2.4.8 - Bowers' -illions

Polytwister Image Created by Jonathan Bowers

(Used with Permission)

2.4.8

Jonathan Bowers'

4 Tiered -illion Series

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Jonathan Bowers

Jonathan Bowers is both an infamous large number enthusiast and polychorist. Like many other amateur mathematicians who have devoted some of their mathematical expertise to generating large numbers, we will be talking about his work quite a lot. In fact Bowers is an exceptional case, and we will have a lot to talk about in the coming section on his work.

Jonathan Bowers is probably best known for his extremely large numbers, and array notation, which he uses to define them. However, we won't be getting into these subjects until at least section III, as these are more advanced topics. He is also known for working with polytopes, which are higher dimensional solids. In addition to these achievements he has also created about a hundred -illion style numbers, which we will call the "Bowers Milestone -illions".

Unlike the systems of Professor Henkle, Conway&Guy and others, Bowers creates a utterly vast system; much more vast than any of the others yet introduced. However he leaves us to construct the intermediate terms.

I contacted him to ask him about his system. He stated that he was more interested in the "milestones" than the intermediate terms. He felt that this could be worked out by those who were interested in the details. This is not that different than the approach taken by Professor Henkles and Mr. Ondrejka when they provided lists for numbers but left the intermediates out. Unlike Professor Henkles system however, Bowers system extends way way beyond the millionth illion. His system is so vast in fact that no complete table could ever be constructed. As we will see, the intermediates for Bowers system are far more complex than anything we have previously encountered.

It should also be noted that Bowers has actually tweaked his list of names at least once. There was an old list of -illions on his older site (now defunct) which differs slightly from his new list. The only difference is in the names he chooses, and how far he extends the system. His new system extends another "tier" beyond his old system. What does that mean? Read on and find out.

Bower's Milestone -illions

Bower's devotes a single web page to his milestone -illions on each of his websites. His old system can be found on his old site. Unfortunately his old site was abandoned quite a few years ago and the service provider eventually terminated it along with all other websites made with AOL hometown. Luckily the website was archived by another server, so that technically his old site can still be found online. The archived web page of Bower's old style illions can be found here:

http://web.archive.org/web/20071205035836/members.aol.com/hedrondude/numbers.html

Bower's eventually moved his site to a different location sometime around 2007. His new site also contains a list of -illion numbers, almost identical to his original list. Some additions were made, and some of the names were changed. His new -illion list can be found here:

http://www.polytope.net/hedrondude/illion.htm

We will be considering both his old and new system simultaneously since they are essentially the same system. Bowers list of milestone members of his -illion system begins with the established -illions. Namely, a million, billion, trillion, ... all the way up to vigintillion. In other words, the first 20 -illions are taken as canon. Next Bowers list begins to jump to the 30th -illion, then the 40th, 50th, ... and so on all the way to the 100th -illion. Next he jumps to the 200th, 300th, 400th, ... all the way to the 1000th -illion, which Bower's coins a millillion. Bowers says he came up with this name independently and later discovered others had chosen it to represent the 1000th -illion. With the exception of the millillion, every other name below this is more or less a reiteration of the Conway&Guy system, with some minor differences. The spellings are almost identical to the Conway&Guy spellings, with only slight alterations.

After the millillion we start getting into unique number names of Bowers own invention. Next he calls the 1,000,000th -illion a " micrillion ". The only other "recognized" name for this is the milli-millillion from Henkles naming scheme. Bowers reasoned that since "milli" is also an SI prefix for 1/1000, he could use the SI prefix "micro" (1/1,000,000) to represent the 1,000,000th -illion. This allowed him to follow up with a nanillion, 10^(3,000,000,003), a picillion, 10^(3x(10^12)+3), a femtillion, 10^(3x(10^15)+3), an attillion, 10^(3x(10^18)+3), a zeptillion, 10^(3x(10^21)+3), and lastly a yoctillion, 10^(3x(10^24)+3). Recall the SI prefixes milli, micro, nano, pico, femto, atto, zepto, and yocto from Chapter 2-2. The problem is that there are no official SI prefixes beyond this point. To continue Bowers opts to transition, somewhat haphazardly, from SI prefixes into a greek based system. However, he chooses to change the offset used in the short scale, and instead the number in greek represents 1000 raised to that number. This is in contrast to the short scale where a latin number represents 1000 raised to that number + 1.

Bower's invents the term xonillion for 10^(3x(10^27)+3), presumably by inventing an additional small scale SI prefix such as "xono". With prefixes exhausted Bower's switches to a system suggestive of greek. His next term is vekillion, which seems to come from "deka", the greek word for ten. After this he follows a similar pattern used to reach the millillion, except this time the greek is indicating the power of a thousand multiplied by 3 in the 2nd exponent. Eventually he exhausts greek once he reaches a killillion for 10^(3x(10^3000)+3), from "kilo", greek for "thousand". Reaching this point can be thought as reaching the end of a kind of second tier, just as reaching a millillion was like reaching the end of the first tier. I sometimes refer to these as "levels".

