It looks like you're using an Ad Blocker.
Please white-list or disable AboveTopSecret.com in your ad-blocking tool.
Thank you.
Some features of ATS will be disabled while you continue to use an ad-blocker.
When I was 9 my Dad told me it was impossible to get from point A to point B because there was no way you could ever cross half the distance as there was no way to ever measure every half. Or something. Same thing?
In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise
“ That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ” —Aristotle, Physics VI:9, 239b10 Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
Originally posted by Astyanax
The solution of Zeno's paradox lies in the blunt fact that there is no such thing at an infinitesimal point in nature – at some point one reaches a 'half-distance' that's shorter than a footprint, or a molecule, or an atom, or a Planck length, or whatever. At that point you have to stop halving distances – you've arrived!
The same solution works for the OP's riddle. At some point, his stepwise zigzag smoothes out into a straight line.
In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise.
Zeno's Paradox
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "In an infinite amount of time."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."
Originally posted by UnixFE
Originally posted by Astyanax
The solution of Zeno's paradox lies in the blunt fact that there is no such thing at an infinitesimal point in nature – at some point one reaches a 'half-distance' that's shorter than a footprint, or a molecule, or an atom, or a Planck length, or whatever. At that point you have to stop halving distances – you've arrived!
The same solution works for the OP's riddle. At some point, his stepwise zigzag smoothes out into a straight line.
Ecaxtly this is my problem. If there is no infinitesimal point in nature and everything is made out of something (no matter how small this part is) than how can I move in a straight line? How can I move in a absolutely straight line if I have to step along those atoms or obey the Planck length. This would result in a zigzag line (although it would look like a straight line as we can't see these absolutely small parts much smaller than an atom) and thus be 20cm long.
I know the formulas, Pythagoras and from my everyday life that the straight line is indeed shorter but wouldn't this require a world without a smallest part. If something in this universe is forced to move along some smallest parts (no matter if you call it strings, atoms or whatever) then it wouldn't be possible to take the straight line.
So it's more a physical problem/question than a math one i quess.