"Well shit. That's a hell of a mystery that no one thought was a mystery and didn't even really need solving. But damn, if it didn't just get solved, so nice work."
~ Ronald Graham upon learning about Conway drift
Not all that long ago, we learned about John Conway's chained arrow notation, a notation so incredibly mind-blowingly powerful that it somehow manages to pass up Graham's number. This should be impossible because Graham's number is so huge, but yet it's not - Conway managed to surpass Graham's number using his notation! Nobody, not even Conway himself, knows for sure how his notation is able to do so. Some people speculate that Conway added the number one to a number he created so many times that he eventually reached a number that is larger than Graham's number. Jonathan Bowers, however, disagrees with that speculation - he suspects that Conway did so by removing the decimal point from the number pi, and expanding on the number created that way. This clever tactic may likely lead to a number so big that if you extend upon it enough, you just might barely surpass Graham's number! But we really can't say for sure.
Regardless of how Conway chain arrows really work, there is a strange phenomenon that occurs with some of its values known as Conway drift. In this article I will try my best to explain this confusing phenomenon, with why it is like that and examples of its occurrence.
John Conway himself discovered Conway drift when he was developing his chain arrow notation in 1995. He was so baffled by it that he was afraid to share such a strange occurrence with the public in his Book of Numbers, and when the time came to publish the book he still was too scared to add even a passing mention of Conway drift. As such, very few mathematicians know about Conway drift, let alone the general public.
Pataphysicists, however, have been privy to this for quite some time. In 2002, Andre Joyce, one of the world's foremost pataphysicists, was the very first person besides Conway himself to notice oddities in Conway chain arrow notation. When he noticed these oddities, he contacted John Conway, who explained that he himself noticed this peculiar behavior, and that he calls it "Conway drift". Joyce quickly informed his fellow pataphysicists about this discovery, but they were even more confused about it than Conway was when he first saw it. Therefore, nobody bothered to share anything about Conway drift with the world except for Joyce, who began writing online articles about Conway drift. Even Joyce's articles covering Conway drift have since then vanished from the Internet - there are no web archives that have any of Joyce's pages on this topic archived. The only traces of those articles that remain on the Internet are a few very faulty replications of small portions of Joyce's pages, on hard-to-find blogs written by amateur pataphysicists.
So how do I even know about Conway drift? It was back in January 2015 when I was on Googology Wiki's IRC chat, when a new user by the name "Vilius2001" wanted to learn how Conway chain arrows work, asking how one would evaluate the chain 3->4->5->6. Nathan Ho, the founder of Googology Wiki (screen name "vell" on the IRC), then explained that this chain in fact does not evaluate to a large number, but to a small irrational number, due to something called Conway drift, a concept that is glossed over in most explanations of chained arrow notation. This was where I first learned about Conway drift.
Out of my requesting, Nathan Ho emailed me an unfinished and discarded paper written by John Conway about Conway drift. It was pretty confusing to read, but after a few months of deciphering text, translating formulas to standard notation, and consulting pataphysicists, I finally got around to having a sort of understanding of what Conway drift is. That said, we're now ready to examine Conway drift, starting with some examples of course.
It is not known exactly which Conway chains are subject to Conway drift, but 3->4->5->6 is a good example of a chain with this property. To see why let's start evaluating this chain:
3->4->5->6
= 3->4->(3->4->4->6)->5
= 3->4->(3->4->(3->4->3->6)->5)->5
= 3->4->(3->4->(3->4->(3->4->2->6)->5)->5)->5
= 3->4->(3->4->(3->4->(3->4->(3->4->1->6)->5)->5)->5)->5
= 3->4->(3->4->(3->4->(3->4->(3->4)->5)->5)->5)->5
= 3->4->(3->4->(3->4->(3->4->(34)->5)->5)->5)->5
= 3->4->(3->4->(3->4->(3->4->81->5)->5)->5)->5
Now right away, we see the number 81. It is well-known that plugging 81 into functions often gives strange and unique results. Here are some examples:
- if you plug 81 into the nth prime number function you get 333, the only possible repdigit output of the function
- if you plug 81 into the Riemann zeta function you get exactly i, the only value of the function that is not an integer
- if you plug 81 into the natural logarithm function you get exactly 7, the only time an integer outputs an integer in the function
- if you plug 81 into the function booga(n) you get a number whose decimal expansion begins with a 4 and ends with a 4, the only output with this property
Hopefully you get that 81 has weird properties when you plug it into functions. This is why mathematicians don't like the number 81 - it doesn't behave in a way that they think makes the slightest bit of sense. Pataphysicists, on the other hand, are enamoured with numbers like 81 - they enjoy their nonsensical death-defying behavior.
