangle, and the determinant in a triangle. Those three numbers, which we'll refer to using the variables b, r, and d, are then plugged in to the expression:
b^b^ ... (r copies of b) ... ^b^d
This means that we can sum up stack notation in a picture as follows:
Since stack notation isn't easy to type, Sbiis Saibian created an ASCII version of the notation, where you put the base inside a pair of square brackets [ ], the replicator also inside square brackets, and the determinant inside a pair of angle brackets < >, so that an expression in ASCII stack notation would look like [b][r]<d>.
For an example of stack notation, take the number 10^10^10^10^10^10^10^10^10^10^10^10^10^303 I mentioned earlier. You would write this in stack notation as <303>: a power tower of 13 10's, topped off with a 303.
It should be pretty obvious that the replicator has the greatest effect on the size of numbers in stack notation. So why not cut to the chase and make the replicator be a massive number ... a centillion perhaps? Or a googolgong? Nope, not big enough. How about making the replicator a number where the replicator is a centillion:
In this, the base is 10, the replicator is a giant number, and the determinant is 303. More specifically, the replicator is a number where the base is 10, the replicator is a centillion, and the determinant is 303: a power tower of a centillion tens topped with a 303. This means that the number shown above is a power tower of "a power tower of a centillion tens topped with a 303" tens, topped off with a 303. You can visualize it as:
10^10^10^10^ ... ... ... ... ... ... ... ... ... ^10^303
10^10^10^10^ ... ... ... ... ... ^10^303
Of course, you can make the replicator be that number as well, giving you the number:
Or even further:
and so on.
You can keep plugging numbers like this into the replicator of stack notation to produce bigger and bigger numbers, but this eventually becomes unwieldy. Why not come up with a compact notation for repeatedly applying this process? That's where we get to the second part of young Saibian's notation: diamond notation.
Diamond notation, or as young Sbiis called it, "scientific notation notation notation", is the 4-argument counterpart of stack notation. While stack notation uses three arguments (the base, replicator, and determinant), diamond notation adds a fourth argument, the second replicator (also called depth). The four arguments are arranged in a diamond like so:
Sbiis Saibian says that he doesn't remember exactly which argument went where in the diamond, except that the second replicator went in the upper-right corner. He guesses that the other three arguments (base, replicator, determinant) went clockwise from where the second replicator went - in other words, the base goes in the bottom right, the replicator in the bottom left, and the determinant in the bottom right. This means that for our purposes diamond notation will look like this:
NOTE: On his article on his childhood large numbers, Sbiis Saibian colors the top-right cell in diamond notation, apparently to signify that the colored cell is the most significant argument as you'll see.
We can write diamond notation in ASCII as:
So what does the second replicator do? To understand it, think back to the sequence:
The members in the sequence would be written as:
/ 303 | 1\
/ 303 | 2\
/ 303 | 3\
/ 303 | 4\
respectively. This is because diamond notation maps to stack notation like so:
with the general rule being:
and pentation using diamond notation:
Eventually these numbers get so big that the second replicator is itself a diamond notation number, or a diamond notation number where the second replicator is a diamond notation number, and so on. Isn't this just like what we saw at the end of the sub-heading on stack notation? It's only logical that we continue from diamond notation with a 5-argument notation, 5-cell notation.
But before we move on, I'd like to note something about the notations and how they compare against up-arrow notation: you may notice that stack notation creates numbers on the same scale as tetration, and that diamond notation creates numbers on the same scale as pentation. It's actually possible to express tetration using stack notation:
You get all this? If so, good. Basically, diamond notation uses recursion to feed an expression in stack notation into the replicator, and repeating that process.
And with diamond notation we can make crazy numbers, where the second replicator gets bigger and bigger:
5-Cell Notation and Beyond
From diamond notation easily follows a five-argument notation, which Sbiis Saibian calls 5-cell notation. In 5-cell notation, a pentagon is split up into five triangles, and each triangle holds a different argument, like so:
The argument r3, in the top-right cell, is called the third replicator. It works like so:
Now that's a really powerful notation - powerful enough that you can't directly relate it back to even stack notation, which is the largest notation in Saibian's system that can be easily understood in terms of standard math notation. In terms of hyper-operators it's on par with hexation. You can express a hexated to b in 5-cell notation by making the base equal to a, the third replicator b, and all other cells 1.