Now "kilo" is also the SI prefix for 1000. Bowers therefore decides to continue by using the large scale SI prefixes to construct his next series of illions. Next we have megillion, gigillion, terillion, petillion, exillion, zettillion, and yottillion. After this we again run out of official SI prefixes, so Bowers invents one so we can have a xennillion.

The next step is a point of divergence from his old to new system. In his old system Bowers, having run out of latin and greek, opts to use english numbers to continue up to the next tier with names like twentillion, hundrillion, and thousillion. In his new system he scraps the english names and opts for a special numbering system of his own invention. Bowers had created a series of special names for polytopes of various dimensions. For example, we call 2-d figures polygons, and 3-d figures polyhedrons. Polychorists study higher dimensional analogies of these figures, and all of these figures are collectively known as polytopes. Because there was no codified system of names for figures in higher dimensions Bowers invented one. In the polychorist community it is generally accepted that 4-dimensional figures should be called polychorons. This however is not recognized in professional circles, and is used mainly by amateur mathematicians who study these objects. Bowers continued these names with 5-d figures being called polyterons, 6-d figures called polypetons, 7-d figures called polyectons, 8-d figures called polyzettons, 9-d figures called polyyottons, etc. From these suffixes bowers derives a whole series of milestone -illions. Amazingly Bowers had extended his polytope suffixes all the way up to the tredecillionth member! This is particularly insane because no human being or current computer can hope to cope with a tredecillion dimensions! In any case, this does allow Bowers to surpass the 3rd tier and enter the next "level" beyond a thousillion. Bower's however does not manage to reach the end of the 4th tier.

In his original system he only gets up to hyper-hyper-exponential class numbers ( as defined by my number ranges described in the article "number ranges"). In his newer system he goes up to lower hyper-hyper-hyper-exponential numbers. I also refer to "hyper-exponents" as "2nd exponents", and growth rates of this kind "exponential exponential growth" or "double exponential growth". "Hyper-hyper-exponents" are "3rd exponents", and growth rates of this kind are "exponential exponential exponential growth" or "triple exponential growth", and so on. Each time we go up another exponent in a power tower, we are going up one "level" , "tier" , or "plateau" in an illion series. Thus we can say Bowers original system reaches the level of triple-exponents (ie. 10^10^10^N) while his newer system reaches one level higher with quadruple-exponents (ie. 10^10^10^10^N). Hence, we can characterize Bowers system as a "quadruple-exponential" illion series. Note that, in comparison Conway & Guys system can only be treated as a "double-exponential" illion series, since it would require exponentially long names in their system to describe triple-exponential illion names, double exponentially long names to describe quadruple-exponential illions, and so on.

The best way to introduce Bowers -illions is probably to provide the list of his milestone illions, more or less as they appear on his website. Although this list can be found on many other websites verbatim, this is my site, and I'm a bit of a completist :)

Unlike other presentations however, we will be reviewing the system and considering the possibility of filling in the gaps and what its limits of extension are.

The following table includes every number that Bowers lists on his -illions pages that is intended as part of his system (his list also includes a few extra names outside of his system for comparison. I will include these in a separate table). These values will be presented in scientific-E notation, where nEm = n * 10^m. Please note that the order of operations will be as follows:

1. All operations within paranthesis resolved first

2. E-notation takes precedence over all other operations

3.Multiplications are carried out from left to right

4. Additions and subtractions are resolved from left to right

Unfortunately Bowers -illions are so large that it is advantageous to have a more compact notation than even Scientific-E notation. For the purposes of -illion numbers I have a series of "prefix functions". I will explain these later in more detail. Specifically we will be using my short scale function:

Let H(n) = 1E(3n+3)

where n is an element of the set { 0,1,2, ... }

The H here stands for "half scale". The reason I use this instead of S(n) for "Short scale" is because I already use S(n) for the "successor function", one of the four fundamental functions. We'll learn what those are in chapter 3-1. The advantage of the H-function is that the number "n" indicates the prefix attached to the -illion in the short scale. For example, an octillion is equal to H(8) = 1E(3*8+3) = 1E(24+3) = 1E27. This is because "oct-" is latin for 8.

Because Bowers changes some of his names I have decided to list both his old and new Nomenclature in a single table. If there is a dash, it means that there isn't a name for that number in that nomenclature. For example, Bowers had no name for a myrillion in his old system. If there is a quotation mark this means the name is the same as in the other system. This makes it easier to see where Bowers has made changes. Only names that have been changed will have rows with two names listed. In every way Bowers new system is an expansion of his old. Not only does it extend out further, but he provides more intermediates than before, and more numbers outside of his system. Thus every number in the old system is included in the new, but not every new number has an equivalent in the old system. Number names in red and italics are numbers that can not properly be attributed to Bowers, even though they appear on his list. This mainly applies to all of the names under a millillion. Bowers most likely got these names from the Conway&Guy system, and they barely differ at all from them. None the less they are included for you inspection.

