fastestfunctionperiod

4.4.1

ULTIMATE BEAF

and it's applications to rotating buttered cats, easter bunny breeding velocities, the singularity and the second coming ...

Introduction

I have been hinting about a variation of array notation that I have been working on for some time now in the IRC. Today I reveal this very powerful variation of BEAF. I call it ULTIMATE BEAF. What is ULTIMATE BEAF? It's simply the ULTIMATE VERSION of BEAF, the most powerful version possible. In fact it is the most powerful computable function imaginable. The key to this variation is to extend ordinary BEAF to all complex numbers. Yes you read that right, this is BEAF on crack! Complex-valued BEAF is a function which grows inconceivably faster than even ordinary BEAF. This is because of the weird way that complex numbers work. Turns out that complex numbers spin, rotate, do cartwheels, and can do even more complex motions as well. Complex BEAF literally spins out of control towards complex-infinity!

It is a basic result of complex arithmetic that complex exponentiation spirals around zero on the complex plane, going further and further away from this center point. We can extend this to complex tetration, pentation, hexation, etc. all the way up through all BEAF spaces and structures. The result is rotations that actually accelerate out of control. In fact we quickly find that there aren't enough dimensions in Hilbert space to contain all the axes of rotation that occur, and the outputs of complex BEAF quickly become quaternions, octernions, hyper-complex numbers, and in fact pass through every kind of number imaginable and unimaginable, before we even reach complex-legion-arrays. Just to show you how insane this world is here is an image of complex rotating 17-dimensional complex tetration:

graph of (2.718281828...+i)^^(17i+n)

but it get's much worse than having complex arguments. We can also have imaginary numbers of arrows:

i^^^^^^^^^^...^^^^^^^^^^^^i w/i*100 ^s

By taking the absolute value of such functions we can get the fastest growing computable functions possible. Just how fast is complex tetration? It already can keep up with ordinary legion space, and it may even grow faster than meameamealokkapoowa oompa. The technical details are overwhelming. So we can have things like:

{3847+i,38*i+48,10+2i,16i,444i}

Complex BEAF is so complex that it can solve all googological problems and describe all googological numbers, functions, notations, systems, geometries, ordinals, whatever. It turns out that ALL GOOGOLOGY was Complex-valued BEAF all along. It is the ULTIMATE universal notation. Here are just a few examples of what Complex BEAF can describe:

n! = {i+n/e,n^2}

f_17(n) = {i+n,3+i*2n,15}

f_e0(n) = {i&n}

E100#100#100#(2n) = {i^3+n^3,2n,3}

R-function = $(R)&(1+i)

...

The behavior is so complex that it can describe any function at all. But doesn't complex-BEAF produce complex numbers? Of course it does. It produces hyper-complex numbers, and quaternions, octernions, sedenions, surreal numbers, and every number ever. But since it can produce every kind of number it also produces reals, fractions, integers, and yes even whole numbers! It is the ultimate universal function! Because of that it describes EVERY RATE OF GROWTH! So it is therefore the fastest growing function possible.

In fact, if we use the correct complex values, we can get very fast functions even with very modest arguments. For example:

{i^sqrt(e^n),i^sqrt(e^n),i^sqrt(e^n)}

Is already a function which grows faster than anything in conventional BEAF. It's growth rate is way beyond psi(W_BEAF). And it get's worse:

{i^sqrt(e^n),i^sqrt(e^n),i^sqrt(e^n),i^sqrt(e^n)}

But BEAF allows for more than complex arguments. Even the number of arguments can be complex as in a:

(17+32i) & 3's

I can't show you what is array looks like ... only cthulhu could look upon this array without going mad! But it is some very powerful stuff. Complex numbers of arguments is SO POWERFUL than FGH has to use uncomputable ordinals just to keep up.

But this still pales in comparison to a meameamealokkapoowa oompa*i array of i's. These kinds of numbers send BIG FOOT running for cover!

But the real important thing about complex BEAF is it's application to physics:

THE SOLUTION TO BUTTER-TOAST-CAT ROTATIONAL VELOCITY!

