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So today we were given the green light to work overtime to catch up on batches. It’s 9:30 right now, and I’ve been keying for roughly one hour. I’ve got 217 items done. Everyone else combined has 99.

“Everyone else” = 2 people.

Tell me again why there are 20 people in this department?

Oh yeah and I just realized the time stamp’s been wrong. It’s corrected now.

When I started work today, I thought our level of batches remaining looked suspiciously like how many were left when we went home early last Friday, so I looked at a stats program and found…yup, 1 whole item had been keyed. That was half an hour ago.

Now, 82 items have been done. I did 71 of them. One other person did the other 11 (this is the same person who had the 1 item done half an hour ago).

Tell me again why there are 20 people in my department? Oh yeah, because none of them does any work at all.

Yet more proof that somehow my job was absorbed into the Federal Government.

Today I got to play the game Dead Space. I wasn’t sure if I would like it or not since I’m not really into sci-fi games that much, but since it was recommended and since it was also really cheap (I should note, I rarely pay full price for a game), I decided to give it a shot. And I’m glad I did, because the game’s pretty good.

Pretty good? Okay, it’s awesome. Not perfect, but it’s got my attention at least. Many games these days just don’t have much “playability” to them. I think of the Halo franchise, for instance, where basically what you do is play the same thing every 30 seconds for six hours. It gets really old, really fast in other words. That the Xbox 360 gives you achievement points is really the only reason why you’d bother to play the solo game more than once. At least with multi-player, you have a human you’re responding to, so it’s not the same thing every 30 seconds then. But if you’re doing the campaign…boring is the only word to describe it.

Not so with Dead Space however. DS is one of those horror shooters, where instead of being Rambo and running toward enemies to blast them to little bits, like in games like the Call of Duty franchise (which, I should note, I do like), you’re freaked out by the enemies that aren’t there. So as you’re going down a hall, you hear the creaks and groans of the ship you’re on, and you can hear the aliens growling and grunting, and you know there somewhere nearby, but you can’t see them. Of course, they’ll jump out at you at some point (although that gets somewhat predictable after a time—but even with the predictability, there are some times that still get you).

In any case, I’ve completed 10 chapters of the game. There are 2 left, according to the achievements section (I could look it up online to make sure, but I figure why bother? I’ll finish it tomorrow either way). And the only complaint that I have about the game is the frustration factor that sets in. That’s when you’re told, “We need to get this goal accomplished” and you go forth and get the goal accomplished only to find out someone behind the scenes has sabotaged something else, so now you have to get ANOTHER goal accomplished. This is okay every once in a while, but when it happens at the end of 10 straight chapters, it gets annoying. Not enough to make me stop playing, but enough to make me note it. I think the game play would be better if they just gave you a list of objectives at the very beginning, and said, “Here’s the ten things you’re going to have to do.” Throw in just two or three sabotage events, and the game play is less frustrating because you feel like you’re actually getting somewhere.

And that brings me to the actual point of this post, which isn’t so much a review of Dead Space as it is using the video game to think about story. Many video games these days are getting story oriented. It happens a lot with RPGs, of course, but it’s becoming more obvious in other games that typically don’t have much story going for them. Now I’ve yet to find a video game story outside of RPGs (like, say Final Fantasy VII) that has any real depth to it, but games are getting better at it. It used to be that you could wow people with awesome graphics and they wouldn’t care about the story; now we’re finding that all the awesome graphics in 1990 are hideous today and, therefore, we need a story.

But the important part of the story isn’t the plot; it’s not the sequence of the events that occur. The important part of the story is the character development. And this you’ll find true of stories in general. Take, for instance, Robert Ludlum’s Bourne trilogy (I mean the books, not the movies). What made the first book so compelling is that you actually cared about Jason Bourne. You struggled with him as he sought to reconcile his good beliefs with all the evidence he was collecting that he was an assassin. How could he be a cold-blooded killer?

The movie, on the other hand, jettisoned the character of Bourne and instead focused on the actions. They weren’t even the actions of the book, but that’s another issue. Bourne became an action figure, and the result was that the first movie was lame. You didn’t care about what happened to Bourne. You had nothing invested in him. The only virtue he brought was that he could do cool moves that you couldn’t do, and even that is something that we subconsciously recognize as a cheat because of trick photography and wire work.

The net result: if you want something good, character always trumps action. If you care about the character in your story, absolutely nothing could happen and it’ll be a great story. If you don’t care about the character in your story, he can win the world and you won’t want to read the story ever again.