So Here is Jonathan Bowers complete list of "milestone" illions:

Jonathan Bowers' Milestone illion Series

In addition to these, Bowers also includes members of the googol series for size comparison, and also includes some erroneous illions from other systems, presumably to show that his system surpasses other popular illion names such as the bentrizillion.

The following table is a complete list of names appearing in both of Bowers lists which are not strictly meant as part of his system. For convenience they are also listed in size order from least to greatest:

Miscellaneous Names on Bowers' Lists

* On Bowers list a bentrizillion is defined as 10^(6*10^(6*10^(6*10^6billion))). It is uncertain whether a billions means 1E9 or 1E12 here.

We'll learn more about the googol series in the next chapter. For some reason no real information seems to exist on the bentrizillion and platillion other than Bowers website and people quoting his website. Bowers himself admits to not knowing where they came from. It is clear however from the definitions that these names must have been coined within a long scale system.

The terms manillion, lakhillion, crorillion, and awkillion do not name unique numbers. Instead they are alternative names for numbers in Bowers newer system. One telling thing about these names is that they occur just after a thousillion. This implies that someone must have seen Bowers list, coined a few extra after a thousillion, and these somehow managed to spread to the point where they got Bowers notice. Perhaps they spurred him on to create his new names.

Excluding these names Bowers new list includes 139 milestone -illions. If we subtract out the 39 names under a millillion, we find that Bowers coined 100 illions exactly!

This number however would pale in comparison to the number of illions that would exist if every gap in his system was filled up to a multillion.

Let's now take a closer look at how intermediate terms might be formed:

Deeper Analysis : Filling in the Gaps in Bowers List

We will now begin to take a closer look at Bowers system. Of prime importance to us here is how to construct all of the intermediate names between Bowers milestone illions. Unfortunately we can not simply list out all the possible names because the number of them would be so staggeringly huge that it could not fit within the known universe. This is not an exaggeration, its an understatement! To prove this consider the following:

If our largest illion has the value H(n), then we must have "n" -illions in order to have a complete system up to H(n). In Bowers case a multillion is the largest illion. It is equal to H(H(H(H(13)-1)-1)) in half-scale notation. Removing an H-function we find that this implies n = H(H(H(13)-1)-1) = H(H(1E42-1)-1) = H(1E(3E42)-1) = 1E(3E(3E42)). That's a 1 followed by 3*10^(3*10^42) zeroes. Compare this to the number of sub-atomic particles in the known universe which is a mere 1E80, or 1 followed by 80 zeroes. Basically even the number of illion names has a hyper-exponential number of digits! There is no hope of listing out all of the intermediates one by one!

Therefore, in order to describe how to construct all of the intermediates we will need to develop a set of rules of construction. This is what the remainder of the article will be about.

Terminology: Tiers and roots

Before we begin I'd like to define what I mean by a "tier" in relation to a general power-based illion series, because we will be using this concept throughout the rest of this article.

A tier is essentially a group of roots used to describe a certain set of names. What is a "root"? A root is either a prefix or a suffix. Essentially a root is a part of a word. Roots can be combined in sequence in order to form words. This practice is most common in naming numbers, however roots play a part even in ordinary language. For example the word "abnormal", is formed from the roots "ab" for "not", and "normal". The definition then follows from the combination of these meanings as "not normal". In a similar way, "numeric roots" can be combined to form number names. For example the number "twenty four" is the combination of the root "Twenty" and the root "Four". Their combined meaning is the sum of the value of the roots. All of the illion series we've been considering can essentially be broken down into complex "root systems". A root system is a set of rules for how roots can be combined and how to interpret them. Roots systems are advantageous because we don't have to remember millions of names in order to construct them all. We only need to remember a handful of roots that can be combined in various ways to form any name within the scope of the system.

We can further break down roots in to various classes each with a different function:

Numeric roots such as the numbers 1~20 are what we might call additive roots. These roots have a specific meaning and when combining the numeric values are simply added together. Another important numeric root type are multiplicative roots. An example is the root "hundred". When it is combined with the root "two" to form "two hundred", the roots are not added to obtain 102. Instead they are multiplied together to obtain 200. Interestingly if the order is reversed to "hundred two" the implied meaning is 102. Multiplicative roots therefore have an important property: order matters. Generally in almost all languages when a smaller numeric root is followed by a larger multiplicative root, they are to be multiplied together. When the smaller roots follows after the multiplicative root it is to be added.

A third kind of root we can call separatrix roots, or separatrices for short. They operate in a similar manner to multiplicative roots but are much more general. A multiplicative root can only be multiplied by the first root to its left. However a separatrix is multiplied by the result of a "group" of roots to its left. A group here refers to a set of legal combinations of roots. These roots may be of any type. The separatrix must be defined to include what kind of groups it will allow. An example of a separatrix is the numeric root "thousand". A thousand can act as a multiplicative root as in "three thousand" but it can also act as a separatrix when it is multiplied not by a single root but by a group as in "two hundred seven thousand".

