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Zeno Boots

 robotic boots to demonstrate the tortoise and Achilles paradox
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Pull on these motorized boots, activate them, and four small mechanical legs pop out of each sole. The robot legs are half as long as your legs and end in identical smaller boots. Also from these boots pop out four even smaller legs with boots from which extend four miniscule legs, etc.

This fractal progression continues towards an infinite number of boot sizes with the wonders of existing near-nano mechanization. When you take a one-yard step with a boot, its four legs simultaneously take an additional half-yard step, their legs take quarter yard step and so forth.

The boots will answer the questions: How long is your total stride? How high are your feet from the ground? How much faster can you walk?

—FarmerJohn, Jan 17 2003

Zeno’s Paradox of the Tortoise and Achilles http://www.mathacad...zeno_tort/index.asp
[FarmerJohn, Oct 05 2004, last modified Oct 21 2004]

[link]






       Wouldn't the really small legs just get crushed under your weight?
—talen, Jan 17 2003
   

       I don't think it works that way [UB]. Your solution assumes each row of legs if facing the opposite direction.   

       I think the example is computed as:   

       1/2^0 (normal stride)+   

       1/2^1 (extension from front row of legs) +   

       1/2^2 +   

       1/2^3 +   

       1/2^4 +   

       1/2^5 +   

       etc.   

       Tends to infinity.   

       Assuming no limitations, the legs would be infinately long, would step an infinately long distance, expending an infinately large amount of energy and a stride would last for an infinate duration.
—FloridaManatee, Jan 17 2003
   

       I like it Farmer dear. I'm reading Godel Escher Bach at the moment (very slooowly) so I'm probably biased though.
—madradish, Jan 17 2003
   

       I agree with UB's take on this. If each successive set of legs steps half the distance of its parent set, you'd never quite make it to twice the stride.   

       You step a full stride (1). The first set of sub-legs step half a stride (1.5 strides total) then the next step half of that, a quarter-stride (1.75 strides total), then an eighth of a stride (1.875), etc... You'll get infinitely close to 2 strides but never reach it.
—waugsqueke, Jan 17 2003
   

       Florida: Nope. Both length of stride and length of leg are bounded series. Maths below   

       Stride = Integral ( 1/2^x) summed between 0 and infinity   

       = [-2^(1-x)] summed btwn 0 and infinity   

       = (-2^-infinity) - (-2^1) = 0 - (-2) = 0 + 2 = 2   

       2* strides will be done, no less, since Farmer has collapsed Zeno's paradox by using the word 'simultaneously'.   

       *[later] ignoring the Quantum Mechanical problems of an infinite number of infinitely small feet.
—st3f, Jan 17 2003
   

       If you ask me, you'll just end up falling over. But then I'm an engineer, what do I know?
—egbert, Jan 17 2003
   

       Support shouldn't be a problem [talen] with so many legs, sort of like standing on steel brushes.   

       I wanted simultaneous movement to double the barefoot speed.
—FarmerJohn, Jan 17 2003
   

       It would take a long time to stop after taking one step wouldn't it? Even though towards the end you would only be moving fractions of cms.
—notme, Jan 17 2003
   

       // But then I'm an engineer, what do I know? //   

       According to Marketing, absolutely nothing.   

       [notme] no, every foot, however small, starts and stops simultaneously.
—8th of 7, Jan 17 2003
   

       How would you walk in these things? The structure involved in even making one is mind-boggling.
—Evil_Baron_Moustachio, Jan 17 2003
   

       Love it, [FJ]. But it would be maddening to try finding the one leg with a squeeky hinge.
—Worldgineer, Jul 09 2004
   

       ALLL HAIL FARMERJOHN!
—DesertFox, Jul 09 2004
   

       Are these like stilts?
—swimr, Jul 09 2004
   

       Or like motorized platform heels almost as long as the legs.
—FarmerJohn, Jul 09 2004
   

       Taking it the other way, have increasingly large sets of legs. You would then have something that didn't prove anybody's paradox, but you'd walk incredibly fast.
—david_scothern, Jul 12 2004
   

       <david> Haven't you heard of Seven League boots? They feature in many Celtic tales. God knows how they managed to build those in days of yore, though.
Maybe after someone makes these legs, they'll invent a time travel machine. That would explain it.........(I've always wondered)
—blueturtle, Jul 12 2004
   

       Haven't seen you in a while, blueturtle.   

       Ohh, and I want these boots.
—DesertFox, Jul 12 2004
   
      
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