We can make more and more notations the same way - 6-cell notation, for instance, uses 6 arguments (all from 5-cell notation plus a fourth replicator), and it is to 5-cell notation what 4-cell notation is to diamond notation. After that comes 7-cell, 8-cell, 9-cell, 10-cell, and so on.
All these notations - stack notation, diamond notation, 5-cell notation, 6-cell notation, and so on - are collectively called the poly-cell hierarchy. It's in its entirety on par with up-arrow notation (or Bowers' 3-entry arrays), and in general, x-cell notation is on par with the (x+1)th hyper-operator (e.g. 10-cell notation is on par with 11-ation or a^9b).
Expanding upon the Poly-Cell Hierarchy
Now young Sbiis Saibian thought even further than that: one day when he was visiting family, he walked outside as he let a train of imagination start blasting through his head. can go up from this and have 100-cell notation, or million-cell, or centillion-cell, or googolgong-cell ... you can have any number of cells you want, and that's where we get to the part of extending the poly-cell hierarchy.
If you can make a poly-cell notation with any number of cells, then just imagine where that can take you. We could have a 10 inside every cell of the notation, then make the number of cells anything. You could have a stack notation number of cells (like [10^303]<303> cells), a diamond notation number of cells, a 100-cell notation number of cells, or a x-cell notation number of cells where x is a 100-cell notation number, or keep feeding in the number of cells into itself
Why not define f(1) = a 100-cell notation number where all 100 cells are filled with 10, f(2) is a number like f(1) but with f(1) cells, f(3) with f(2) cells, and so on - f(1) would be comparable to the 100th hyper-operator, f(2) would be comparable to the f(1)th hyper-operator, and so on. With that direct expansion upon poly-cell notation, you could surpass Graham's number quite easily: f(1) is a 100th hyper-operator number much greater than G(1) which is a small hexational number, f(2) is a f(1)th hyper-operator number much greater than G(2) which is a G(1)th hyper-operator number, and so on: f(64) surpasses the so-called "world's largest number" quite easily. And you can also have not the 64th member of the sequence f(n), but the f(64)th member, the f(f(64))th member - that also forms a sequence, which you can jump to whatever member: if f(64) is the first member, f(f(64) the second, and so on, you can have the f(f(f(f(f(64)))))th member or whatever. You can progress further from that by defining maybe g(x) = f(f(f(...f(x)...) with x copies of f, define h(x) = g(g(g(...g(x)...) with x copies of g, i(x), j(x) ... or just f1(x) = f(x), f2(x) = g(x), and so on. Or fn(x) where n is f100(100)? It can go on and on...
Young Sbiis Saibian imagined all these sequences going further and further, so far that he became lost in his train of thought. He wondered: did he reach infinity? He felt like all these crazy large number sequences would eventually lead to what infinity really is, but part of him couldn't help but have the feeling that infinity really is unreachable. He previously had thought of infinity as something real at the end of the number line, and he remarks that it's hard for him to appreciate how he thought of infinity back then. However, the pay-off was describing numbers way bigger than almost all others would ever dream up, leaving the measly googolgong in the dust! He says that he told his dad about stack notation and possibly diamond notation, but all the larger numbers were only something he knew about.
After that day where he thought up numbers that may go as far as Bowers' linear arrays (according to personal communication), he promised to himself that one day he will return to large numbers. That didn't really happen until 2004, when he researched large numbers, and was surprised to learn that others have devised large numbers much like he did, with notations like up-arrow notation and chained arrow notation. He fiddled around with the notations and devised his own extensions to them, and also compared them to his childhood large numbers. Eventually, in 2011, he decided to make a modernized version of his childhood poly-cell notation, and called it Hyper-E notation.
Parts 2 and 3: Coming soon!