The last kinds of root we will be considering we can call modifiable roots, or mods for short. A modifiable root, or mod becomes augmented by a "modifier group" to augment its own meaning. A mod/modifier group combo can be used to form a separatrix. For example the separatrix, "quintillion" is formed from the modifiable root "illion" combined with its modifier "quint" which is latin for 5. The meaning is therefore the 5th member of the illion series.

We now have sufficient terminology to describe what a tier is properly:

A tier represents all of the constructions that can be formed using a set number of separatrix classes, that can't be described using constructions with one less separatrix class.

A separatrix class can be any positive integer and represents the rank of the separatrix. The higher the number the higher the rank. A class 1 separatrix separates ordinary groups. A class 2 separatrix separates modifier groups within a class 1 separatrix. A class 3 separatrix separates modifier groups within a class 2 separatrix ... and so on.

So a tier is basically the measure of the highest separator class allowed. To give you a better understanding of what we're talking about I'll describe the first few tiers.

Let's begin with "tier 0" constructions:

Tier 0 constructions contain no separatrices what so ever. This means that tier 0 only includes numbers from 1 to 999. "one thousand" requires the use of the separatrix "thousand", so this number can not be included in tier 0, nor any higher number.

Tier 1 allows the use of only class 1 separatrices. This means the class 1 separatrix can only use additive and multiplicative roots as modifiers. A class 1 separatrix can not contain any separatrices within its modifier group. Allowable separatrices in Tier 1 include, a thousand, million, billion, trillion, ... all of the standard illions up to a centillion, all the way up to the 999th illion. Tier 1 ends not at H(999) but at H(1000)-1. In other words, the largest tier 1 number is one less than the smallest tier 2 number.

Tier 2 allows the use of class 2 separatrices. These act as separators within the modifier group of class 1 separatrices. The smallest tier 2 number would be the 1000th illion, called a millillion in Bowers nomenclature. Here "milli" is a modifier for the mod root "illion". Numbers such as micrillion, nanillion, picillion, femtillion, attillion, etc. are all example of tier 2 numbers. Tier 2 ends at a killillion-1, or the number just before the smallest tier 3 number.

Tier 3 allows use of class 3 separatrices. In the case of Bowers system it begins with a killillion and ends with a kalillion-1.

Tier 4 allows use of class 4 separatrices. In Bowers system tier 4 begins with a kalillion (formally a thousillion). Bowers system does not completely fill out tier 4, and ends somewhere within it. We can therefore classify Bowers system as a "4 tiered illion series".

Keep in mind that the boundary points between tiers depends on the system of separatrices used. For example, there are actually 1000 class 1 separators in Bowers system, not 999 because besides the 999 legal class 1 illions, we must also recognize that a thousand is also a class 1 separatrix. This is not always the case. In Russ Rowletts greek based system, for example, there are exactly 999 class 1 separatrices, because the greek modifier prefixes represent the power of a thousand. This means that the smallest tier 2 number in Russ Rowletts system would be 1E3000, while the smallest in Bowers would be 1E3003. Depending on how we define things we might even say that the smallest tier 2 long scale number would be 1E6000. For this reason, understand that the boundaries are dependent upon the system considered.

We will now go over each tier of Bowers system, one by one, in much greater detail, to develop a complete system for describing any Bowers' Style illion.

The 0th Tier Intermediates

Since Bowers is extending the standard American number names, his Tier 0 numbers are simply the ordinary english names from 1 to 999. Although forming these numbers should be elementary we will quickly be going over the mechanics of them to elucidate the basic principles we will be using at all of the higher tiers. To form the first 999 numbers we actually need only 24 roots. These are:

one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thir, fif, teen, twenty, thirty, forty, fifty, sixty seventy,eighty, ninety, hundred

The following table includes all tier 0 roots:

Tier 0 Roots

The above table is what we can call a root table. It is different from a list of numbers, and instead provides us with the "components" to construct numbers. This will be our primary way of making sense of Bowers system, without having to list every single name that can be constructed within his system.

The basic principles of a root table is that it should include all the necessary combinations of letters (roots) that are needed to construct any number within a given range. Special rules can be included to deal with nuances. Roots that are part of a "tier group" can be broken up into ones, tens and hundreds. Under these columns roots can take any value from 1 to 9. A roots base value is dependent on where its located on the table. For example, if the root is in the tens column in row 5, then its value is 50. Normally there should be at least one unique root for every position on the table. These roots are usually of the additive type, and can be listed either in decreasing or increasing order. For english we list the roots in decreasing order.

One special feature of english, which breaks slightly from what is described above, is that we don't have unique names for 200, 300, 400, etc. as we do for 20,30,40 etc. Instead "hundred" acts as a multiplicative root, and must be combined with a ones root to its left to give it a definite value.

