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Introduction
We have finally gotten to section 3, where we examine in depth, for the first time in this site, the work of Jonathan Bowers, and other lesser known large number notations. Jonathan Bowers' work in large numbers is something of a founding work for the spirit of modern googology, and it has a cult-like popularity, so we will be focusing very much on covering his work in this website. In this article we will learn about the work of Jonathan Bowers, and why it's important to googology, as a prerequisite to examining his famous large number notation, Bowers' Exploding Array Function (BEAF for short).
Who is Jonathan Bowers?
Jonathan Bowers is an American amateur mathematician who primarily studies polytopes (shapes in higher dimensions) and large numbers, and he also has an affinity for creative pursuits. He publishes all of his work on his personal website, which can be found here - his old website, which has since disappeared from the Internet, is archived here on the web archive. Although his work is quite popular, not much is known about the man himself, besides some basic biographical details: he was born November 27, 1969[1], he lives in Tyler, Texas[1], he has two brothers named Jeff and Justin[2] (the former of whom has an interest in making art out of wood[3]), he has a Master's degree in mathematics[4], and he appears to be a Christian[5]. A picture of him is shown to the right. Besides those biographical details, we don't really know much about Bowers himself, which is a bit strange considering his work's popularity.
We will mostly be focusing on Bowers' googological work, his work in large numbers, but he made a variety of other works that are worth noting. The field he focuses on the most is shapes in higher dimensions beyond the familiar three, mainly polychora (shapes in fourth dimensions) and polytera (in 5 dimensions), but also in dimensions higher and lower than that.
He explains on this page that he began researching polychora in 1990, when he was only 21 years old and in college. He documented all the polychora he found in a blue notebook he carried every day to college, which he often showed to others, and within 3 years he had documented over 1000 polychora. By 2002 he had found a total 8190 uniform polychora (polychora where all vertices are congruent and all their cells are uniform polyhedra), and gave many of them short names such as "Spript" and "Garsixhi", which are today known as Bowers style acronyms. He published a catalogue of these polychora on his website, with a lot of visualizations of the higher dimensional figures. Over the years, Bowers has continued to expand upon his catalogue of polychora and other polytopes (shapes in any dimension), and around 2008 he slightly modified the requirements for a polychoron to be uniform, changing the count down to 1865 and later to 1869. To this day Bowers is a pretty well-known figure among the polytopists, the community of amateur mathematicians who study polytopes.
Bowers's second biggest area of study is large numbers, and his reputation among the googologists (the people who study large numbers) is similar to that of his among the polytopists. In fact, many googologists (including myself) consider Jonathan Bowers to be the father of modern googology. Bowers invented a simple and aesthetically pleasing but powerful large number notation he called "array notation" and further extended upon it in a way nobody else really did before. He also named a variety of large numbers with his notation, and his work in large numbers gradually gained a cult-like popularity. We'll learn more about this history of his googology later in this article.
Jonathan Bowers was the first to create large numbers in the way googologists do today, not just creating large numbers just to show how easy it is to define really big numbers, but to seriously try to define and make names for the biggest numbers you can! Googologists soon followed in spirit of Bowers, and the googology community gradually formed. Before Bowers' googology there wasn't much information on the Internet relevant to large numbers, mostly short pages written by math professors about the "classic" large numbers - an exception is Robert Munafo's large number site, created 1997 (5 years before Bowers published his notation). Now there is very much information on very large numbers on the Internet, such as Sbiis Saibian's site, Googology Wiki, my own large number site, Peter Hurford's blog posts, Adam Goucher's blog posts, Chris Bird's large number papers, and several other sites created by members of the googology community - I have compiled a list of anything on the Internet I find to be relevant to large numbers here. All this was caused by the "large number revolution" Bowers started.
Jonathan Bowers' other work is mainly artistic work, visual or otherwise. He has written and drawn some cartoons in a series titled "Another Reality" (one comic of his is shown to the right). He has also written a story titled "Welcome to the Fourth Dimension" about a man who stumbles into a 4-dimensional world, and a rather creepy story relating to large numbers titled "Forever Endeavor" about an ordinary man who takes an opportunity to earn a million dollars but it ends up being an incredibly bad idea, as he is forced to do a seemingly eternal task, each part unimaginably longer than the previous, without a means of escape. His graphics of polytopes also fall under the category of "Bowers' artistic work".
So now that we have a perspective on who exactly Jonathan Bowers is, let's go into more detail about the history of his googology.
