## 0687: "Dimensional Analysis"

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eboyce
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### Re: "Dimensional Analysis" discussion

While revising for a physics doctoral exam, I realised that fuel economy has units of inverse area, and so miles per gallon was equivalent to cm^{-2}, inverse barns etc.
My dimensional abuse was topped by a fellow student who explained the meaning of this area. If you take the fuel burned by a car during a trip, and spread it out in a column along the length of the trip, the cross-sectional area of the column is the (inverse of the) fuel economy.
Today Randall puts us both to shame.

Zorlin
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### Re: "Dimensional Analysis" discussion

Salamnzer wrote:
England drifted out to sea...

And nothing of value was lost.
ouch, that's cold man
Just like England.
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sabik
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### Re: "Dimensional Analysis" discussion

eboyce wrote:While revising for a physics doctoral exam, I realised that fuel economy has units of inverse area, and so miles per gallon was equivalent to cm^{-2}, inverse barns etc.
My dimensional abuse was topped by a fellow student who explained the meaning of this area. If you take the fuel burned by a car during a trip, and spread it out in a column along the length of the trip, the cross-sectional area of the column is the (inverse of the) fuel economy.
Today Randall puts us both to shame.

That's actually quite a neat visualisation, and might even occasionally have use; for instance, some proposed spacecraft launch techniques lay out the fuel in front of the craft for use by the first stage. The Prius consumptions quoted earlier would be equivalent to 10 human hairs.

sabik
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### Re: "Dimensional Analysis" discussion

squelart wrote:I bet this is related to the "God Equation" nonsense.

Doubt it... more likely the Fine Structure Constant. Which has pretty much the form given in the comic.

macrocephalic
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### Re: "Dimensional Analysis" discussion

He failed to define that he was using the ton-pace-minute system to get the figure for Planck energy. Is he being a typical American and implying that the use of metric units will destroy the earth?

Salag
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### Re: "Dimensional Analysis" discussion

Howdy people,

I've been reading this comic almost since it started and only just registered to say that surely these calculations should be done with mpg converted to km?

Khorbin
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### Re: "Dimensional Analysis" discussion

squelart wrote:I bet this is related to the "God Equation" nonsense.

That was my first thought, except, as PZ points out in that article you linked to, the "God Equation" uses arbitrary made up values like Omega (0.0123456789), the units don't cancel in that equation, and the value for C is in "megalithic yards" which is basically a fudge-factor to get the value that they wanted.

Therefore, XKCD's God Equation is far superior.

hthall
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### Re: "Dimensional Analysis" discussion

macrocephalic wrote:He failed to define that he was using the ton-pace-minute system to get the figure for Planck energy. Is he being a typical American and implying that the use of metric units will destroy the earth?

Salag wrote:Howdy people,

I've been reading this comic almost since it started and only just registered to say that surely these calculations should be done with mpg converted to km?

We've been through this before. The whole point of the dimensional analysis is that it does not matter what units you choose to measure the various quantities with, the dimensionless constant on the right will still have the same numerical value, as long as you cancel the units correctly and consistently. (For example, 1 mile / 1 kilometer does not equal one, it equals the dimensionless quantity 1.609344.) The number on the right has the same units as the numerical constants of mathematics; it has units of one per one. That's what it means to be dimensionless, and only once you've found a way to cancel all of the units does it make any sense to look for any special meaning in the number you get.

EDIT: Doh. km/mi is what cancels the 1.6.

Professor: "So we see that this alternating series of rational numbers sums to pi/4." Student: "Wait, is that pi/4 in kilometers, or miles?"
Last edited by hthall on Mon Jan 11, 2010 6:48 am UTC, edited 1 time in total.
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SocialSceneRepairman
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### Re: "Dimensional Analysis" discussion

Of course, he really ought to be using SI units anyway.

(137th post!)

Although there is one minor problem, in that most countries measure fuel economy in volume per unit distance (i.e. L/100km), not the other way around. So it should be on the denominator.

dtilque
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### Re: "Dimensional Analysis" discussion

westrim wrote:I thought it was California that would drift out to sea first. No fair, England...

