Talk:Benderama

God Entity
I believe the voice for the God entity is used in this episode, I'm wondering if it's a purposeful reference to that character. When Bender is defeated in the fight against the ugly giant, light appears on Bender and a voice is heard saying "Walk into the light Bender." and Bender replies saying "Aww, man, do I have to walk?!" Polantaris 07:56, 24 June 2011 (CEST)
 * I'm not sure. It sounds more like a general, deep voice to me. I don't see why the Galactic Entity would bother talking to the big Bender. - akitalk 08:08, 24 June 2011 (CEST)
 * Seeing as no-one replied for quite a while, I took away the information about Galactic Entity appearing. Before re-adding it, please restart this discussion. - akitalk 13:30, 26 June 2011 (CEST)

The Formula
The mass of an individual generation is just 60% the mass of a bender in the previous generation. So given the original mass M_0, the mass of a Bender in generation n should be M_0*0.6^(n-1) (assuming n starts at 1). So, given 2^(n-1) Benders per generation, the total mass should be the sum from n=1 to N of M_0*(1.2)^(n-1) = M_0*(1.2^n -1/)0.2 which does not converge as N -> infinity. (1.2 > 1). The article currently says that the mass changes by the cube of the scaling factor. Did I miss something in the episode, or should the mass of a Bender in generation n just be the 60% the mass of a bender in generation n-1 (consider the first duplication: In generation 2, there are 2 Benders, each massing 60% of the original. So, the mass of this generation is M_0*0.6*2; adding this to the original M_0 matches the formula above, 2.2*M_0).
 * Mass is directly proportional to the cube of length; that is, mass scales directly with volume. If Bender's height is scaled by 60% (and all other linear measures are similarly scaled by 60%), then Bender's volume is 60%^3 = 21.6% of the volume of the original Bender, thus 21.6% of the mass of the original.  This has nothing to do with anything in the episode, but is rather a basic result from geometry or physics.  Imagine that you have a perfect cube of material that measures 1 m on a side, and weighs 1 kg (thus having a density of 1 kg/m^3).  Now make a scale copy of that cube that is 50% smaller.  The new cube is half a meter on a side, and still has a density of 1 kg/m^3.  Mass is density times volume, thus the new cube has a mass of (1 kg/m^3)*(0.5 m)^3 = 0.125 kg.   --71.83.120.33 17:19, 25 June 2011 (CEST)
 * I interpreted it as 60% the volume, which would imply 60% as massive. I'll watch the episode again, but you're probably right that it's a 3/5 scale length ratio.
 * In general, when people talk about scale, they are referring to linear scale. Hence a 1:45 scale model of a train would be a model train that is 1/45th of the length of a full sized train.  Thus typical usage would suggest that Bender is scaled by a linear factor of 60% (i.e. all linear measurements are scaled to 60%).  If we assume that volume is scaled by 60%, then linear measures would be scaled by about 84% (the cube root of 60%).  Visually, it would appear to me that the copies are 60% of the height of the original, rather than 84%.  --71.83.120.33 19:27, 25 June 2011 (CEST)
 * I understand what you're saying; I probably just wan't paying too much attention to detail the first time around (it is a comedy cartoon, after all!). Thank you for pointing it out.
 * Maybe I missed it, but was there a rule that each Bender could only reproduce once to make 2 "children"? Because if the Bender's could reproduce indefinitely, the formula doesn't apply.
 * Each Bender could multiply into 2 more benders, but since the copying machine was in Bender, it too would get duplicated, so they could duplicate infinitely as long as they find matter to stuff into the machine. Polantaris 11:55, 26 June 2011 (CEST)
 * I don't think that it was ever explicitly stated that each Bender could only make two copies. However, if that limit is not in place, it doesn't make any sense to talk about limiting series.  The original Bender could reproduce over and over again, consuming an infinite amount of matter.  So, while such a rule was never explicitly stated, it seemed implicit in the way that the situation was described. --71.83.120.33 19:31, 26 June 2011 (CEST)

Convergence or Divergence?
Hi, guys! In spite of being a total math geek, I have tried to stay away from this discussion. However, it has sort of started to mess up the actual article, with contradictions and personal incredulity and whathaveyou. So, here goes: Why don't you guys try to settle things here on the talk page, create a badass text that everyone agrees on, and then put it back on the article? Until then, please don't mention the formula in the trivia or goof sections. Thanks!

