1162: "Log Scale"
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1162: "Log Scale"
Alt text: "Knuth PaperStack Notation: Write down the number on pages. Stack them. If the stack is too tall to fit in the room, write down the number of pages it would take to write down the number. THAT number won't fit in the room? Repeat. When a stack fits, write the number of iterations on a card. Pin it to the stack."
First time making a thread, surprised I'm the first one. Visualizing large numbers is always fun
Last edited by The Synologist on Fri Jan 18, 2013 5:28 am UTC, edited 1 time in total.
 rhomboidal
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Re: 1162: "Log Scale"
Burning all the paper required could keep you rolling in joules.
Re: 1162: "Log Scale"
Damn you rhomboidal, you made me curious...
I went and looked and found a site quoting for biomass details 17 MJ per kg of paper (ref) & an A4 sheet will average about 5g (ref). If we assume they're right and that the sheet at the front is A4 (Admittedly it looks a bit bigger, but I'm going to guess the stick figure is standing a long way into the background ), the sugar line at 19 takes up 5px of the available 194px of the page, so it's safe to assume it's about 1/20th of a page height, and the uranium cropoff looks to be approximately 1/3rd of the sheet's width, so 19 MJ/kg from uranium would up 1/60th of an A4 sheet. Ignoring the heading since it's not that big relatively speaking, the uranium in pages of paper would be 76,000,000 / 19 / 60 = 66666 2/3 sheets of paper. * 17 MJ/kg * 0.005 g per sheet = 5666 2/3 MJ. It'd only be about 59.4km long of paper hanging off the top of the page. The odds I've gotten something wrong above is probably around 100%
I went and looked and found a site quoting for biomass details 17 MJ per kg of paper (ref) & an A4 sheet will average about 5g (ref). If we assume they're right and that the sheet at the front is A4 (Admittedly it looks a bit bigger, but I'm going to guess the stick figure is standing a long way into the background ), the sugar line at 19 takes up 5px of the available 194px of the page, so it's safe to assume it's about 1/20th of a page height, and the uranium cropoff looks to be approximately 1/3rd of the sheet's width, so 19 MJ/kg from uranium would up 1/60th of an A4 sheet. Ignoring the heading since it's not that big relatively speaking, the uranium in pages of paper would be 76,000,000 / 19 / 60 = 66666 2/3 sheets of paper. * 17 MJ/kg * 0.005 g per sheet = 5666 2/3 MJ. It'd only be about 59.4km long of paper hanging off the top of the page. The odds I've gotten something wrong above is probably around 100%
Last edited by fyjham on Fri Jan 18, 2013 5:39 am UTC, edited 1 time in total.
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Re: 1162: "Log Scale"
Sif not just make an excel chart and enlarge it to suit with A3 paper.
Re: 1162: "Log Scale"
Edward Tufte would be proud.

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Re: 1162: "Log Scale"
Alternatively, log scales are for people who don't have printers with high enough DPI to scale the largest bar to the page and still print the others.
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Re: 1162: "Log Scale"
this comparison was bad, because the energy density in uranium is due to nuclear fission, while the others are due to chemical reaction.
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Re: 1162: "Log Scale"
xkcd wrote:Alt text: "Knuth PaperStack Notation: Write down the number on pages. Stack them. If the stack is too tall to fit in the room, write down the number of pages it would take to write down the number. THAT number won't fit in the room? Repeat. When a stack fits, write the number of iterations on a card. Pin it to the stack."
What do you do if the number of iterations doesn't fit on the card?
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Re: 1162: "Log Scale"
skeptical scientist wrote:xkcd wrote:Alt text: "Knuth PaperStack Notation: Write down the number on pages. Stack them. If the stack is too tall to fit in the room, write down the number of pages it would take to write down the number. THAT number won't fit in the room? Repeat. When a stack fits, write the number of iterations on a card. Pin it to the stack."
What do you do if the number of iterations doesn't fit on the card?
It's a large stack. Multiple cards should work.
Re: 1162: "Log Scale"
Uranium is for quitters who can't find enough antimatter to power a Kardeshev Type V civilization properly.
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Re: 1162: "Log Scale"
It doesn't seem like a roll of toilet paper unrolled.
Re: 1162: "Log Scale"
Or for those primitives who haven't learned how to handle antimatter yet.
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Re: 1162: "Log Scale"
Graphical representations are for quitters who can't comprehend number sizes.
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Re: 1162: "Log Scale"
It's called a "log scale" because that's what you need to make all the paper required for the stacks.
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Re: 1162: "Log Scale"
zukenft wrote:this comparison was bad, because the energy density in uranium is due to nuclear fission, while the others are due to chemical reaction.