Nah, California is going to slide into the ocean, not drift out to sea. There's a difference.

Seriously, the parts of CA west of the San Andreas Fault are moving northwest while the rest is moving west. The friction between the two is what causes all the earthquakes. But this also means that in 10 million years or so, LA (that part west of the fault) will be in the San Francisco area. In 50m years, it'll be in the far north Pacific being subducted under the Aleutian Islands.

The English Channel, on the other hand, was eroded by a glacial meltwater lake which cut through a ridge connecting Great Britain to the continent some half million or so years ago. So England is not moving relative to France. However, global warming will raise the sea level, meaning the channel will become wider. So the Prius will have to increase its mileage to keep the equation valid.
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Red Hal
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### Re: "Dimensional Analysis" discussion

dtilque wrote:However, global warming will raise the sea level, meaning the channel will become wider. So the Prius will have to increase its mileage to keep the equation valid.
Which is completely consistent with improving technology.
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dennisw
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### Re: "Dimensional Analysis" discussion

petiejoe wrote:The Pi on the right side of the equation isn't lowercase pi (a particular constant that equals 3.14159... and is most notably used in a variety of geometric and trigonometric formulas, but also has strange applications elsewhere), it's uppercase Pi as used in the Buckingham Pi Theorem http://en.wikipedia.org/wiki/Buckingham_Pi . The actual value of this Pi depends completely on the units used, but if you're consistent with the units then the constant will be valid (if the systems truly are similar, which obviously these aren't - but that's part of what makes it a joke).

And yes, the dimensional analysis appears to be correct.

It looks like a lowercase pi to me.

$\text {lowercase:} \; \; \pi \; \; \; \text {capital:} \; \; \Pi$
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westrim
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### Re: "Dimensional Analysis" discussion

dtilque wrote:
westrim wrote:I thought it was California that would drift out to sea first. No fair, England...

Nah, California is going to slide into the ocean, not drift out to sea. There's a difference.

Seriously, the parts of CA west of the San Andreas Fault are moving northwest while the rest is moving west. The friction between the two is what causes all the earthquakes. But this also means that in 10 million years or so, LA (that part west of the fault) will be in the San Francisco area. In 50m years, it'll be in the far north Pacific being subducted under the Aleutian Islands.

The English Channel, on the other hand, was eroded by a glacial meltwater lake which cut through a ridge connecting Great Britain to the continent some half million or so years ago. So England is not moving relative to France. However, global warming will raise the sea level, meaning the channel will become wider. So the Prius will have to increase its mileage to keep the equation valid.

Actually all of LA and it's metropolitan area (though not the entire county) are on the west side of the fault. 'Tis fun living by a strike-slip fault, and at least we won't get subducted or mashed into mountains for a while. But hey, in 10 million years Sacramento might actually start paying attention to our issues. Still won't do us any good in the Fed's graces (about 90 cents is returned to California for every dollar in taxes- income and otherwise- we send to DC. Alaska gets 1.40.)

Even with the lakebursts, the Channel was still tundra until the end of the last ice age. But regardless, it is geologically very definitely part of Europe, so It's not drifting anywhere without everything in its vicinity coming along for the ride.

...Ok, I'm done waving my penis and making sure you know that I knew what I was talking about, but was playing along with the comic and the 2012 references in the thread that I expected. Don't make me whip out the Mediterranean and the Black Seas too.

cheezewizz2000
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### Re: "Dimensional Analysis" discussion

Another fun one is this:

acceleration due to gravity (g) = 9.81ms^-2
length of the year (y) = 31,556,736s
speed of light (c)= 299,792,458ms^-1

g * y = ~c

ohki
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### Re: "Dimensional Analysis" discussion

hthall wrote:
Salag wrote:Howdy people,

I've been reading this comic almost since it started and only just registered to say that surely these calculations should be done with mpg converted to km?