Dead Space doesn’t have much characterization in it, but it does have more than the average FPS. You’re given more than just a set of objectives to fulfill. Even so, the game would have been so much better had the gamers tried to invest a bit more in building empathy for the characters in the story and the people who play the game. So it’s a good first step, but an even better first step was F.E.A.R. which, incidentally, is of a similar horror game genre—in fact, the first time you play that game if you don’t know what’s happening, you’ll end up firing about 20 shots at nothing (the first level is a training level so there’s no real threat, but it’s set in the story so you think there’s a threat).

In any case, there is perhaps hope for the future. Game developers are reaching a state where they can no longer hide deficient story with the “coolness” factor of better graphics or sounds. We’re pretty much maxed out there (about the only thing that’ll be better is when they get a 3-D game). This means that story is becoming more important, and as a result I predict that in the next five years or so we’re going to finally get the types of good stories found in RPGs over in action games.

So we got to go home early from work today. Because of the snow. Yeah, we were told at noon that we’d close at 2. And at 2, there were blue skies to the south. Typical. We never did get the 6-24 inches that were forecast (and seriously, what’s with a forecast that says you’ll get 6-24 inches? Soon it’ll be forecasts of anywhere from trace to nine feet.) because it melted on impact with the ground. Still, Global Warming is to blame. And Bush. But mostly, well, Bush actually.

As I came to work this morning, something dawned on me about the Factor Field. Long story short, the value I assigned for *! would actually be 0, since I had already established that *! = 0; and therefore, *! can’t be the place where infinity switches its sign.

But that’s okay, because I think I figured out where that switch would be anyway. To demonstrate this, I have to start with the fact that if *! = 0, as I showed in this post, then we are faced with what seems to be a contradiction. Numbers get larger and larger, yet at some point they have to switch to become smaller and smaller in order to get us back down to zero. And, in fact, this happens if you think about infinity switching signs.

So I thought to myself, Self, is there another place on the graph where we can extrapolate from something we see to form an analogy of what would happen at infinity? And I responded, Why, yes, Self, there is!

That’s because basically what we are looking for are the properties when x =1, 2, 3, 4, 5…infinity are all valid. So we can look at a place where some of these properties are valid and deduce what would happen if they were all valid. So I picked the graph at 420.

That’s because 420 has factors 1, 2, 3, 4, 5, 6 and 7 (it also has lots more, but this make a nice “spike” in the graph for us to look at). Here’s what it looks like in part:

Now here’s the important thing to examine: the radial arms. The radial arms match the spike that shoots out from the center in terms of length. They also go at a 45 degree angle (a slope of 1/1 and -1/1) from the spike at y = 420.

Why is that important? If we ignore everything after x = 7, the graph looks exactly the same at y = 420 as it does at y = 0:

So the value of y at the maximum point will be equivalent to the length of the radial arms, and the length of the spike—so it’s the infinity formed by adding 1 + 1 an infinite number of times.

This means that the value of *! needs an adjustment. While it is true the *! = 0 if we define *! as 1 x 2 x 3 x 4 … x infinity, that’s not the only infinite value that equals zero!. Consider for a minute something about even numbers. If 4 is a factor of a number, then 2 must also be a factor of that number. If 16 is a factor of a number, then 2, 4, and 8 are all factors too!

Suppose for a moment that we’re looking for the graph of a number that has 2, 4, 6 and 8 as factors. If we have 8 as a factor, then we must already have 2 and 4 as factors. That means that we only need to multiply 6 x 8 to get a number that has 2, 4, 6, and 8 as factors. 6 x 8 = 48, which does indeed have 2, 4, 6, and 8 as factors. Now it is still true that if we multiplied 2 x 4 x 6 x 8 we’d also have a number that contains 2, 4, 6, and 8 as factors (that number would be 384). But 384 is a lot larger than 48! If the key that we were searching for was the first time we got 2, 4, 6, and 8 as factors, it would happen at 48 and we’d never get to 384.

So that means that we’re not actually going to use the infinity of 1 x 2 x 3 x 4 … in our problem. We can throw out 2 no matter what, because any even number greater than 2 will automatically have 2 as a factor. That alone halves the size of the infinity we’re dealing with. In reality, the number we’re dealing with is simply the number that is made by multiplying the highest values of numbers close to infinity that are composed of all the other factors below infinity. That number is, of course, impossible to write down. But let’s call it m, for maximum. m = 0, for the same reasons that *! = 0. Namely, m has all the factors of all the numbers, so the graph of m + 1 looks exactly like the graph of 0 + 1, so m + 1 = 1, and therefore m = 0.