The current text (that I now removed) was this:


 * The formula that the professor shows is incorrect. The total mass of all of the Benders should be the sum of the number of Benders in each generation times the mass of each Bender in that generation. The number of Benders in each generation is twice the number of Benders in the previous generation, thus there are 2^n Benders in the nth generation.  Mass changes as the cube of the scaling factor, thus each Bender weighs 0.6^3=21.6% of what its parent weighs.  If we assume that the mass of the original Bender is M_0, then the mass of an nth generation Bender is M_0*0.216^n.  Thus the mass of all of the Benders after an infinite number of generations is the sum of terms of the form 2^n*(M_0*0.216^n), where n goes from 0 to infinity.  This series does converge.
 * In fact, the math does work out correctly. The equation shown by Farnsworth correctly describes a (divergent), so the only question would be whether this sequence accurately describes the scaling of Bender. With the second generation (n=1), the formula predicts that each Bender will weight 1/4 of the original Bender. If that is true, the linear scale of each 2nd-gen Bender would be the cube root of 1/4, which works out to about 0.63 with rounding. Therefore, Bender's "60 percent scale" statement is roughly accurate for the first generation. The criticism above assumes that "each" generation is a 60 percent scale replica of its parent, yet Bender never claims that to be the case; he only remarks on the 1st generation, for which the statement is roughly correct with rounding.

Again, thanks. - akitalk 13:15, 27 June 2011 (CEST)


 * I agree that discussion of the mathematics should remain here, rather than on the page---at the time I made my edit, there did not seem to be much controversy, and there was no discussion here, so I didn't really think anyone cared all that much.


 * With regard to the series that Farnsworth presents, it seems that the show tells us that (a) every time the machine is used, it produces two copies of the original object, each at 60% scale, and it gives us no reason to believe that a duplicate duplicator would behave differently; (b) every Bender has a duplicator, which should produce 60% scale copies of the Bender making copies; and (c) for whatever reason, no Bender uses his duplicator more than once (I kind of assumed this is because Bender is really lazy, rather than because the machine only functions once). Even if we assume (as the most recent editor has) that (n+1)st generational Benders are 1/4 the mass of nth generational Benders, I don't see how we can reconcile the series that Farnsworth presents with the points above---there is an extra factor of n+1 in the denominator which does not make sense to me.


 * That being said, perhaps all of the mathematical wankery is not of general interest, and should be elided a bit in the actual entry. Perhaps something like this would be better:
 * The formula that Professor Farnsworth presents simplifies to the harmonic series multiplied by a constant. This series is divergent, though it is unclear how this is meant to model the infinitely multiplying Benders.
 * --134.197.0.22 16:51, 27 June 2011 (CEST) (and my IP at work is, of course, different from the one at home---I am the same as 71.83.120.33, above)


 * That statement would work fairly well, I believe. The only mystery regarding the Benders is exactly why they will scale according to a harmonic series; however, it is not a contradiction at any point. As noted, Bender's 60% scale comment is a one-off in reference to his immediate successors, and is accurate (according to the formula) in that case, even though the formula predicts that it would not be accurate for future generations. No one in the episode, at least according to my memory, claims that 60% is some sort of general rule for the replicator.