You may notice in the above table the use of the slash, " / ". This means that there is more than one root to chose from within a particular position. Which is used depends on the situation and the rules. The roots, listed from left to right, are referred to as the 1st option, 2nd option, etc.

The parentheses , " ( ) " , contain "optional" letters. These can either be dropped or added based on the situation and the rules. Sometimes commas may be used to separate letters within the parentheses. This simply means that there is more than one optional letter to be added.

The above table of roots can be used to construct any number from 1 to 999 by following these rules:

1. If hundreds & tens = 0 , use 1st option ones roots, with all optional letters included to represent the number.

2. If hundreds = 0 & tens = 1, use "ten" if ones = 0. If ones = 1 or 2 , use 2nd option ones root, without any tens root. Otherwise use 2nd option ones root with all optional letters removed followed by 2nd option tens root (teen).

3. If hundreds = 0 & tens > 1, use tens root followed by 1st option ones root with all optional letters included.

4. If hundreds > 0 & tens = 0, use 1st option ones root (optional letters included) equal to value of hundreds followed by root "hundred". Follow this by 1st option ones root equal to value of ones.

5. If hundreds > 0 & tens = 1, use 1st option ones root (optional letters included) equal to value of hundreds followed by root "hundred". Follow this with "ten" if ones = 0, with "eleven" if ones = 1, with "twelve" if ones = 2; otherwise follow with 2nd option ones roots with optional letters removed followed by "teen".

6. If hundreds > 0 & tens > 0, use 1st option ones root (optional letters included) equal to vlaue of hundreds followed by root "hundred". Follow with appropriate tens root and finish with 1st option ones root with optional letters included.

These rules should in theory be all that is necessary to explain the mechanical construction of naming numbers from 1 to 999. Note that this particular naming scheme makes no use of "and".

The following table shows the names of the numbers using the above scheme. You might want to compare it to the above rules to get a feel for how this works:

List of Tier 0 Numbers (1-999)

The largest number we can express at Tier 1 is not novemnonagintinongentillion. Instead it would be ...

nine hundred ninety nine novemnonagintinongentillion nine hundred ninety nine octononagintinongentillion nine hundred ninety nine septennonagintinongentillion

... ... ... ... ... ...

nine hundred ninety nine trillion nine hundred ninety nine billion nine hundred ninety nine million

nine hundred ninety nine thousand nine hundred ninety nine

In other words, its the number generated by filling all the groups we can generate with 999. The value of this number in scientific notation would be (1E3003)-1. The "name" for this number is so long that it would take somewhere between 1 to 2 hours just to say. No one could ever count to such a number as a person would most likely die before even reaching a billion. A trillion is utterly beyond human counting already! and it only gets a thousand times worst with every additional separator ... and there are a thousand separators!! Just for the heck of it, how long would it take some nigh immortal being to count to this number counting at normal human speed? It would take about 1.14*10^2999 years. Even black holes don't take anywhere near this long to completely evaporate. There is no human time frame I can readily compare this to.

Just understand that already at the end of tier 1 we already have more numbers than we could actually ever use, or have use for. Although numbers of this size and larger are used in the more theoretical areas of physics, by the time we get to numbers of this size exactitude down to the nearest integer is virtually never used. Instead approximate values are used. In this way there are numbers within the first tier range that have never been used and may never be used, even though each one of them individually could be written out in full. Yet we aren't even close to being done with counting in Bowers' Counting System! Let's now continue to Tier 2!

The 2nd Tier Intermediates

There aren't really many guidelines for how to continue after this. We now enter Bowers system and vastly unexplored territory, so we can't use Conway&Guy as a guide. I contacted Bowers to try to get some feed back. One the reasons he cites for not being interested in the intermediate terms is that they become unwieldingly long. Personally I think that's the most fascinating aspect of them (you'll see what I mean later). In any case he said that it might be worth mentioning some of the shorter ones. He mentions a centiattillion, and a decifemtillion.

This makes it fairly clear what Bowers probably has in mind, at least for the 2nd tier, where the index of the illion is itself an illion class number.

Recall earlier that Bowers' defined the thousandth illion as a millillion. He then decided to use the small scale SI prefixes to continue. Namely: micro, nano, pico, femto, atto, zepto, and yocto. Basically what Bowers' comment suggests is that we can use the canonical small scale SI prefixes as Class 2 separators.

Recall that Class 1 separators were used between "groups" equivalent to Tier 0 constructions. We can call these "Class 0 groups". The Class 0 groups are simply the english number names from 1 to 999. Now the Class 2 separators (small scale SI prefixes) will be used between "modifier groups" which are equivalent to Tier 1 constructions. Tier 1 constructions are the "latin numbers" we used to describe all of the Tier 1 separators. We can call these "Class 1 groups".