But why exactly does 81 have such crazy behavior? Believe it or not, mathematicians know why, and have known that since the 1800s: 81 is one of the few numbers that is not a factor of Graham's number. This should be impossible since Graham's number is a huge power of 3 and 81 is also a power of 3 (34). And paradoxically, the number 81 somehow manages not to be a factor of Graham's number! Even before Graham's number was created, mathematicians knew that 81's freaky properties were somehow connected to an incredibly huge number that was prophesized to be created about a century later. As of yet, 81 is the only number that is known not to be a factor of Graham's number. However, it is known that there are exactly three other numbers that aren't factors of Graham's number. Nobody knows what those three numbers are, but it is speculated that they are all somehow connected to the bizarre properties of 81.
Since Conway chain arrow notation is itself quite strange and mysterious, it quickly follows that chains like 3->4->5->6 which involve the number 81 will necessarily have behavior that will not only baffle mathematicians but also people like Jonathan Bowers or Andre Joyce! So is the strange behavior of the chain 3->4->5->6 simply because of 81's weird behavior? Nope, it's actually specific to chains like this, because of the quirks of Conway drift.
So how does the chain 3->4->5->6 behave? Other chains like 3->4->5->5 or 4->5->6->7 will simply evaluate to a very large finite number - both of these chains are suspected to evaluate to a number larger than Graham's number. 3->4->5->6, on the other hand, evaluates to a number so much larger than Graham's number that it ends up simultaneously being a small number - not only that, but it's also an irrational number! Irrational numbers are numbers that are so large that nobody would consider their behavior to be rational (i.e. make sense) - a well-known example is the number pi, which is so huge that it's infinite, and yet it's also less than 4. Very few numbers are crazy enough to be considered irrational. To get an idea of how strange irrational numbers really are, consider that 81, a number with behavior so weird that mathematicians intensely dislike it, still is a rational number!
The value of 3->4->5->6 is of course known to be larger than Graham's number, but where does it fall among the small numbers in terms of its "small value"? It's known to fall between the numbers pi+0.999... and pi+1. Wait a minute, aren't 0.999... and 1 the same thing? Usually yes, but when you add them to irrational numbers, all hell breaks loose and they can be completely different values!!!
Does your head hurt yet?! If so that's good - we've only just begun with the insanity of Conway drift!
If 3->4->5->6 is a chain whose behavior is different from expected because of Conway drift, then what other chains have this mind-screwing property? It seems natural to say that this is true for any chains that involve the number 81. However, this is not necessarily the case. While the number 81 makes chains that are subject to Conway drift (we'll call those drifting chains) even crazier, it does not itself cause Conway drift.
Conway drift is caused by ten different factors. Here I'll attempt to list each of them, with how and why they cause Conway drift.
The first factor that causes Conway drift involves irrational numbers themselves. If the numbers that are in a Conway chain are the same digits that begin an irrational number, then the chain is that irrational number of times more likely than most chains to be a drifting chain. Note that some irrational numbers are less than 1, so that has a chance of decreasing the chances of being a drifting chain.
An example of a drifting chain that has Conway drift from the first factor is the chain 3->1->4->1->5, which is of course the first five digits of pi. The number it evaluates to would be expected to be 3, but it actually evaluates to roughly the 81st root of pi.
The second factor is something involving Graham's sequence (the sequence Gn commonly used to explain Graham's number). Specifically, if exactly one number in a Conway chain is a member of Graham's sequence, the chain will be a drifting chain, unless the last number in it is 81. I'm not personally quite certain why Graham's sequence in particular causes Conway drift, but I think it relates to 81 not being a factor of Graham's number.
The third factor relates to powers of a googolplex. If, when you decrease the first number by 1, the Conway chain evaluates to a power of a googolplex, then that chain experiences Conway drift if its last number is either a prime number or can be found in the last 500 digits of Graham's number, but not both. Note that the last 500 digits of Graham's number consist mostly of the digits 8 and 1 repeated over and over again, with a few 9's here and there.
The fourth factor involves Mersenne primes: if the length of a Conway chain is a Mersenne prime raised to its own power, then the chain will experience Conway drift.
