I just meant in a semi-magical, non-physical way, only for visualising scale. Like a computer simulation of the world that scales up everything other than you twenty orders of magnitude, then uses some hacked-in rendering convention that lets you “see” without trouble from stuff like wavelengths.
Or if you want something more physical-like, imagine looking from “floor level” at a human statue 10 million light-years (relative to our c) in size, of correct proportions and colors (but no universe-crushing gravity), in a non-relativistic universe (to get around light-speed issues). Do you think you could tell the difference between that and a 10000 light-years one without seeing them side by side nor using instruments?
I just meant in a semi-magical, non-physical way, only for visualising scale. Like a computer simulation of the world that scales up everything other than you twenty orders of magnitude, then uses some hacked-in rendering convention that lets you “see” without trouble from stuff like wavelengths.
Then I’d see something like the ball-and-stick models that chemists build. We already know the shapes of molecules, and the photographs made of them in the last few years look just like that.
OK, sorry. It appears I’ve rolled a critical failure in communication :-)
I wasn’t referring to the small scale structure, just the ability to comprehend scale. Something like the way that when you’re at the foot of the mountain, the brain doesn’t really capture the difference between a 1km-tall and a 8-km tall one. Or how the distinction between a 10-story building and a 100-story one isn’t really manifest in the mind unless they’re side by side. Now take that and multiply both scales by enough orders of magnitude to span molecule-to-human scales.
Let me try a better example. Take this image. Without using symbolic math (i.e. actually figuring orders of magnitude and doing arithmetic with them), what can your brain do that simultaneously includes numbers of the scales “the width of one of the galaxy’s arms”, “the diameter of one of the stars” and “the height of a person on one of the planets”?
I mean, I don’t have to resort to math to know that ten people in a normal car would be crowded, or that a bucket of nails are hard to fit in a typical person’s pockets. I can have an intuitive comprehension (albeit inaccurate) of how much work might be needed to dig a small ditch. But I have no intuitive feel for similar problems posed at astronomical scales other than “intuition overflow, use math”. E.g., I’ve no chance of estimating the number of people needed to crowd just the solar system, let alone the galaxy, within a couple of orders of magnitude, unless I actually do at least a few back-of-the-envelope calculations.
Something like the way that when you’re at the foot of the mountain, the brain doesn’t really capture the difference between a 1km-tall and a 8-km tall one.
I’ve walked up a 1 km hill. 8 km is Everest. I’ve only seen mountains that big in pictures.
Or how the distinction between a 10-story building and a 100-story one isn’t really manifest in the mind unless they’re side by side.
10 storeys is the height of some of the more substantial buildings (other than skyscrapers) in central London. 100 is a skyscraper. I’m not sure there are any buildings that tall in London.
Let me try a better example. Take this image. Without using symbolic math (i.e. actually figuring orders of magnitude and doing arithmetic with them), what can your brain do that simultaneously includes numbers of the scales “the width of one of the galaxy’s arms”, “the diameter of one of the stars” and “the height of a person on one of the planets”?
From general knowledge I’m guessing 1000 to 10000 ly for the thickness of an arm and 100,000 miles for the diameter of a star. Then it’s just counting zeroes. 1 ly is 10^13 km, which is 10^13 miles. So that’s 8 zeroes from the star to the arm, and 8 zeroes from a person to a star: 100,000 miles = 100,000 km, and a person is 2 m, which is equal to 1 m. (“If anyone asks, I did not tell you it was ok to do math like this.”)
Ok, I’m figuring orders of magnitude and doing arithmetic with them, but that is intuitive to me.
For numbers of zeroes up to 15, a while back I posted some handy visualisations which I can’t find, so here they are again. Take the solid copper earth conductor from some mains cable, which is around 1mm^2 cross-section, and cut a little piece just 1mm long. Roll it between your fingertips. That’s a cubic millimetre. In your other hand pick up a 1 litre bottle of milk. You’re looking at a million. One million of those copper fragments will fill the bottle. (They will weigh 10 kg, and if you do any weight training, you’ll know what a 10 kg weight feels like.) One billion of them is enough to fill the space between the top of a largish dining table and the floor (3/4m high, top surface 1m by 4⁄3 m). One trillion will fill a few lanes of an Olympic swimming pool (50m long, 10m wide, 2m deep). Get another factor of 1000 by using coarse sand (0.1mm grain size) instead of diced copper wire, and that’s 10^15.
But I have no intuitive feel for similar problems posed at astronomical scales other than “intuition overflow, use math”. E.g., I’ve no chance of estimating the number of people needed to crowd just the solar system, let alone the galaxy, within a couple of orders of magnitude, unless I actually do at least a few back-of-the-envelope calculations.
As I say, there isn’t a boundary to me between intuition and calculation. As in, 10^24 just is, to me, about a mole, the relationship between one molecule and a handful of stuff. It’s also a lower bound on the number of operations of individual transistors you can expect a computer to perform without a single error. A billion transistors clocked a billion times a second for a million seconds, a million seconds being 1⁄30 of a year, or 12 days.
