## 1162: "Log Scale"

This forum is for the individual discussion thread that goes with each new comic.

Moderators: Moderators General, Prelates, Magistrates

The Synologist
Posts: 41
Joined: Tue Sep 25, 2012 5:50 pm UTC

### 1162: "Log Scale"

Alt text: "Knuth Paper-Stack 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
Posts: 797
Joined: Wed Jun 15, 2011 5:25 pm UTC
Contact:

### Re: 1162: "Log Scale"

Burning all the paper required could keep you rolling in joules.

fyjham
Posts: 240
Joined: Wed Nov 21, 2007 1:16 am UTC

### 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 crop-off 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.

Quicksilver
Posts: 437
Joined: Wed Apr 29, 2009 6:21 am UTC

### Re: 1162: "Log Scale"

Sif not just make an excel chart and enlarge it to suit with A3 paper.

zzyss
Posts: 5
Joined: Fri Sep 02, 2011 6:48 am UTC

### Re: 1162: "Log Scale"

Edward Tufte would be proud.

algorerhythms
Posts: 44
Joined: Fri Nov 02, 2007 4:23 pm UTC

### 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.
The nerdy webcomic that I update on Saturdays: Cesium Comics

zukenft
Posts: 44
Joined: Mon Dec 20, 2010 11:34 am UTC

### 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.

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

### Re: 1162: "Log Scale"

xkcd wrote:Alt text: "Knuth Paper-Stack 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?
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

Djehutynakht
Posts: 1546
Joined: Thu Feb 10, 2011 1:37 am UTC

### Re: 1162: "Log Scale"

skeptical scientist wrote:
xkcd wrote:Alt text: "Knuth Paper-Stack 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.

keithl
Posts: 660
Joined: Mon Aug 01, 2011 3:46 pm UTC

### Re: 1162: "Log Scale"

Uranium is for quitters who can't find enough antimatter to power a Kardeshev Type V civilization properly.

alvinhochun
Posts: 54
Joined: Wed Nov 14, 2012 3:07 pm UTC

### Re: 1162: "Log Scale"

It doesn't seem like a roll of toilet paper unrolled.

ijuin
Posts: 1110
Joined: Fri Jan 09, 2009 6:02 pm UTC

### Re: 1162: "Log Scale"

Or for those primitives who haven't learned how to handle antimatter yet.

Bernkastel
Posts: 13
Joined: Fri Nov 30, 2012 9:04 am UTC

### Re: 1162: "Log Scale"

Graphical representations are for quitters who can't comprehend number sizes.

The Old Wolf
Posts: 22
Joined: Fri Aug 21, 2009 7:28 am UTC
Location: Not Denver, but we have better powder.

### 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.
"The greatest insanity is surely to see the world only as it is, and not as it might be."
-Miguel de Cervantes

Condor70
Posts: 18
Joined: Wed Jan 11, 2012 8:11 am UTC

### 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)?

emilv
Posts: 4
Joined: Wed May 25, 2011 9:36 am UTC

### 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.

K^2
Posts: 68
Joined: Mon Feb 18, 2008 6:33 am UTC

### 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.

nathanmacinnes
Posts: 11
Joined: Fri Jun 24, 2011 8:51 am UTC

### 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.

Pfhorrest
Posts: 5447
Joined: Fri Oct 30, 2009 6:11 am UTC
Contact:

### Re: 1162: "Log Scale"

Yeah, if we wanted to compare absolute strict theoretical energy density we would just measure the total mass-energy per volume. And we already have a term for that measure. It's called density.
Forrest Cameranesi, Geek of All Trades
"I am Sam. Sam I am. I do not like trolls, flames, or spam."
The Codex Quaerendae (my philosophy) - The Chronicles of Quelouva (my fiction)

philip1201
Posts: 201
Joined: Tue Nov 03, 2009 6:16 am UTC

### 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.

if4124l
Posts: 2
Joined: Mon Nov 12, 2012 3:07 pm UTC

### 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 Paper-Stack Notation again. If the number of iterations of that stack still doesn't fit in its note, repeat.

gmalivuk
GNU Terry Pratchett
Posts: 26767
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: 1162: "Log Scale"

Paper-Stack 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 base-3 logarithms and up-arrow 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 base-b 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 base-3 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 gg_26, where g64 is Graham's number, and g65 already surpasses the xkcd number.