If the pattern is not readily apparent, an example is likely to help the idea along. After a novemnonagintinongentillion would be a millillion. "milli" here is acting just like "thousand" did before. To go to the next illion (the 1001st one), we simply place "one" to the right of milli. How this is done is not exactly spelled out, however I would suggest using milli-untillion. Follow this with milli-duotillion, and this with milli-tretillion. After this, we can simply add "milli" to all of the illions we've already constructed. Next would come milli-quadrillion, milli-quintillion, ... etc. all the way to milli-novemnonagintinongentillion.

To continue, we now place a Class 1 group to the left of milli. The 2000th illion would be duomillillion. Again we could append a Class 1 group between "milli" and "illion" to represent the "ones group". So we would continue with duomilli-untillion, duomilli-duotillion, duomilli-tretillion, duomilli-quadrillion, etc.

Beyond all this would be tremillillion, quattuormillillion, quinmillillion, etc. Basically we can describe the first 999,999 illions in the form:

G(1)-milli-G(0)-illion

where G(1) and G(0) are latin numbers (as described in Tier 1) from 1 to 999.

The value of this expression is then defined as H(G(1)*1000+G(0)) where G(1) and G(0) are interpreted as numeric values

Using all of the canonical small scale SI prefixes we can form names for the first 999,999,999,999,999,999,999,999,999 illions. They will generally be of the form:

G(8)-yocto-G(7)-zepto-G(6)-atto-G(5)-femto-G(4)-atto-G(3)-nano-G(2)-

micro-G(1)-milli-G(0)-illion

Just as before we can drop group/separator combos when the group = 0, and G(0) can be dropped when G(0) = 0. When any Class 2 separator is not followed by a group, but by illion, its last vowel can be dropped, as is the case with millillion, micrillion, nanillion, etc.

It's important to note that any group G(k) where k > 0 can never use the 1st option ones roots. The only time these get used is in the names million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, and nonillion. This applies only to G(0). For any other group, rule 2 must be changed so that the 2nd option ones roots are used, Thus we would have duomillillion, tremillillion, quattuormillillion, quinmillillion, sexmillillion, septenmillion, octomillillion, and novemmillillion. Also note that for any other group other than G(0), when G(k) = 1, a blank is used. For example, the 1,001,000th illion would be micro-millillion. G(0) also acts differently when there are Class 2 separators involved. In this case when G(0) = 1,2,3, we must use "unt", "duot", "tret" respectively instead of "m", "b", and "tr". If G(0) > 3 however, all of the normal rules for Tier 1 apply.

As far as I can tell the above describes all the rules we will ever need for Class 1 groups from this point onwards. For clarity here is a table with some examples, so you get a better idea of how the above would work:

List of Bowers' Style Low Tier 2 illions

The last entry is the largest illion we can name with the official large scale SI prefixes. You can also bet that this name is also so long that no human being will ever say the whole thing. Roughly speaking you need something like an 100 quintillion years to say it. This says nothing of the largest number the we can name so far. To say that number would take 10^(10^19) years to say. To count to it would take something like 10^(10^(10^19)) years! This numbers are not in the least humanly obtainable, and yet the system we are constructed was constructed by us. We can devise a system which can generate more than a googolplex names for numbers, without actually being able to ever construct them all!

To go further still Bowers continues the large scale SI prefixes, with some prefixes of his own devising. He decides to adapt his polytope suffixes to use as prefixes for his illions. Let's look into how these prefixes work.

As stated at the beginning of this article, Bowers is best known for his work with polytopes. Polytopes are multi-dimensional analogies of polygons. For our purposes here we can define a n-dimensional polytope as an mathematical object created by connecting polytopes of one less dimension together to enclose an n-dimensional space. The polytopes of one less dimension used to form an n-dimensional polytope are known as the polytopes "cells". "poly" means many, and a Polytope is essentially an object formed from many cells. A polytope is a generic term for objects of any dimension. However, in ordinary mathematics we also have names for polytopes of a given dimension.

For example polygons , which exist in 2-dimensional space, are made up of 1-dimensional line segments which enclose an area. A polygon can be thought of as a 2-dimensional polytope. You may also have heard of polyhedrons. A polyhedron is a 3-dimensional polytope formed by joining 2-dimensional polygons at their edges to enclose a 3-dimensional space. The name "polyhedron" actually means "many faces", where the "faces" here refer to the cells. The terms polygon and polyhedron have been around for a long time and are officially recognized by the mathematical community. However there is no officially recognized singular name for 4-dimensional polytopes. Professional mathematicians are not particularly hung up on names. They are sometimes referred to in mathematical literature as 4-polytopes. Some names suggested for the 4-dimensional polytopes are polyhedroids or polycells. Norman Johnson, George Olshevsky, and Jonathan Bowers have advocated the term polychoron. "Choron" here refers to "space" in greek. "Polychoron" therefore translates as "many spaces". This makes sense since each cell of a polychoron is a polyhedron enclosing a 3-dimensional "space". The name also follows a similar naming scheme from polygon and polyhedron. I personally like the term polychoron best of the options proposed. The term is not widely used outside the circle of "polychorists" as far as I know. Although both Jonathan Bowers and George Olshevsky are amateur mathematicians, it is significant to note that Norman Johnson is a professional mathematician and has advocated the use of the term polychoron. Much like the names for very large numbers, naming of the polytopes and there various dimensional categories is esoteric subject matter. For this reason we can expect most of these naming conventions to never be officially recognized. However I might make this suggestion: just because certain esoteric terms may never make it into some kind of universal dictionary, its clear that with esoteric subjects there are only a handful of specialists even interested in such matters. Therefore the names and conventions that such specialists can agree upon should be considered canonical within this limited scope. With that in mind we can say that "polychoron" is the official term amongst those who study them most fervently.