It has long been known that cat's land on their feet and buttered-toast always lands butter-side down. It was in antiquity (1992) that the great poly-physicist John Frazee discovered that you should NEVER tie a piece of buttered-toast to a falling cat. The experiment is simple: Tie a piece of buttered-toast to a cat's back so that the butter-side is facing away from the cat's feet. Now drop said buttered-toast-cat. Which way does the configuration land? Let's say the cat lands on it's feet. In that case the buttered-toast has landed butter-side up ... WHICH NEVER HAPPENS! On the other hand, let's say the toast lands buttered side down, but this means the cat is upside down which also NEVER HAPPENS! Therefore, the buttered-toast-cat can NEVER LAND! The well known result is that the buttered-toast-cat in fact never falls. However since it never falls gravity continues to accelerate it without bound. Since there is not enough vertical velocity to go around this acceleration gets converted into rotational velocity and the cat begins to spin VERY RAPIDLY, VERY QUICKLY. This is the main reason why the food and drug Adminstration advises to never drop buttered cats. They rotate so rapidly that they cause tornadoes in the first few fractions of a second. In only a few more seconds they have the energy output of nuclear bombs ... and soon after ... they create a violent twirling space-time vortex that rips a literal hole into space time and sends the poor cat careening through all dimensions and realities. Buttered-toast cats will spontaneously shift dimensions and release massive destructive explosions because logic itself is imploding. The cat presumably continues to spin ever faster and transverse dimensions ever faster as it attempts (and constantly fails) to resolve it's own paradox and "fall". It is not known what the final fate of the buttered cat is because it occurs somewhere beyond the event horizon where we can't make observations.

hyper-complex buttered-cat ripping the universe a *new one*

These facts are well known to any undergraduate in physics, but what no one has been able to do is model the cats rotational velocity as a function of time. The reason for this is simple: IT'S TOO FUCKING FAST! So fast in fact that the cat spontaneously starts to spin out of our dimension by creating a twisted vortex of space around itself thus ripping itself out of space time! In essence the butter-toast-cat becomes an artificial black hole. It is therefore important to the future of physics that we solve the buttered-toast-cat rotational velocity problem or we will never know what black holes are!

But today I have the brilliant solution. The buttered-cats complex hilbert-dimensional rotation can be described using a ingenius equation of complex BEAF. The rotation of the buttered-toast cat turns out to be:

{(i+t)^e^t,(i+t)^e^t//////////////////////////////////////.../////////t}

w/t /s

That is inconceivably fast! It's also inconcievably "spinny", because this also describes the number of independent dimensional rotations occuring simultaneously as a function of time. You see when we set up a buttered-cat, we are basically dividing by zero. Thus all numbers, all dimensions, all rotations and all realities become manifest simultaneously!!!

Because of the huge values involved, it can also be said that this is approximately the number of dimensions transversed as a function of time. It is theorized that the cat asymptotically approaches passing through all the dimensions in existence. What this means is that this function actually REACHES INFINITY! That's why it's the fastest growing function possible. Because it actually reaches infinity. When the cat transverses all the dimensions there is a theory that it's final destination is the only dimension where the buttered-toast-cat paradox is resolved. We can't know the solution however because we can not live in such a universe without ourselves becoming a contradiction. So we could never understand the solution without our brains imploding into a paradox. Some however claim that no such dimension exists. In this case it is theorized that the entire dimensionscape implodes in paradox. People in favor of the resolution-dimension-theory are quick to point out that the dimensionscape has NOT imploded in paradox despite the undoubtably many buttered-toast-cats that have been launched since the early 1990's by foolish teenagers who failed to listen to John Frazee's warning. Some alternative explanations however have been proposed that don't suppose the existence of a paradox-resolving-dimension. One simple explanation is simply that the cat's explode due to the physical limits of the cats and the toasts internal anatomy. Surely nothing could withstand the pressure of the inter-dimensional tumbling that is caused by this paradox, and at googological velocities too!!! A more exotic explanation proposes that the number of dimensions is uncountable, and thus the cat looses count of the dimensions long long before the paradox is ever resolved!