 * Given the strangeness of the duplicating technology, it isn't such a stretch to imagine that matter duplication doesn't work according to a linear scale, and that a replicated replicator itself will not be a simple ratio of its parent in terms of power and capability. In fact, to assume that the second replicator has the same capabilities as its parent replicator, only scaled, would mean making some kind of real link between physical scale of the device and its output (replicating) power. Evidently, that is not the case, for the first replicated replicator, while presumably being a 63% scale of the original, is capable of producing duplicates that are 69% scale of their parent objects (using Farnsworth's formula, if my quick math is correct). 130.160.45.36 17:21, 27 June 2011 (CEST)


 * As I see it, the problem with your interpretation is that you are required to make a lot of extra assumptions, one of which seems to be directly controverted by the show itself, i.e. that Farnsworth said 60%, but really meant 63%. He does not seem to be the kind of character that would make that kind of rounded statement.  If the writers had actually thought it through as much as you have, I would imagine that they would have had Farnsworth refer to a 63% scaling factor, rather than a 60% scaling factor.  As I see it, there are two things going on:  (1) the writers wrote out the situation without thinking about the mathematical model at all (and who would expect them to) and (2) someone picked a divergent series at random, and flashed it on the screen as though it were meaningful.  It would not be the first time that something like this happened in a television show, though I would have expected more from Futurama. --134.197.0.22 17:32, 27 June 2011 (CEST)


 * Where did Farnsworth say anything about 60%? Perhaps I'm just forgetting a scene, but I only recall Bender saying 60% with regards to his first-generation replicas, which -- according to the equation shown -- would in fact be scaled at 63%, so that line makes perfect sense (and while Farnsworth wouldn't be expected to round, it does make sense for Bender, particularly given the comedic context of that exchange with Leela). 130.160.45.36 19:47, 27 June 2011 (CEST)


 * Oi... you are correct---Farnsworth never mentions the 60% number. That being said, I don't think that changes anything else about what I said previously.  There is no reason to expect that the machine would behave differently at different scales, and it is still unclear to me what the n+1 in the denominator is meant to model.  --134.197.0.22 20:08, 27 June 2011 (CEST)


 * I do get the skepticism, but here's how I see it: we know nothing of the functioning of this bizarre device, and the only specific information Farnsworth produces is the harmonic equation which shows the mass of any generation of Benders. Therefore, the primary knowledge we have of this replication process is that self-replications -- ie. those in which the replicator itself is copied and then reused -- follow a harmonic series. Bender's single line about scale should then be measured against this "scientific" account, and in fact it works out to be roughly correct. Yes, they are 63% scale replicas of him, so it's perfectly reasonable that he introduces them as 60% replicas to Leela. After all, Bender himself doesn't understand how the device works, he simply sees that they appear to be roughly 60% of his size, and we all know he's not a good calculator (See Transcript:The_Cyber_House_Rules)


 * Now, the equation itself is certainly going to involve a little absurdity given the very nature of a replicating device replicating itself; yet it's still a relatively clever joke about the standard harmonic series, written so as to explain the generations of Benders yet also so that you can simplify it and understand it very easily.


 * I know you already get this, but just to reiterate the basics of the joke: The 2^n outside of parentheses indicates the number of Bender's per generation, while the part in parentheses indicates the mass of each Bender in that generation, relative to the starting gen. The existence of that same 2^n term in the denominator makes it cancel itself out, and therefore you get the recognizable M0/(n+1) which produces an obviously divergent series like 1, 1/2, 1/3, etc. That satisfies helping their niche of math-y audience members to "get" the joke. As you indicated, the sci-fi part is the assumption that in fact a bizarre duplication device such as the one shown would follow this sort of unusual equation when self-replicating... yet I honestly also see no grounds to assume that such a device would produce a consistent scale for every generation. After all, the level of absurdity of these duplicates reaches bizarre proportions when you see that Benders are manipulating atoms, meaning the duplicates themselves are clearly not composed of ordinary matter as we know it. 130.160.45.36 20:18, 27 June 2011 (CEST)


 * I guess the way I see it is that the formula was given as a somewhat obvious math joke (i.e. the harmonic series is easily recognized), but that the situation presented is not well modeled by the formula, or rather that one must invent additional facts to explain the formula. As I said above, I think it is the case that the writers didn't really think things through---the device works in whatever manner is required by the plot, and the formula is a somewhat unconnected math joke.  Given that the writers have gone to great lengths in the past to produce believable mathematics, I find the formula presented in "Benderama" to be disappointing.  --134.197.0.22 20:46, 27 June 2011 (CEST)