We've been through this before. The whole point of the dimensional analysis is that it does not matter what units you choose to measure the various quantities with...

Now now, let's not get hasty. He said to represent miles-per-gallon with kilometers. I'm not sure how you convert distance/volume to just a distance, but it if you did, the final result of that equation would be an area, rather than be dimensionless

I suppose converting mpg to km would actually imply another term with the dimensions 1/len^2 and that the end result would be dimensionless after all, but where's the humor in that?
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TheoGB
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### Re: "Dimensional Analysis" discussion

So Wales, Scotland and Ireland stay where they are? England's not going to move much, then...

Blackblaze00
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### Re: "Dimensional Analysis" discussion

Well, using the averaged EPA fuel economy of a 2010 Toyota Prius (49.5 mpg), and an ~11% error of the averaged pressure of the earth's core (assuming the average is taken with the maximum and minimum values of the approximate range of 330 GPa - 360 GPa), and then using the standard values for Planck energy and the minimum width of the English Channel, you get to pi within ~3 decimal places, i.e. 3.141

In all honesty, that's simply brilliant. One of the many reasons why I love xkcd; I learn something amazingly awesome, yet completely useless, 3 times a week.

http://www.wolframalpha.com/input/?i=%280.5433+MWh+%2F+385.47+GPa%29+*+%28%28%2851%2B48%29%2F2%29+mpg+%2F+34+km%29

AL_
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### Re: "Dimensional Analysis" discussion

Isn't "mpg" really a "distance per unit energy" term if you're strict about what it is representing, rather than a "distance per unit volume"? It's a measure of how far you'll move the vehicle with the potentially available energy from a given volume of fuel, or thereabouts.

tahrey
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### Re: "Dimensional Analysis" discussion

Zorlin wrote:
Salamnzer wrote:
England drifted out to sea...
And nothing of value was lost.
ouch, that's cold man
Just like England.

As a resident of one of the slightly darker grey patches in the middle of that mottled white polygon, having lazily drifted my way into work perpendicular to the direction of travel on summer tyres with 130mph showing on the clock, may I say.... damn! Beat me to it!

Also on the point of "so what of Wales, Scotland, and Berwick on Sea?" (Ireland, Man, Scillys, Orkneys, Shetlands, Wight, Skye, Hebrides both inner and outer and Channel Is. etc all being for the sake of this argument seperate landmasses). It is indeed the English channel (or la Manche if you prefer your onions on the soupy side), but it's the British Isles, in this particular case the island of "Great" Britain.

So there.

I do like this though

Coupla three other things tho:
Complaining that the equation doesn't account for countries where it's L/100km and all: So? That's irrelevant. The equation is given in terms of "EPA Gas Mileage", which is provided as a US MPG figure, ie a single number that uses imperial miles (1609.3m) and minuscule heathen United States "Standard" gallons (3.8 litres or so... whatever's 4/5ths of 4.546) as its basis. If you want to use L/100km either pre-convert it (i think that requires dividing 235 into your figure, if I've not ballsed up my calculations?) or shove in your own additional terms to fudge it compensate.

Britain has already drifted out to sea: Not so much. The English channel, if you remove the water, looks more like a river gorge, possibly of glacial origin ultimately. The river between the monstrous lake of what has become the north sea, and the atlantic, flowed, carved out the land, produced some reasonably impressive cliff canyons where the ground was soft chalk (and floodplains either side)....... then the big melt came, and/or the ground sank somewhat, and the sea rose to permanently flood those areas. We're not exactly a seismically active area, at least not any more. I'd be fairly confident on saying that France, GB and Ireland are all really just mountainous plateaux on the same bedrock.