Completely true! Somebody care to calculate the energy density of sugar if used as nuclear fusion source (assuming all atoms could be fused to form iron)?
Re: 1162: "Log Scale"
zukenft wrote:this comparison was bad, because the energy density in uranium is due to nuclear fission, while the others are due to chemical reaction.
The title says "fuel energy density", which may be somewhat correct given how we use these energy sources as fuel today.
Can we use the others for nuclear fission? I ask because I know very little about this subject.
Re: 1162: "Log Scale"
I tried writing the number of iterations on some paper, but the stack of pages with that number won't fit in the room. Please advise.
Fission? No. But hydrocarbons consist of elements that can be used for fusion fuel. And they actually contain significantly more nuclear energy than Uranium does. Per weight, of course. Per volume, Uranium still wins.
emilv wrote:Can we use the others for nuclear fission? I ask because I know very little about this subject.
Fission? No. But hydrocarbons consist of elements that can be used for fusion fuel. And they actually contain significantly more nuclear energy than Uranium does. Per weight, of course. Per volume, Uranium still wins.

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Re: 1162: "Log Scale"
zukenft wrote:this comparison was bad, because the energy density in uranium is due to nuclear fission, while the others are due to chemical reaction.
The comparison is fine. It's comparing the energy density using the methods available to current technology.
Re: 1162: "Log Scale"
Yeah, if we wanted to compare absolute strict theoretical energy density we would just measure the total massenergy per volume. And we already have a term for that measure. It's called density.
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Re: 1162: "Log Scale"
keithl wrote:Uranium is for quitters who can't find enough antimatter to power a Kardeshev Type V civilization properly.
Antimatter? Antimatter is for quitters who can't maintain a stable Hawking radiating black hole. (100% mass > radiation conversion)
Black holes? Black holes are for quitters who can't harvest dark energy by collecting the work of two galactic superclusters expanding away from each other. (violates energy conservation)
More importantly though, there simply is no significant amount of natural antimatter in the accessible universe. It's the most efficient method of energy storage, but you can't power anything with it, unless you have FTL travel, wormholes, or something similar, and can find antimatter somewhere out there in the multiverse.
Re: 1162: "Log Scale"
K^2 wrote:I tried writing the number of iterations on some paper, but the stack of pages with that number won't fit in the room. Please advise.
Just use the Knuth PaperStack Notation again. If the number of iterations of that stack still doesn't fit in its note, repeat.
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Re: 1162: "Log Scale"
PaperStack Notation for a number n really just gives you log* n (the number of times you need to take its logarithm before you get something less than or equal to 1), scaled and shifted by a couple constants depending on the base you're using and how many digits you can fit in one room's worth of stacked paper.
For truly large numbers, this method doesn't even come close to being useful. Using base3 logarithms and uparrow notation:
log* (3^3) = log* (3↑↑2) = 2
log* (3↑↑3) = 3
log* (3↑↑↑3) = log* (3↑↑(3↑↑↑2)) = 3↑↑↑2 = 3↑↑3 = 3^27 = 7,625,597,484,987
3↑↑↑↑3, which can also be represented as 3 → 3 → 4, is g_1
g_2 is 3 → 3 → g_1, and when we continue this to g_64 we have Graham's number. But even g_1 is not really expressible using simple paper stack notation.

Note: as you may have noticed, log* (b^^k) = k if we're using baseb logarithms, just as log (b^k) = k. So to invert triple arrows, we'd need a new operation, log**, which corresponds to the number of times you need to iterate log* before getting something less than or equal to 1.
g_1 = 3↑↑↑↑3 = 3↑↑↑(3↑↑↑↑2)=3↑↑↑(3↑↑↑3), so log** g_1 = 3↑↑↑3, which as established above is a number with so many digits that you'd have to take the base3 logarithm more than 7 trillion times before you got something manageable. And to deal with g_2 in a similar way, you'd need log******...***, with a total of roughly g_1 stars.
And g_64 itself was blown right the fuck out of the water at the end of the very first page of the My number is bigger! thread, by a number bigger than g_{g_26}, where g_{64} is Graham's number, and g_{65} already surpasses the xkcd number.
There have been 1210 more posts in that thread since then.
For truly large numbers, this method doesn't even come close to being useful. Using base3 logarithms and uparrow notation:
log* (3^3) = log* (3↑↑2) = 2
log* (3↑↑3) = 3
log* (3↑↑↑3) = log* (3↑↑(3↑↑↑2)) = 3↑↑↑2 = 3↑↑3 = 3^27 = 7,625,597,484,987
3↑↑↑↑3, which can also be represented as 3 → 3 → 4, is g_1
g_2 is 3 → 3 → g_1, and when we continue this to g_64 we have Graham's number. But even g_1 is not really expressible using simple paper stack notation.