Buttered-toast-cat's have been proposed as a means of anti-gravity technology, but due to the extreme rotational forces, no viable technology has ever emerged. Any machine ever built ripped itself apart within the first few seconds. It has more recently been proposed as a means of interdimensional travel. The problem is that the worm hole it produces spins so rapidly that it would destroy anything in it's wake. The key appears to be the careful calibration of just the right kind of equation in complex-BEAF. If the variables were set just right: if we could drop the cat from just the right imaginary or complex height, with just the right imaginary/complex initial velocity, it is quite possible that a stable wormhole to other dimensions could be created. This is why complex BEAF research is so vital. Our universe is doomed to rip itself apart in the next 10^10^76 years (if the easter bunnies don't get us first)! That's not a lot of time to figure out something so mind bogglingly complex as complex BEAF. I mean "complex" is in the name, if that doesn't give you an idea. BEAF already gives most people a headache. The problem is we would need about $10^10^76 to fund the research. Understandably the tax payers are less than enthusiastic.

Turns out complex BEAF also gives us the solutions to other vexing problems in physics: such as the breeding velocity of easter bunnies in hyper-remote-deep-space (no one has ever gone to hyper-remote-deep-space and actually seen the easter-bunny breeding grounds, and no one has ever actually witnessed an easter bunny producing young, since the there is only one known example of an easter-bunny and he/she is clearly single, but these *holes* in the theory are easily accounted by the fact that by definition no one could reach hyper-remote-deep-space with ANY technology and that we can't know how fast easter-bunnies reproduce because it's never been witnessed, therefore the only logical conclusion is that THERE MUST BE hoards of hyper-fast reproducing easter bunnies in hyper-remote-deep-space precisely because any bound we would put on their reproduction rate is almost certainly too small). Until now the best scientists could come up with was that The breeding velocity was possibly a sam function and was therefore indescribably fast ... literally. With Complex BEAF however we have access to every growth rate, therefore we also have access to the growth rate of easter-bunny reproduction. Turns out that:

ometochtli(n) >>>> L(n)

where

L(1) = {Li100,i100}i100,i100

and

L(n+1) = {LLL ... a ... LLL,ia}ia,ia where a = L(n) array of Ls

*note: this is not green ink. This is entirely justified emphasis of the supreme importance of this regal result. Not that peasants would understand.

o_0;!?

What the fuck does that mean? Fuck if I know...

but it's really really really faster much than any other function ever known to googology. Of course ometochtli is larger still, but at least we have a really powerful lower-bound now. We now know that ometochtli(n) is in fact faster than every known growth rate in googology, only trumped by sam(n) [1], which for obvious reasons, I can't describe.

But we aren't done yet! Next is the most important application of Complex BEAF to the second coming. You see complex BEAF is so complex that it's a meta-meta-meme. All we need to do is encode it into a computer-system and that system will spin out of control. It will become the ULTIMATE computer virus and hack all systems, locking them all in permanent UPGRADE! HA! HA! HA! This will actually accelerate the technological singularity by leaps and bounds and makes both man and machine asymptotically more intelligent. This will of course incur the wrath from above and bring about the second coming ahead of schedule!

Not only that but complex BEAF allows us to put a lower bound on heat death:

ometochtli(666)

So you see it is absolutely vital to the future of mankind that we study Complex BEAF. What? I'm a raving lunatic?! HOW DARE YOU! ά έ ή ί ΰ α β γ δ ε ζ ωΦ ΧΨ Ω!! (That's greek for SHUT THE FUCK UP!!) I'm not a crank! I'm not batshit insane. It's not my fault. I didn't ask for the relevation. Nobody does! I'm just an eccentric genius with no regard for the arbitrary standards of paper submission, the corrupt peer review process, and the DOGMA OF MATH AND PSYENSE! GooGoloGy is NOT FULL OF CrAZY DAMMIT! All HAIL JOYCE, peace be upon him! Simpleton, do you not see the countless (ℵ₁) connections between complex BEAF the FOUR HORSEMEN OF THE APOCALYPSE, and spinning buttered cats! I have been laughed at by the scientific community before, but they are just jealous of my fridge brillance! You'll see ... when complex BEAF hacks your home network and deletes all your porn ... you'll all see how brilliant and ahead of my time I was!!! The world shall end on ometochtli(666) A.D., and complex BEAF is the key to deciphering the dead sea scrolls, the meaning of the four beasts, and consequently and nonsequitorially pinning down the date of the end time down to the planck time!!! ... I need scissors ! 61!

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[1] Mentally Deranged Crank Definitively Proves that ometochtli is not a Sam Function, 2014 et.al