Last edited by gmalivuk on Fri Jan 18, 2013 10:58 am UTC, edited 1 time in total.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

Condor70
Posts: 18
Joined: Wed Jan 11, 2012 8:11 am UTC

### Re: 1162: "Log Scale"

emilv wrote:

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.

squonk
Posts: 131
Joined: Fri May 21, 2010 12:25 pm UTC

### 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.

Tass
Posts: 1909
Joined: Tue Nov 11, 2008 2:21 pm UTC
Location: Niels Bohr Institute, Copenhagen.

### 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.

Crosshair
Posts: 33
Joined: Mon Aug 02, 2010 11:45 am UTC

### 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.

K^2
Posts: 68
Joined: Mon Feb 18, 2008 6:33 am UTC

### 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 Paper-Stack 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.

Klear
Posts: 1965
Joined: Sun Jun 13, 2010 8:43 am UTC
Location: Prague

### Re: 1162: "Log Scale"

gmalivuk wrote:Paper-Stack 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 base-3 logarithms and up-arrow 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 Paper-Stack Notation is already a parody of Knuth's up-arrow notation...

gmalivuk
GNU Terry Pratchett
Posts: 26767
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: 1162: "Log Scale"

gmalivuk wrote:Paper-Stack 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.
Actually, let's go ahead and figure this out. And by "let's" I mean "I'm now going to".

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 base-10, 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 Paper-Stack Notation results in a note card with floor((log10* n)/3) written on it.

Roughly.

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.
Until we leave the googolplex in the dust, which already happened above with the comparatively quite small 3↑↑↑3
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

jgh
Posts: 145
Joined: Thu Feb 03, 2011 1:04 pm UTC

### Re: 1162: "Log Scale"

Crosshair wrote:

Gasses? What's a gass?

peewee_RotA
Posts: 504
Joined: Mon Dec 12, 2011 1:19 pm UTC

### 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.
"Vowels have trouble getting married in Canada. They can’t pronounce their O’s."

rmsgrey
Posts: 3632
Joined: Wed Nov 16, 2011 6:35 pm UTC

### 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...

Invertin
Posts: 127
Joined: Fri Mar 20, 2009 8:08 pm UTC

### 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.

eculc
Wet Peanut Butter
Posts: 451
Joined: Mon Jun 27, 2011 4:25 am UTC

### 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.

Invertin
Posts: 127
Joined: Fri Mar 20, 2009 8:08 pm UTC

### Re: 1162: "Log Scale"

It might be both

Angelastic
Posts: 700
Joined: Thu Nov 03, 2011 8:36 am UTC
Location: .at (let's see what's through here!)
Contact:

### 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.
Knight Temporal, and Archdeacon of buttermongery and ham and cheese sandwiches. Nobody sells butter except through me.
Smiley by yappobiscuits. Avatar by GLR, buffygirl, BlitzGirl & mscha, with cari.j.elliot's idea.
Haiku Detector
starts a trend to make way for
my robot army.

Posts: 57
Joined: Fri Aug 21, 2009 1:29 pm UTC

### 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 crop-off 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!

What about this Thorium stuff? I know it's been tried for ages with little proof of success, but it keeps showing up on my FB stream as THE thing.

RAGBRAIvet
Posts: 129
Joined: Tue Apr 05, 2011 12:50 pm UTC
Location: 43° 53' 03" -91° 14' 06"

### 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 non-fissionable materials! (or if they are, we haven't yet figured out how to do it)

Condor70
Posts: 18
Joined: Wed Jan 11, 2012 8:11 am UTC

### Re: 1162: "Log Scale"

RAGBRAIvet wrote:Oh, that's right — things like fat and coal are non-fissionable 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.

rcox1
Posts: 112
Joined: Wed Nov 11, 2009 11:23 pm UTC

### 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.