However, m < *! in terms of the size of the infinity, since 1 x 2 x 3…etc. is going to be way larger than simply multiplying the highest numbers that contain all the lesser factors together! Furthermore, it would place us in the position where *! falls outside the range of the Factor Field. I think this is just Gödel making his appearance, so I’m prepared to dismiss it—but of course that may not be the rigorous mathematical thing to do!

Now if the distance from 0 to m is one oscillation, then the sign switch of infinity will occur at a distance of m/2. That’s because you go out m/2 distance, then you return m/2 distance. Let’s call m/2 by the variable M.

Let’s try to visualize what’s going on. Imagine the graph that goes up from y = 0 somehow meeting the graph that goes down from y = 0. At the point they meet, infinity switches signs. The numbers are getting larger and larger as the graph goes up; then it hits the merge point (at the sum of all whole numbers) and instantly we go from +infinity to –infinity—exactly as what occurs on the tangent graph. Then the points fall back down (or rather, still increase but now from their negative state) to the value of zero, where they switch signs and go back up toward infinity.

Now for why I think this M point is where the sign of infinity switches. Again, think of the graph at y =0. We move up in the positive values to reach infinity at y = M. But at y = M + 1, the graph has inverted. If M – 1 is positive infinity – 1, then M + 1 is negative infinity + 1. This is exactly what you’d expect, because M + 1 looks exactly the same as the line you’d get at negative infinity when the line reaches the negative value of M – 1.

I maintain that this is exactly what happens at the tangent graph, and the reason that we cannot actually graph it when cos x = 0 is because M is simultaneously positive and negative, just as zero is simultaneously positive and negative.

Just some quick notes (more for me to keep track of things to think about, rather than me trying to prove anything here). I think that given my previous two posts, -*! corresponds to the positive number line while +*! corresponds to the negative number line. That is, since zero is a reflecting point, and *! Is functionally equivalent to 0, then *! is also a reflecting point. However, if you try to picture this mentally, in order to keep positive integers from being cancelled out by the graph created at *!, you’d have to invert the signs in one of them.

Secondly, where functions are undefined (such as the tangent function wherever cos x = 0), I think that’s where the sign switches at the infinity mark. In other words, just as hitting zero on a number line switches our sign, so hitting *! switches the sign. This is sort of like saying the function actually is continuous, but in a different dimension that we cannot draw on the 2D grid. In other words, if we flip it and instead graph it from the x-axis crossing y at *!, then we’d see the line go through the point at *! and we’d be referring to the positive and negative zero, rather than the positive and negative infinity, and we’d be saying that the function is undefined at 1/n as n-> infinity (instead of how it is now, when it’s as n -> 0).

In other words, I think that the only reason these functions are considered undefined there is that we cannot visualize the two infinities meeting. Except that we can once we realize that it would look exactly like the zero line. We still couldn’t draw it, but we could at least visualize what it ought to look like, at least as far as we can mentally think of any infinite.

Last night through this morning, I’ve thought of some more implications of what would happen if we viewed infinity in the same manner as we view zero. First off, we can ask a simple question: what happens to the number line at zero? I’ll give an ASCII representation here:

< ------------------->
     -2 -1  0  1  2

So we see that as we move left to right, the numbers are negative. Ignoring the negative sign, the numbers appear to be getting smaller (they are in fact getting larger because of the negative sign, but think about the concept for a moment). Then it hits zero. At that point, the sign of the numbers change from negative to positive, and the numbers printed begin to get larger again.

The result is that we can view the negative numbers as an inversion of the positive number, and the inversion point is at 0. This is more clearly seen in the graph I showed in the previous post, which I’ll reproduce here:

If we ignore the “name” that we give to the numbers and just look at the picture, we see that the zero line gives us a “reflection surface” such that what is above the line is reflected by what is below the line. Above the line goes in one direction; below, in the opposite.

If we maintain the equivalency of zero and infinity, then when we hit the infinity line, we will exactly recreate the graph you see above, with only the “names” of the numbers different. Instead of seeing “1” and “-1” we would see infinity -1 and infinity + 1.