And what's this stuff about economy being an area? OK, I figured out that technically it's true (a standard-ish hatchback of about 8L/100km (35mpg UK, 29 US) gives an area of 0.08mm2, or a square 0.28284mm on a side... but what the hell does it actually represent? It's not a figure you can actually use for much, and it's not like you can pick up 2-dimensional squares and throw them around. The only thing I can see is it being a universal economy figure... but that's not going to happen, it'll end up being a fifth little-used standard (besides 2x MPG, L/100km, and km/L). What MPG (and L/100km) really represent are comparitive rate measurements - your distance along a 1-dimensional line (how far you've driven, or in fact how many times your differential has rotated) vs a total flow of a 1-dimensional stack of petroleum molecules (as volume is but a certain molar quanitity at a particular temperature and pressure - in this case 1 bar and 20 celcius).
So really, what?

tahrey
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### Re: "Dimensional Analysis" discussion

AL_ wrote:Isn't "mpg" really a "distance per unit energy" term if you're strict about what it is representing, rather than a "distance per unit volume"? It's a measure of how far you'll move the vehicle with the potentially available energy from a given volume of fuel, or thereabouts.

Ah but it's not, unless you count on the same mix of the same grade of the same fuel type. EG in the UK, 95RON petrol (gas), 98RON, Diesel (and normal/city-clean/"ultimate" types of each) and LPG (CNG) are all sold by the litre (with different prices), and economy for vehicles using these disperate fuels all given as MPG... despite each having significant differences in usable stored energy per litre at STP, and in the case of LPG not even a fixed volume if someone chooses to mess with the compression level... But people will still complain about how much extra diesel costs or vaunt the improved consumption of vehicles using it, even though it's more energy dense --- and the opposite for LPG.

So in absolute, literal, government economy figure terms, it's distance per volume. I've had vehicles tuned to run on 91RON or worse petrol, again with markedly different energy characteristics, and not actually able to work to their best on our standard juice, but uk.gov (and usa.gov, etc) would still give it a same-as-the-rest MPG rather than a "miles per gigajoule" or whatever.

starabo
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### Re: "Dimensional Analysis" discussion

dtilque wrote:
westrim wrote:ns that in 10 million years or so, LA (that part west of the fault) will be in the San Francisco area. In 50m years, it'll be in the far north Pacific being subducted under the Aleutian Islands.

The English Channel, on the other hand, was eroded by a glacial meltwater lake which cut through a ridge connecting Great Britain to the continent some half million or so years ago. So England is not moving relative to France. However, global warming will raise the sea level, meaning the channel will become wider. So the Prius will have to increase its mileage to keep the equation valid.

I see a catch 22 coming on here.... if the prius fuel economy increases, then (perhaps) you could assume that similarly other cars (and industries) are reducing their emissions also. This (may) then reduce the onset of climate change meaning that the channel wouldn't increase with sea levels. Therefore Toyota would have to start making a thirstier Prius ...

Ozzah
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### Re: "Dimensional Analysis" discussion

Note sure why this isn't working, but could probably work this out on paper if I wasn't too lazy:
http://www.wolframalpha.com/input/?i=Solve[(0.5433+MWh+/+345+GPa)+*+(((48%2Ba*51)/2)+Mpg+/+34+Km)%3D%3DPi,+a]

Instead of working out a simple average between highway and city mileage for the Prius, you could work out a weighted average (Say, if you drive more on the highway than the city, for example).

Substituting this weighted average for some given ratio of highway to city driving, we can solve to find a solution which is exactly Pi.

Dobblesworth
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### Re: "Dimensional Analysis" discussion

Could someone get to quantifying the likely enlargement distance of the English channel for Prius unit efficiency improvements? As a resident of North-East England, and assuming only England is drifting and that the island of Ireland will be getting a kick up the arse from Cornwall, the south coast of Wales with Cardiff and Swansea will take a beating from Bristol's advance, and that the Flower of Scotland will be a tectonic immovable object... I'd just like to know how much of that vehicle's development I need to sabotage to stop my home getting mauled by contact with the Grampians.

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### Re: "Dimensional Analysis" discussion

sab39 wrote:Either Randall (via his stick man avatar) means Britain will drift further out to sea, or Scotland and Wales are about to get very squished.
Or he means the United Kingdom and what's left of the dispute over Northern Ireland is about to get a lot more interesting.