What does any of this have to do with large numbers? Well the roots, -gon, -hedron, and -choron, form a series of suffixes for the polytopes of various dimensions. Now Bowers has studied polytopes beyond the polychorons. He is responsible for providing a great number of short names for all sorts of polytopes. His naming conventions can be thought of as an evolution from the standard way to name polytopes. Usually a greek prefix is attached to the dimensional suffix to form a polytope name. For example a "pentagon" is a 5-sided polygon. A "tetrahedron" is a 4-celled polyhedron. Bowers uses a similar naming convention with polychorons, where a polychoron is an "n-choron". Since Bowers studies polytopes beyond polychorons he naturally needed higher dimensional suffixes. Originally he used the term polytetron for 5-dimensional polytopes. For technical reasons, the using of latin numbers for the dimensional suffix leads to some ambiguity for the names since it clashes with the use of latin prefixes. As Bowers tells it, Wendy Keiger came up with a solution to this dilemma. She came up with the term polyteron from "tera" the large scale prefix for 1E12. We can assume polyteron to mean "many terons" where a teron is merely another term for a polychoron. Wendy Keiger also came up with the following continuation:

polypeton for 6-dimensional polytopes

polyecton for 7-dimensional polytopes

polyzetton for 8-dimensional polytopes

and

polyyotton for 9-dimensional polytopes

Bowers then took Wendy's suggestion and went on to extend this up to tredecillionth-dimensional polytopes! Yes you read that correctly. He creates something in the way of an extended system for polytope suffixes. Eventually he decided to adapt these suffixes into prefixes for his illion series, discarding his old "english" roots for new non-standard polytope roots. It seems advisable to avoid using "english" roots in an illion series for one simple consideration. If the Tier 0 roots are being named in english it is likely to cause ambiguity if they are also used at any higher Tier.

Since the polytope suffixes up to polyyottons are simply based off the large scale SI prefixes it means that they needn't replace the use of large prefixes that we used in the construction of the last table. To see what new large scale prefixes we can devise we must refer to Bowers original list of polytope names. The original list can be found at:

http://www.polytope.net/hedrondude/topes.htm

For convenience here is a similar table. It shows how we can extract prefixes for the third plateau from his polytope names:

This is all we have to go on. None the less, even from this small list of examples a rule can be obtained. We simply drop the last vowel in the Class 3 Group, and follow it with the Class 4 Separator with the leading consonant removed. This practice may even eliminate potential ambiguities, because it strongly implies that the group and its following separator are linked. One ambiguity that could result from my suggestion would be something like meji-daka-kalillion. Is this the 1,000,010th Class 3 Separator followed by the 1000th? or is it the 1,010,000th Class 3 Separator. This could be avoided by altering "daka" as "daki" to imply it is part of a continuing sequence, but Bowers' system works too.

As far as I can tell this is it. We now have all the rules to complete Bowers' system. If ambiguities should be present, we can change "a"s to "i"s in the Tier 3 roots to imply continuation. We can now construct the Tier 4 root table. This table will not be completely filled out, and we will need to set up a special set of rules for how they work, since they don't follow the usual pattern. None the less here it is:

Bowers' Tier 4 Roots

The rule here is unusual but simple. If the value is 1~9 just use the 1st option ones root as always. If the value = 10, just use "gloc". For 11~14, don't include a tens root, and simply use the 2nd option ones roots. That's all there is to it.

So we are now ready to explore the 4th and final Tier. One note of interest before I begin another massive list of examples. The "multillion" is NOT the largest number we can now name, nor is it the largest illion we can form in Bowers' system. We can now actually surpass it. However, Bowers' system doesn't extend all the way to infinity. We are fast approaching a kind of dead end, as you will see shortly. With that in mind, I present the final list of examples before we wrap this up:

We are now nearly at the end of our journey. Of coarse the last entry above is not the largest number we can name in Bowers' system; It merely represents the largest Class 3 Separator. If we form the largest Class 2 separator, and thereby the largest illion, we obtain the number 1E(3E(3E(3E45))). If we form the largest Class 1 Separator, and thereby the largest number, we obtain 1E(3E(3E(3E45))+3)-1.