Note: as you may have noticed, log* (b^^k) = k if we're using baseb logarithms, just as log (b^k) = k. So to invert triple arrows, we'd need a new operation, log**, which corresponds to the number of times you need to iterate log* before getting something less than or equal to 1.
g_1 = 3↑↑↑↑3 = 3↑↑↑(3↑↑↑↑2)=3↑↑↑(3↑↑↑3), so log** g_1 = 3↑↑↑3, which as established above is a number with so many digits that you'd have to take the base3 logarithm more than 7 trillion times before you got something manageable. And to deal with g_2 in a similar way, you'd need log******...***, with a total of roughly g_1 stars.
And g_64 itself was blown right the fuck out of the water at the end of the very first page of the My number is bigger! thread, by a number bigger than g_{g_26}, where g_{64} is Graham's number, and g_{65} already surpasses the xkcd number.
There have been 1210 more posts in that thread since then.
Last edited by gmalivuk on Fri Jan 18, 2013 10:58 am UTC, edited 1 time in total.
Re: 1162: "Log Scale"
emilv wrote:zukenft wrote:Can we use the others for nuclear fission? I ask because I know very little about this subject.
The other fuels only contain hydrogen, carbon and oxygen. To my knowledge only atoms heavier than iron produce more energy when split than what is needed to split them.
Using nuclear fusion with these fuels would generate huge amounts of energy, but the technology for doing that reliably (on earth) doesn't exist yet.
Last edited by Condor70 on Fri Jan 18, 2013 11:40 am UTC, edited 1 time in total.
Re: 1162: "Log Scale"
Instead of juryrigging the Base 10 system of counting to do a job it's not cut out for, why not just use Base Googolplex?
Then, no matter how big the stack of cards, its quantity can be written out in a single symbol.
Then, no matter how big the stack of cards, its quantity can be written out in a single symbol.
Re: 1162: "Log Scale"
philip1201 wrote:keithl wrote:More importantly though, there simply is no significant amount of natural antimatter in the accessible universe. It's the most efficient method of energy storage, but you can't power anything with it, unless you have FTL travel, wormholes, or something similar, and can find antimatter somewhere out there in the multiverse.
Or you could just find a practical way of violating CP symmetry.
Re: 1162: "Log Scale"
I try to do the same thing when I try to point out the people why electric cars have been dead for 100 years and will still be dead in 100 years and that the time and money is better spent perusing Algae Diesel or other alternatives while continuing to develop current reserves.
Re: 1162: "Log Scale"
if4124l wrote:K^2 wrote:I tried writing the number of iterations on some paper, but the stack of pages with that number won't fit in the room. Please advise.
Just use the Knuth PaperStack Notation again. If the number of iterations of that stack still doesn't fit in its note, repeat.
The room, well, more of a warehouse, is now almost completely filled with stacks of pages. The number for the last stack still, stubbornly, refused to fit in the card. While trying to get more paper to write on, I have discovered that I no longer know where the exit is. All the stacks look the same, and every time I thought I recognized the passage, it lead to a dead end. GPS on this phone is nowhere near precise enough to be of any use. I have considered leaving a paper trail to mark visited passages, but I can't find any stack short enough to get some pages off the top, and I dare not pull pages from the bottom of any of these stacks for fear if being buried underneath. None of these look stable. Naturally, climbing isn't an option either.
Please advise.
Re: 1162: "Log Scale"
gmalivuk wrote:PaperStack Notation for a number n really just gives you log* n (the number of times you need to take its logarithm before you get something less than or equal to 1), scaled and shifted by a couple constants depending on the base you're using and how many digits you can fit in one room's worth of stacked paper.
For truly large numbers, this method doesn't even come close to being useful. Using base3 logarithms and uparrow notation:
log* (3^3) = log* (3↑↑2) = 2
log* (3↑↑3) = 3
log* (3↑↑↑3) = log* (3↑↑(3↑↑↑2)) = 3↑↑↑2 = 3↑↑3 = 3^27 = 7,625,597,484,987
(...)
The Knuth PaperStack Notation is already a parody of Knuth's uparrow notation...
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Re: 1162: "Log Scale"
Actually, let's go ahead and figure this out. And by "let's" I mean "I'm now going to".gmalivuk wrote:PaperStack Notation for a number n really just gives you log* n (the number of times you need to take its logarithm before you get something less than or equal to 1), scaled and shifted by a couple constants depending on the base you're using and how many digits you can fit in one room's worth of stacked paper.