Two way misunderstanding:

Brits: "There certainly is a big difference between English, Welsh, Scotch, ... . Why don't these non-Brits understand?!"
non-Brits: "Whatever difference between English, Welsh, Scotch, ... . Why do these Brits think we care?!"

spekkeh
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### Re: "Dimensional Analysis" discussion

I never believed much of the global warming bubble, but if buying a new Prius means more sea in between us decent people and the Britons, then I'm ordering one this instant.

Felstaff
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### Re: "Dimensional Analysis" discussion

sab39 wrote:Either Randall (via his stick man avatar) means Britain will drift further out to sea, or Scotland and Wales are about to get very squished.
Or he means the United Kingdom and what's left of the dispute over Northern Ireland is about to get a lot more interesting.

Two way misunderstanding:

Brits: "There certainly is a big difference between English, Welsh, Scotch, ... . Why don't these non-Brits understand?!"
non-Brits: "Whatever difference between English, Welsh, Scotch, ... . Why do these Brits think we care?!"

I'm sure there's a difference between the the Dutch and the Belgians, but they're pretty much the same. No-one can name 3 famous Belgians, however. Except Poirot.
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ianf
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### Re: "Dimensional Analysis" discussion

Felstaff wrote:I'm sure there's a difference between the the Dutch and the Belgians, but they're pretty much the same. No-one can name 3 famous Belgians, however. Except Poirot.

Actually, it's pretty easy to name 3 famous Belgians - the problem is that many people would not recognise them as Belgian. For example Magritte, Jacques Brel, Plastic Bertrand are all famous Belgians, but many people would think they are French.

Nanaki09
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### Re: "Dimensional Analysis" discussion

Prius combined MPG* = 49.5 MPG
English Channel minimum width = 34 km

am I the only one that sees an inconsistency between miles per gallon and kilometres?

surely it should either all be imperial, -or- SI units? (kilometres per litre)

sabik
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### Re: "Dimensional Analysis" discussion

tahrey wrote:And what's this stuff about economy being an area? OK, I figured out that technically it's true (a standard-ish hatchback of about 8L/100km (35mpg UK, 29 US) gives an area of 0.08mm2, or a square 0.28284mm on a side... but what the hell does it actually represent? It's not a figure you can actually use for much, and it's not like you can pick up 2-dimensional squares and throw them around. The only thing I can see is it being a universal economy figure... but that's not going to happen, it'll end up being a fifth little-used standard (besides 2x MPG, L/100km, and km/L). What MPG (and L/100km) really represent are comparitive rate measurements - your distance along a 1-dimensional line (how far you've driven, or in fact how many times your differential has rotated) vs a total flow of a 1-dimensional stack of petroleum molecules (as volume is but a certain molar quanitity at a particular temperature and pressure - in this case 1 bar and 20 celcius).
So really, what?

Well, if, instead of having a fuel tank your car was consuming a "string" of fuel that had been laid out for it, it would have to be as big as 10 human hairs.

At the other end, if you do emission volume per km, you can visualise how big a trail a car would leave if it were a solid trail rather than particulates and gas.

sabik
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### Re: "Dimensional Analysis" discussion

Nanaki09 wrote:am I the only one that sees an inconsistency between miles per gallon and kilometres?

surely it should either all be imperial, -or- SI units? (kilometres per litre)

Well, the thing is that it doesn't matter for the calculation. You just need to make sure you keep track of the units throughout — and today's software will do that for you, so it's no real effort.

At that point, it's just a stylistic thing; and if you're looking up the various quantities in different sources which use different styles, it's probably better to quote them as they are in the first instance and convert them later (as part of the calculation).

flguy1980
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### Re: "Dimensional Analysis" discussion

Time to get out Mr. TI-89:

(sqrt( (h/(2*pi))(c^5/G) ) / 330 GPa) * ( 50 mpg / 34 km ) = 3.706

My TI-89 only knows about the reduced Planck's constant, so a trip to Wikipedia fixed that.