Jonathan Bowers' Googology
It all started around 1987 when Bowers read a book about the hyper-operators (he doesn't remember the name) - this is around the time Bowers became interested in polytopes. What Bowers did next was something nobody else had done before in the same way he did. He discovered a natural way to extend upon the hyper-operators, and developed a notation for that. He later found out that although the notation he created used only four arguments, it was already able to keep up with Conway chain arrows, a notation that takes an arbitrary number of arguments. He then generalized his four-argument function to an aesthetically pleasing notation he called array notation (also called linear array notation) that can take any number of arguments, and easily transcends any of the popular large number notations.
Then Bowers took his notation a step further by combining his large numbers with his study of the higher dimensions - he extended his notation to allow arrays in the higher dimensions. That extension is known as extended array notation or dimensional array notation. Bowers further extended upon his notation, and along with those extensions, Bowers came up with whimsical names for numbers defined in his notation, like gaggol, tetratri, iteral, xappol, gongulus, trilatri (visualization by Bowers to the right), tridecatrix, and the largest googolisms of his at the time, the guapamonga and guapamongaplex.
Slowly Bowers' work in large numbers began to gain popularity and recognition, for its appeal and whimsicality that was very unlike anything anybody else has done. People would often hail Bowers' googolisms as the largest numbers in the world, so when someone mentions a popular large number like Graham's number on the Internet, someone else will likely jump in and mention Bowers' named numbers. While people recognized a merit in Bowers' googology, people also started to note imperfections in his work, as much of his googology is rather vaguely described. Take this webpage for example. It is an incomplete review of Bowers' array notation and his named numbers, noting both its strengths and its weaknesses. One thing the author of that page notes repeatedly is that Bowers' notation is "by no means practical", since it surpasses Graham's number - which the author (erroneously!) claimed to be the largest number to have any mathematical use - very early on.
But nonetheless, with Bowers' work the googology community as we know it today gradually started to form. People would start to invent large number notations and names similar to those of Bowers'. A particularly notable example is a notation devised by Chris Bird as a sibling to Bowers' array notation, called Bird's array notation. Bird's notation starts off almost identical to Bowers' notation, but with a different base rule (which Bowers himself later adopted). However, Bird's notation later diverges from Bowers' notation in the way it works, and unlike Bowers' notation, Bird's notation has a precisely made formal definition, without ambiguities like Bowers' notation has. On the other hand, Bird's array notation doesn't have any googolisms defined with it or quite the same aesthetic appeal as Bowers', so Bird's notation is much less known than Bowers'. We'll examine Bird's array notation starting later in section 3.
More recently (around 2007), Bowers revised his array notation, with a better, clearer definition, and some aspects borrowed from Bird's notation. He further extended upon his notation as well, and named this generalization BEAF (Bowers' Exploding Array Function). With that, his notation continued to gain popularity, and his new largest googolism, the infamous meameamealokkapoowa oompa, became commonly hailed as the largest number in the world (although there are in fact larger googolisms existing, such as Rayo's number and the current record holder, BIG FOOT).
When the googology community started to form around 2008 (when Googology Wiki was created), BEAF became honored as a sort of "flagship notation" in googology. However, members of the googology community also also began to notice issues with the notation, debating about how powerful it is and its ambiguities. Some googologists invented their own variant array notations with a better definition.
Eventually, a large portion of the members decided to give up on trying to figure out how the higher parts of Bowers' notation work because of its ambiguities, and many refused to recognize his higher googolisms (such as his largest, meameamealokkapoowa oompa) as numbers at all. There are some exceptions to this, most notably Sbiis Saibian, who is a self-described "fan of Bowers' work", but many googologists still have lost interest in trying to figure out the mysteries of Bowers' notation.
So why bother covering the work of Jonathan Bowers? Because, as I said, Jonathan Bowers is clearly a founding father in googology, so his work is most definitely worth examining in detail. I think that all googologists can agree that Jonathan Bowers' work (at the VERY least the lower parts of his notation) is worth covering in a site devoted to googology. In the next article we'll learn about linear array notation, the simplest subset of Bowers' notation.
Sources
[1] http://polytope.net/hedrondude/home.htm
[2] http://polytope.net/hedrondude/graphics.htm
[3] http://polytope.net/hedrondude/knotboxes.htm
[4] http://web.archive.org/web/20040628144501/http://members.aol.com/hedrondude/home.html
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