Yes, it’s possible our intuitions simply function differently.
I do the same kinds of calculations, more or less intuitively. I can juggle zeros too if I need to. But my point is that for most human-scale things I don’t need to do that. Maybe it’s just learned behavior, I’m sure an astrophysicist has better intuitions in his area of expertise. The fact that intuition triggers even in situations that are not often encountered seems to indicate there’s more to it than that, though.
I just meant in a semi-magical, non-physical way, only for visualising scale. Like a computer simulation of the world that scales up everything other than you twenty orders of magnitude, then uses some hacked-in rendering convention that lets you “see” without trouble from stuff like wavelengths.
Or if you want something more physical-like, imagine looking from “floor level” at a human statue 10 million light-years (relative to our c) in size, of correct proportions and colors (but no universe-crushing gravity), in a non-relativistic universe (to get around light-speed issues). Do you think you could tell the difference between that and a 10000 light-years one without seeing them side by side nor using instruments?
Then I’d see something like the ball-and-stick models that chemists build. We already know the shapes of molecules, and the photographs made of them in the last few years look just like that.
OK, sorry. It appears I’ve rolled a critical failure in communication :-)
I wasn’t referring to the small scale structure, just the ability to comprehend scale. Something like the way that when you’re at the foot of the mountain, the brain doesn’t really capture the difference between a 1km-tall and a 8-km tall one. Or how the distinction between a 10-story building and a 100-story one isn’t really manifest in the mind unless they’re side by side. Now take that and multiply both scales by enough orders of magnitude to span molecule-to-human scales.
Let me try a better example. Take this image. Without using symbolic math (i.e. actually figuring orders of magnitude and doing arithmetic with them), what can your brain do that simultaneously includes numbers of the scales “the width of one of the galaxy’s arms”, “the diameter of one of the stars” and “the height of a person on one of the planets”?
I mean, I don’t have to resort to math to know that ten people in a normal car would be crowded, or that a bucket of nails are hard to fit in a typical person’s pockets. I can have an intuitive comprehension (albeit inaccurate) of how much work might be needed to dig a small ditch. But I have no intuitive feel for similar problems posed at astronomical scales other than “intuition overflow, use math”. E.g., I’ve no chance of estimating the number of people needed to crowd just the solar system, let alone the galaxy, within a couple of orders of magnitude, unless I actually do at least a few back-of-the-envelope calculations.
I think our intuitions work differently.
I’ve walked up a 1 km hill. 8 km is Everest. I’ve only seen mountains that big in pictures.
10 storeys is the height of some of the more substantial buildings (other than skyscrapers) in central London. 100 is a skyscraper. I’m not sure there are any buildings that tall in London.
From general knowledge I’m guessing 1000 to 10000 ly for the thickness of an arm and 100,000 miles for the diameter of a star. Then it’s just counting zeroes. 1 ly is 10^13 km, which is 10^13 miles. So that’s 8 zeroes from the star to the arm, and 8 zeroes from a person to a star: 100,000 miles = 100,000 km, and a person is 2 m, which is equal to 1 m. (“If anyone asks, I did not tell you it was ok to do math like this.”)
Ok, I’m figuring orders of magnitude and doing arithmetic with them, but that is intuitive to me.
For numbers of zeroes up to 15, a while back I posted some handy visualisations which I can’t find, so here they are again. Take the solid copper earth conductor from some mains cable, which is around 1mm^2 cross-section, and cut a little piece just 1mm long. Roll it between your fingertips. That’s a cubic millimetre. In your other hand pick up a 1 litre bottle of milk. You’re looking at a million. One million of those copper fragments will fill the bottle. (They will weigh 10 kg, and if you do any weight training, you’ll know what a 10 kg weight feels like.) One billion of them is enough to fill the space between the top of a largish dining table and the floor (3/4m high, top surface 1m by 4⁄3 m). One trillion will fill a few lanes of an Olympic swimming pool (50m long, 10m wide, 2m deep). Get another factor of 1000 by using coarse sand (0.1mm grain size) instead of diced copper wire, and that’s 10^15.
As I say, there isn’t a boundary to me between intuition and calculation. As in, 10^24 just is, to me, about a mole, the relationship between one molecule and a handful of stuff. It’s also a lower bound on the number of operations of individual transistors you can expect a computer to perform without a single error. A billion transistors clocked a billion times a second for a million seconds, a million seconds being 1⁄30 of a year, or 12 days.
Yes, it’s possible our intuitions simply function differently.
I do the same kinds of calculations, more or less intuitively. I can juggle zeros too if I need to. But my point is that for most human-scale things I don’t need to do that. Maybe it’s just learned behavior, I’m sure an astrophysicist has better intuitions in his area of expertise. The fact that intuition triggers even in situations that are not often encountered seems to indicate there’s more to it than that, though.