If we use an appropriately sized font, let's suppose we can fit 10,000 digits on a single piece of paper, and let's assume that each piece of paper is 0.1mm thick. So in a 3m high room, we can fit 30,000 sheets in a stack, or 300,000,000 digits. So if we're writing in base10, the number on the note card is log* n when using base 10^300000000 logarithms. (This is assuming that when the original number already fits in the room, we consider it the first iteration and write a 1 on the card. If we start at 0, then obviously it'll be log*n  1.)
If we play fast and loose with our approximations (as in, we round orders of magnitude of orders of magnitude), we can simplify this by rounding off to a font size 1/333 as big, and guesstimate about 10 billion digits per room. Each logarithm in base 10^(10^10) is worth three in base 10, so PaperStack Notation results in a note card with floor((log_{10}* n)/3) written on it.
Roughly.
Until we leave the googolplex in the dust, which already happened above with the comparatively quite small 3↑↑↑3squonk wrote:Instead of juryrigging the Base 10 system of counting to do a job it's not cut out for, why not just use Base Googolplex?
Then, no matter how big the stack of cards, its quantity can be written out in a single symbol.
Re: 1162: "Log Scale"
Crosshair wrote:
Gasses? What's a gass?
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Re: 1162: "Log Scale"
The Old Wolf wrote:It's called a "log scale" because that's what you need to make all the paper required for the stacks.
We might be barking up the wrong forum with these jokes.
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Re: 1162: "Log Scale"
squonk wrote:Instead of juryrigging the Base 10 system of counting to do a job it's not cut out for, why not just use Base Googolplex?
Then, no matter how big the stack of cards, its quantity can be written out in a single symbol.
Get back to me when you have the list of possible symbols...
Re: 1162: "Log Scale"
jgh wrote:Gasses? What's a gass?
gasses plural of gas (Noun)
Noun
The English language is funny sometimes with plurals.
Re: 1162: "Log Scale"
I always thought it was just "gases"
Um, this post feels devoid of content. Good luck?
For comparison, that means that if the cabbage guy from Avatar: The Last Airbender filled up his cart with lettuce instead, it would be about a quarter of a lethal dose.
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Re: 1162: "Log Scale"
Crosshair wrote:I try to do the same thing when I try to point out the people why electric cars have been dead for 100 years ...snip...
I don't get what you mean when you say you try to do the same thing; sure, you're presenting it without using a log scale, but who would ever present that graph on a log scale anyway? It doesn't even go up to 11.
Edit: d'oh... it does go up to 11. So much for that joke. But still, it doesn't have anywhere near the sort of ratios that people would use logs for.
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Re: 1162: "Log Scale"
fyjham wrote:Damn you rhomboidal, you made me curious...
I went and looked and found a site quoting for biomass details 17 MJ per kg of paper (ref) & an A4 sheet will average about 5g (ref). If we assume they're right and that the sheet at the front is A4 (Admittedly it looks a bit bigger, but I'm going to guess the stick figure is standing a long way into the background ), the sugar line at 19 takes up 5px of the available 194px of the page, so it's safe to assume it's about 1/20th of a page height, and the uranium cropoff looks to be approximately 1/3rd of the sheet's width, so 19 MJ/kg from uranium would up 1/60th of an A4 sheet. Ignoring the heading since it's not that big relatively speaking, the uranium in pages of paper would be 76,000,000 / 19 / 60 = 66666 2/3 sheets of paper. * 17 MJ/kg * 0.005 g per sheet = 5666 2/3 MJ. It'd only be about 59.4km long of paper hanging off the top of the page. The odds I've gotten something wrong above is probably around 100%
I don't think you compensated for the weight of the ink. Slacker!
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Re: 1162: "Log Scale"
zukenft wrote:this comparison was bad, because the energy density in uranium is due to nuclear fission, while the others are due to chemical reaction.
You are free to compare the amount of energy that is able to be obtained by nuclear fission of fat, coal, gasoline, etc.
Oh, that's right — things like fat and coal are nonfissionable materials! (or if they are, we haven't yet figured out how to do it)
Re: 1162: "Log Scale"
RAGBRAIvet wrote:Oh, that's right — things like fat and coal are nonfissionable materials! (or if they are, we haven't yet figured out how to do it)
Yes, we have figured it out! We can split carbon and oxygen atoms (on a very small scale). But splitting atoms lighter than iron (and nickel) requires energy instead of generating it.
The reverse is true for nuclear fusion. And yes, we have figured out how to do that too (just not on a scale that actually delivers usable energy).
Last edited by Condor70 on Tue Jan 22, 2013 9:44 am UTC, edited 1 time in total.
Re: 1162: "Log Scale"
So should I be driving a nuclear powered car or an electric car whose power comes from nuclear power. I can certainly imagine a business model in which the former is leased to the end user and then returned. The advantage would be that fuel and maintenance of the power plant would be covered by warranty for the time of the lease. Miles should not matter.