I guess it's time to redefine Pi. Oh wait... http://en.wikipedia.org/wiki/Indiana_Pi_Bill

The ironic part here is that the proper pi is involved in the equation.
Last edited by flguy1980 on Mon Jan 11, 2010 12:58 pm UTC, edited 1 time in total.

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### Re: "Dimensional Analysis" discussion

Except, most places I've seen have the pressure at more like 360GPa, and gas mileage under 50, which gives a number much closer to Pi.
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Big_Nose
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### Re: "Dimensional Analysis" discussion

I can confirm that the numbers (roughly, depends where you get your values from) works out if you do the conversion from miles and gallons to metres and cubic metres respectively. I got 3.68 based on an MPG of 55.

I set about trying to work out the maximum Prius gas mileage allowed before Britain drifts into something, but noticed that the island is small enough to drift out of europe, round the cape of good hope and over the other side of the world to the pacific ocean. Assuming that the upper bound for the width of the English channel (at sea level, great circle distance) is approximately half the earth's circumference and the earth's pressure stays constant, if a future Prius has a gas mileage in excess of 31000mpg, Britain will be forced to take off into space!

Curiously tho, thanks to this wonderful law, a Prius will always be able to drive on a hypothetical bridge between England and France in less than one gallon of fuel. It also follows that if someone built a bridge between England and France, a Prius should never be allowed to have it's mileage measured there (because as the mileage increases, so will the length of the bridge, exponentially in-fact, as the distance from the car to it's start point will also be increasing, thus increasing the effective mileage whilst the car drives...dangerous!)

I clearly think these things through way too much...

flguy1980
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### Re: "Dimensional Analysis" discussion

gmalivuk wrote:Except, most places I've seen have the pressure at more like 360GPa, and gas mileage under 50, which gives a number much closer to Pi.

Change the pressure to 360 GPa, and we're at 3.398. I used a 2010 Prius's new combined MPG from fueleconomy.gov

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### Re: "Dimensional Analysis" discussion

ianf wrote:
Felstaff wrote:I'm sure there's a difference between the the Dutch and the Belgians, but they're pretty much the same. No-one can name 3 famous Belgians, however. Except Poirot.

Actually, it's pretty easy to name 3 famous Belgians - the problem is that many people would not recognise them as Belgian. For example Magritte, Jacques Brel, Plastic Bertrand are all famous Belgians, but many people would think they are French.

Well, actually there are as many of these misunderstandings as there are layers:
...
Belgians: "Of course there is a difference between Vlaanderen (=Dutch speaking) and Wallon (=French)".
Dutch: "Thats just language. But we live in another country!"
Europeans: "We don't care, you both play the same roll in the EU. Only need to know the big nations".
Other earth inhabitants: "Wasn't Belgium the capitol of Amsterdam, that little country where drugs are legal?"
aliens: ...

SEE
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### Re: "Dimensional Analysis" discussion

England: 51.5 million people.
Great Britain: 61.5 million people.
Percentage of a damn the distinction is worth giving: 16.3%

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### Re: "Dimensional Analysis" discussion

SEE wrote:England: 51.5 million people.
Great Britain: 61.5 million people.
Percentage of a damn the distinction is worth giving: 16.3%

Note that the corresponding quotient is dimensionless.

lawrencedavies
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### Re: "Dimensional Analysis" discussion

Come on guys.
It's not England. It's GREAT BRITAIN (emphasis on GREAT ).
Either that, or let's work out how many prius' it'll take to actually detatch England from the rest of the British Isles.

SolkaTruesilver
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### Re: "Dimensional Analysis" discussion

I think the people who posted their calculation understood something wrong. They say the calculation are not equal to Pi, well I personally think they are not using the best Prius available. Maybe there are some current prototypes that makes the equation equal pi?

BioTube
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### Re: "Dimensional Analysis" discussion

lawrencedavies wrote:It's GREAT BRITAIN (emphasis on GREAT ).
Shouldn't that be "Formerly Great Britain"?
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