Now for the important part. I maintain that at that point, the sign of the numbers (in relation to infinity) changes, so we can say that the sign of infinity changes. In other words, if we say all numbers > 0 are positive and all number < 0 are negative, then we can also say that all numbers > the infinity line are positive infinity, and all numbers < the infinity line are negative infinity.

(Quick side note: when I say infinity above, I am specifically referring to the infinity that is formed by 1 x 2 x 3 x 4 x 5 … etc. Since there are different infinities, then it is critical that this be kept in mind. To help do so, and because 1 x 2 x 3 … etc. are factorials, then I will refer to this specific infinity as “infinity factorial.” Since WordPress won’t let me paste in the infinity symbol, I’m going to use the * symbol as the infinity symbol. Therefore, *! = infinity factorial. Hope that’s not too confusing.)

Back to the point now. As I was saying, if we say there is a sign switch at the *! line, then we’d have positive *! and negative *!, or +*! and -*!. The immediate question is, is there any evidence that such a thing exists? That is, is there evidence that there’s a single point where positive and negative infinity switch at all? And the answer is…yes.

Consider the tangent curve. You can see a picture of it here (look at the graph called y = tan x). The important feature is that there are “discontinuities.” What’s a discontinuity? It’s the part of the graph where the line suddenly jumps from positive infinity to negative infinity. Since tan = sin/cos, then this happens anytime that cos x = 0. This is due to the fact that you can’t divide by zero. I would argue that, given what I’ve stated above, division by zero = *!, and that’s why you get the jump between positive and negative infinity on the graph.

If this is true, then we have a startling relationship. It is loosely stated that 1/infinity = 0 (in reality, to be more precise, you have to talk about the limits, and say that 1/n = 0 as n -> infinity). What I’m proposing is that 1/0 = *!. Algebraically, this makes sense:

If a/b = c, then this is equivalent to:

a = cb

b = a/c

Now in the above, let a = 1, b = *!, and c = 0. We see that if 1/*! = 0, then it’s also true that 1/0 = *!.

Yes, I realize that division by zero is something that you ought to rebel against because it can be abused to prove many things that are contradictory. For instance, you can prove that 2 = 7 in the following manner:

a = 1
b = a
2(a – b) = 7(a – b)
Divide both sides by (a – b) to remove the common factor:
.: 2 = 7

This doesn’t work because (a – b) = 0, and this is division by zero which is officially “undefined” and not allowed.

But note that if my theory is correct, the math still works! I argue that any number (x) would solve the equation x/0 = *!, so 0/0 = *!. Therefore, what you have is 2*! = 7*! and I would say that this does not violate any rules of math, any more than 2 x 0 = 7 x 0 violates rules of math. Of course, just because i would say that doesn’t make it true…so let’s examine it a bit further.

My claim would be that x*! = y*!, no matter what x and y are. So this is equivalent to saying x*! = *!. Is there a way to prove this? Well, we could try it this way (note: it still keeps the denominator as zero, but we don’t actually do any division by zero in the following so I think it doesn’t violate any rules, but I may be wrong on that count).

*! = a/0, where a can be any number.
x*! = xa/0
.: y*! = ya/0

Now the claim is x*! = y*! so:

xa/0 = ya/0

We can multiply both sides by 0/1, which is the typical way to try to remove the denominator of a fraction:

(0/1)(xa/0) = (0/1)(ya/0)
[0(xa)]/(1 x 0) = [0(ya)]/(1 x 0)
0/0 = 0/0

Since the final line is an obvious truism, insofar as both sides have the same symbols, then it follows that x*! = y*!, and therefore it would not be a violation to say 2*! = 7*!.

The net result is that I think we may have ways to divide by zero now, although it only replaces saying “don’t divide by zero” with “divide by zero but now you have to think about a whole bunch of infinity properties.”

As I’ve done some more looking at the Factor Field, I’ve discovered an interesting phenomenon. Of course, when I say “I discovered” this I don’t think it means that I’m the first person to think this, but only that I’ve come up with it on my own. Of course, that doesn’t mean I’m not the first person to think of it, but I’ll be the first to admit that I’m not a professional mathematician and only dabble in it for fun, so I’m pretty much clueless as to where the vast majority of theorists are on this subject!

In any case, I’ve come to the conclusion that zero and infinity are functionally equivalent. This isn’t just due to the fact that you can’t divide numbers by either zero or infinity (e.g., 1/0 is “undefined”), but because when you look at the Factor Field you can SEE that these things are graphed IDENTICALLY.