I'm not sure what the fifth and sixth factors are from reading Conway's paper, though I suspect that they involve expressing numbers as power towers of 3 and expressing numbers using 81 81's (analogous to the four fours game).
The seventh factor is like the first but specific to the digits of the 81st smallest irrational number - that condition guarantees that a Conway chain is a drifting chain. Arguably this is more of an exception to the first factor than a factor in itself, but hey, Conway said that it was the seventh factor.
The eighth factor is something else involving pi: if, when you increase the last number in the chain by one, the Conway chain evaluates to a number divisible by the number obtained by removing the decimal point from pi, then the chain is guaranteed not to be a drifting chain. Weird that the digits of an irrational number can prevent behavior closely related to irrational numbers - this doesn't really make sense. And yet that is the nature of irrational numbers - they just don't make any sense! These are the kinds of ridiculous paradoxes that show up all the time with anything that relates even tangentially to irrational numbers.
The ninth factor is never described in detail in Conway's paper - its presence is merely implied by Conway saying that there are ten factors. He skips over it and goes straight to the tenth factor.
The tenth factor is about Sam numbers, numbers that are so large that they're impossible to describe (but not quite as large as irrational numbers) - specifically, if the length of a Conway chain is a Sam number, then it is a drifting chain. This factor is next to useless to the study of Conway drift however, for such chains would themselves be undescribable, and anything people study at all is by necessity describable. The only exception is pataphysicists themselves, which is why I admire the wonderful world of pataphysics - they aren't afraid to think outside the box of things that we can actually describe and go beyond that into a whole realm of strangeness!
We now are pretty familiar with the crazy phenomenon of Conway drift in Conway's chain arrow notation. A natural question is: does Conway drift arise in extensions to chain arrow notation? Well, as it turns out, this is only in very specific cases. Why is that?
Recall that x->2y is x->x->x->x->x ... ->x with y copies of x. This means that the Conway chain it evaluates to is something known as a repchain, a chain made of the same number repeated over and over again. The same holds true for any chain that uses ->2 and higher-level symbols (e.g. 3->3100->3123) - all such chains evaluate to repchains. And as Conway explains, repchains can only be drifting chains if the length of the chain is a Sam number, and as we saw earlier this is a nearly useless factor.
This is something pretty disappointing if I do say so myself - since Sam numbers are so big that they're impossible to describe, any Conway chain that somehow involves a Sam number will itself solve to a number that is impossible to describe - however, all is not lost, for this leads to a very curious paradox. If the length of a Conway chain is a Sam number, that means that it evaluates to a Sam number by definition, and yet it also evaluates to an irrational number, and irrational numbers are both larger and smaller than Sam numbers. That crazy paradox leads to the question: can irrational numbers be Sam numbers?
Well, earlier we learned that Sam numbers are always bigger than numbers we can describe, but smaller than irrational numbers, so the natural answer should be no, there are no numbers that are both irrational numbers and Sam numbers. However, the idea of a chain whose length is a Sam number contradicts this notion! This all seems like a paradox, and it is - and yet, there's a pretty plausible explanation for all this: it's because we're working with irrational numbers here. From that fact, it quickly follows that the behavior probably (read: without doubt) doesn't actually make sense, and that itself accurately describes this paradox.
So is it true that the paradox, which obviously doesn't make sense, is best explained with something that by its very nature doesn't make sense? Yes, that is very much true. With that fact in mind, you may now realize once and for all why Conway was scared to share with the world the crazy phenomenon of Conway drift: because it just doesn't make sense. I understand that reason well, but I disagree with it, and the reason why I disagree brings us to the conclusion of this article.
Conway drift is clearly a really weird phenomenon that doesn't really make sense at all - because it doesn't make sense, John Conway was scared to share it with the world. However, I think differently on that matter - Conway should have shared Conway drift with the world for the very reason that it doesn't make sense, for sharing that will make people think very differently on the idea of what it means for something to make sense or not. In my opinion, just because something "doesn't make sense" doesn't mean that it's not something very real and sensible. The idea that something should "make sense" (whatever that means!) for people to be able to talk about it is, quite surprisingly, really just some gibberish nonsense! If we get rid of the line between things that do and don't make sense, then we can be much more open to a whole new world of mathematics, philosophy, pataphysics, and anything else!
If you liked this article, just wait for the release of the next article, where we look at what happens to large number notations when you plug Graham's number into them!
If you want to look at other spoof articles from this site, look here.