4th dimension question

Steven G

Steven G

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The other day I was sitting in front of a table that had a few quarters on it. So I placed one in front of me and then proceeded to put the next two coins around this one. Then I asked myself how many quarters will go around this centered one. My quick answer was 2pi but quickly realized that the answer was 6 after seeing the equilateral triangles. So then I upped the problem to how many ping pong balls can be placed around a centered ball. I easily got that answer. Then of course I considered the 4th dimension case and realized that I could NOT visualize this situation.

Can some of you mathematicians or Physicists (can't forget about them!) please explain how to see this problem and solution as well as possibly going to higher dimensions?
 
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Heck, I can't even visualize in 2D. My Engineering Drafting class was a serious problem for me.

-Dan
 
Jomo said:
The other day I was sitting in front of a table that had a few quarters on it. So I placed one in front of me and then proceeded to put the next two coins around this one. Then I asked myself how many quarters will go around this centered one. My quick answer was 2pi but quickly realized that the answer was 6 after seeing the equilateral triangles. So then I upped the problem to how many ping pong balls can be placed around a centered ball. I easily got that answer. Then of course I considered the 4th dimension case and realized that I could NOT visualize this situation.

Can some of you mathematicians or Physicists (can't forget about them!) please explain how to see this problem and solution as well as possibly going to higher dimensions?
Click to expand...
One thing about higher D than 3-D is that us humans cannot really visualize it (including you Jomo). Forget arranging circles or spheres in close pack, think about cubes. A cube in 4_D will be a hypercube -- with 16 (=24) corners. The shadow (projection) of a hypercube in 3-D will be a regular cube (just like the projection of a 3-D cube in 2-D will be a square). We cannot visualize a hypercube - however we can feel its existence through mathematics (just like curved space can be proven to exist by a 2-D scientist through measuring interior angles of a triangle drawn in the curved space).

All this "deep" knowledge was gathered through one of my most favorite books - "One, Two, Three, Infinity…" by George Gammow.

Anyway, visualization does not work in advanced sciences. It does not work for 4_D, and it does not work for Quantum Mechanics. Lots of time wasted by good scientists trying to explain what is that "electron" doing in an atom. In Brian Green's words (quoting Charles Lutwidge Dodgson), it might as well be "gyre and gimble in the wabe" (Jabberwocky).
 
:love: Gammow's book.

And you forgot to mention that the electron is a "slithy tove." ? (I love how Jabberywocky's terms are used to describe QM.)

-Dan
 
In mathematics, "dimension" just refers to how many numbers you need to specify the object of interest. If I were studying spheres in space, to specify a given sphere I could give the coordinates of the center and the radius of the sphere. That is three numbers to specify the center and one number giving the radius. That is a "four dimensional" situation.

In physics we study "events", things that happen at a given point, at a given time. That is "four dimensional" because we need three numbers to specify the point and one number to specify the time of the event.
 
I have been thinking about how the idea of getting an intuition through visualization, even with simple proofs, is sometimes not really a coherent idea. For example, in an indirect proof, you assume the negation of the thing you want to show. Ultimately that initial “reductio” assumption (or set of assumptions) is equivalent to a contradiction if its negation is true. But then what does it mean to visualize a logical impossibility? I take this to be part of the intuitionists’ motivation for preferring “constructive” proofs where possible
 
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....us humans cannot really visualize it (including you Jomo)....

Jomo can too! :)
 
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