Odd nD Space thot

Discussion in 'Physics & Math' started by Dinosaur, Feb 6, 2015.

  1. Dinosaur Rational Skeptic Valued Senior Member

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    Consider an n-dimensional hypercube with a side one foot long. Volume is one unit for all values of n.

    If you consider the side to be 12 inches long, the volume is unbounded as n increases.


    If you consider the side to be 1/3 yard, the volume approaches zero as n increases.


    I am not sure how to avoid the above anomaly. Is it an anomaly or merely an annoyance requiring care in matching units to context?
     
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  3. James R Just this guy, you know? Staff Member

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    Think about the 2-dimensional hypercube, i.e. the square.

    Take a side length \(L\). The area is then \(L^2\).

    If \(L<1\) then \(L^2<L\). If \(L>1\) then \(L^2>L\).

    But notice that if we're talking lengths and areas, they have different units, so the above numerical comparison doesn't really mean anything in physical terms.

    You could look at some quantity such as surface area to volume ratio. For the square, this is \(1/L\), which tells you something that might be physically useful, although it still has units.

    The problem with your "volume approaches infinity" kind of argument is that it is completely dependent on units. For example, if the side length is 2 cm, that's less than 1 inch, so you get seemingly contradictory results depending on which length units you choose.
     
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  5. Dinosaur Rational Skeptic Valued Senior Member

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    It seems obvious that choice of units should not result in volume computation being unbounded, approaching zero, or neither.

    Yet it appears to do so. How can the apparent anomaly be resolved?
     
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  7. Jason.Marshall Banned Banned

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    You are all geniuses I know you can figure this one out am waiting for the answer but am willing to settle for a learning opportunity I feel something may come out of this thread.
     
  8. QuarkHead Remedial Math Student Valued Senior Member

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    Although I know very little about "yards", "feet" and "inches", it seems that here there is a confusion about the arithmetic base being used.

    Suppose an hypercube with sides 1 and therefore volume 1, I assume it is fair to say I am using base 10 arithmetic?

    Suppose the same hypercube with sides 12 and volume 1 then we are using base 12 arithmetic. That is where \(1 \sim 12 \sim 24.....\). But since in base twelve, \(1 \sim 12\) , then \( 1 \sim 12^n\)

    Then again, the same hypercube with volume = 1 and sides \(3^{-1}\) means we are assuming base 3 arithmetic where \(1 \sim (3^{-1})^n\)

    Of course I am assuming the result to be proved, but it might help........just possibly
     
    Last edited: Feb 8, 2015
  9. Jason.Marshall Banned Banned

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    654



    Ding ding ding ...I think you are the winner judges cast your votes now??? we should let Dinosaur test your hypothesis himself and see if he still gets the same anomaly after he makes the adjustments.
     
    Last edited: Feb 8, 2015
  10. someguy1 Registered Senior Member

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    727
    A cubic foot is a lot bigger than a square foot. Just visualize it. One is a 2-dimensional square and the other is infinitely many of these squares stacked up. In inches, the square is 144 square inches and the cube is 1728 cubic inches. It's larger.

    In yards, the square is 1/9 square yards, and the cube is 1/27 cubic yards. But nothing's really going to zero. 1/27 of a cubic yard is (1/27) * (36^3) = (1/27) * 46656 = ... wait for it ... 1728 cubic inches, exactly as it should be.

    This is a cute misdirection. You can't sensibly compare square inches to cubic inches.
     
  11. Dinosaur Rational Skeptic Valued Senior Member

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    There are apparent paradoxes which can be dealt with. For example, the series

    1 + 1/2 + 1/3 . . . .: 1/n diverges.​

    There is at least one apparent paradox resulting from rearrangement of the terms & some apparently valid arithmetic on the rearranged terms.

    The rearrangements required are deemed to be verboten.

    Might there be some similar rule preventing the paradox due to choice of units?
     
  12. RJBeery Natural Philosopher Valued Senior Member

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    4,222
    someguy1 gave a good response. As n grows for a given unit of choice, conversion back to other units will produce consistent results just as expected. The math works, and unit choice can remain arbitrary, which means that our impression that any value approaches zero as n increases isn't valid.
     

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