That last part is important to emphasize. These are identical graphs. Which means that if we stipulate that if method A = Graph 1, and method B = Graph 1, then method A = method B, then we have to say that zero = infinity.

So let me show you what I mean. Here’s a cross section of the Factor Field. I’m getting this by using the following VBScript code, which you can run at home if you have Windows and Microsoft Excel:

set objExcel = CreateObject("Excel.Application")
Set objWorkbook = objExcel.Workbooks.Add()
Set objWorksheet = objWorkbook.Worksheets(1)
objExcel.Visible = True

' Use if you want to fill up entire spreadsheet for 2007 format
'MaxWidth = 16384
'MaxDepth = 1048576	

' Use if you want to fill up entire spreadsheet for 2003 format
'MaxWidth = 256
'MaxDepth = 65536   

' Use for a practical demonstration
MaxWidth = 201
MaxDepth = 201

a = 1

for x = (MaxDepth-1) to 1 step -1
	objExcel.Cells(a,1).Value = x
	a = a + 1
next

for x = 1 to (MaxWidth - 1)
	for i = (MaxDepth-x) to 1 step -x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

a = MaxDepth
for x = 2 to (MaxWidth)
	objExcel.Cells(a,x).Value = (x-1)
next

for x = MaxDepth to (MaxDepth * 2)
	for i = MaxDepth + x to 1 step x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

a = -1

for x = MaxDepth + 1 to ((2 * MaxDepth) - 1)
	objExcel.Cells(x,1).Value = a
	a = a - 1
next

for x = 1 to (MaxWidth - 1)
	for i = (MaxDepth + x) to ((MaxDepth * 2) - 1) step x
		objExcel.Cells(i,x+1).Interior.ColorIndex = 1
	next
next

wscript.Echo "Done."

Now I should note that this code is fairly “hacked together” so it’s not elegant, but it does the job. And what it does is make this graph (if you scroll to row 201 of the Excel spreadsheet):

Now you can see what I’ve done to modify the Factor Field is to extend the pattern into the negative numbers. But with one important thing missing. There are no cells filled in on the y = 0 line (which holds the number for the x-axis of the graph in the picture). But clearly the pattern continues, because if we look at it with the y = 0 line filled in, it fits the patterns established in every column:

Clearly, if we just look at the pattern itself, the graph wants the y = 0 line to be filled in its entirety. But this causes an immediate problem, because what the Factor Field represents is factors of numbers. That is, when you look at y = 1, you see only x = 1 as a valid answer. That’s because 1 has only 1 as a factor. When you look at y = 2, you see both x = 1 and x = 2, because 2 and 1 are both factors of 2, etc. That’s why when you see y = 9, for instance, you see x = 1, x = 3, and x = 9, because 1, 3, and 9 are the only factors of 9.

But when you look at y = 0, you get x = 1, 2, 3, 4…to infinity.

Because this graph functions to show us factors, that means that you can take the furthest out x value and it will be equivalent to the y value. Or, to state it the other way, y is equal to the highest x value. So if the highest x value is 9, then y = 9. Therefore, if the highest x value is infinity, then the graph holds for when y = infinity.

Infinity = zero on this graph.

But it’s not just that. I’ve been thinking of the mini “proof” I gave about an infinitely long prime number, and it turns out that the “proof” exists only because I stipulated that a number must have itself as a factor by virtue of the identity axiom. But let’s jettison that for a moment and ask ourselves what the proof really requires. Let me restate it here, modified slightly:

1. Let n = 1 x 2 x 3 x 4 … x infinity.

2. Let w be a factor of some number, c.

3. If w is a factor of c, then the first number greater than c that w can also be a factor of is c + w. (e.g., 7 is a factor of 14; the next number greater than 14 that 7 can also be a factor of is 14 + 7, or 21.)

4. Since n is the product of every positive whole number, every positive whole number is a factor of n.

5. n + 1 can have only the factor of 1 (via 3). (Reasoning: if 2 is a factor of n + 1, then 2 cannot be a factor of n; since 2 is a factor of n, the next possible number after n that 2 can be a factor of is n + 2. This holds true for all numbers, therefore the only possible number that can be a factor of n + 1 is 1.)

6. Since n + 1 has only the factor of 1, then n + 1 MUST BE the number 1.

7. If n + 1 = 1, then n = 0.

8. Conclusion: 1 x 2 x 3 x 4 … x infinity = 0.

Pretty amazing, huh?