There are no larger numbers we can name in Bowers' system past 1E(3E(3E(3E45))+3)-1. To continue we would need a Class 4 Separator past the "multi". I mentioned this to bowers and suggested either using "omni" for omniverse to form "omnillion", or "cosmosillion" from "cosmos". Writing back he said that we preferred the use of "cosmillion". What then? We would need yet another Class 4 Separator to continue. We could have course fill out the Tier 4 Table, so that we could have the maximum of 999 Class 4 Separators. Yet the largest number expressible would still be finite, or put more poignantly, there would still be numbers without name! That's right Bowers' system can not name all numbers. Even if we continued to the 5th, 6th, 7th, 8th, 9th, 10th Tier, still there would be the nameless hordes beyond. Is counting, in the absolute sense, impossible then?! Will there always be numbers we can not name, and therefore can not count! That is, is it impossible to give unique names to all the counting numbers?! The surprising answer is, ... no, it is not impossible. We could for example name every number by reading out its digits. The problem is that even smallish large numbers would be difficult to say and understand. For example a 1,000,000 would be "one zero zero zero zero zero zero". One would have to count to write it, say it, read it, and hear it. Not an ideal form of communication by any means. Yet every number would have a name.

The problem therefore is NOT giving every number a name, but rather giving it a name that is comprehensible. What is meant by "comprehensible"? Well one definition might be that the name is short enough to be able to say in a reasonably short amount of time, say 10 seconds or less. The problem is that there are only a finite number of combination's of sounds that can fit within 10 seconds. Inevitably, no matter how we choose to name ALL numbers, there will only be a finite number of them that can fit within some reasonable temporal or spatial interval, and an infinite number that can not. The solution that english is pointedly trying to achieve is not to give every number a short name, but to provide certain milestone numbers with short names. Bowers' system is simply an evolved and advanced form of the same impulse. English allows us to give a tremendous number like 1E63 a relatively short name such as "vigintillion". The catch is that names within its vicinity will be very long. Note however, that after awhile even the use of illions becomes long for certain names such as novemnonagintinongentimicro-novemnonagintinongenti-milli-novemnonagintinongentillion. When this happens there is yet a second level of name shortening, then a third, a fourth, and so on. Yet in these types of systems we don't have the same open endedness as other approaches. Is it possible to devise a system which names every number, and for every "short name" there is always, eventually some very very large number later on that must have an equally short name. This too must be impossible, because there are only a finite number of such short names. The best we can do is delay the inevitable. However at some point, in any open ended naming scheme, there must be an infinite horde of unpronounable seemingly endless named numbers, without a single short name among them!

Does that seem mind boggling? Well here is an even more mind boggling truth, ... we can describe some of these numbers, but we can't do it with language alone. As long as we attempt to "name" EVERY number we are actually holding ourselves back. Once we move beyond into the nameless horde there is only one proper way to continue ... mathematics. This is the ticket required on the train ride to the infinite. Even in our attempts to avoid anything technical on our way to the very very large, mathematical logic was sneaking in. The way that the names are constructed is very regular and when seen properly is really a recursive process. It is recursion that will let us continue beyond the named and into the nameless. In fact we can already think about these numbers if we let go of our attempts to provide roots and names. Imagine a number that would require a 100 Tier root table, a 1000 Tier root table, a million, a billion, ... I'll let you imagine where that thinking leads :)

That very thought is recursive in nature. Note that there is nothing so difficult or strange about it. Anyone is capable of recursive thinking. You don't need to be a mathematical wizard to understand it, I believe it is an innate form of thinking.

We have reached the limits of our amateur excursion into the very large. To progress further we will need to explore the mathematical implications of what we have begun. What we have learned about exponents, scientific notation, and extending the illion series has a definite mathematical consequence. The results of this will lead to a very surprising revelation: numbers much larger than you might ever have imagined possible!

In the next chapter we will explore recursion for the first time, and begin generating numbers so large that everything in the first two sections will pale in comparison.

If your prepared to take this game to the next level, check out chapter 3-1.

More content for Chapter 2-4 and 3-1 coming soon: Winter 2011 ! Stay tuned ...

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Some links of Interest. These are links to various pages on Bowers old website. They are listed here for archival purposes:

http://web.archive.org/web/20070303130639/members.aol.com/hedrondude/scrapers.html : Link to page 1 of infinity scrapers on Jonathan Bowers old website.

http://web.archive.org/web/20070303131827/members.aol.com/hedrondude/scrapers2.html : Link to page 2 of infinity scrapers on Jonathan Bowers old website.

http://web.archive.org/web/20071205035836/members.aol.com/hedrondude/numbers.html : Link to Jonathan Bowers -illions on his old website

http://web.archive.org/web/20071206101451/members.aol.com/hedrondude/array.html : Link to Jonathan Bowers old website where he explains his array notation for the first time.