| Thank you,
David Klein
KLEIN@HUJIVMS
And here is the solution
------------------------
The correct answer is lim Rn = infinity. This counter intuitive
result is due to the fact that the spheres in the individual quadrants
only bound the center sphere along the direction of the cubes diagonal.
Since the length of the diagonal goes to infinity as root(n) we see
that the central sphere really isn't very well bounded.
To be specific, the radius of each bounding sphere is clearly 1/4.
The center of the sphere is at a distance of root(n)/4 from the origin.
Along the diagonal of the cube the center sphere just touches the corner
spheres so its radius is root(n)/4 - 1/4. In 9-dimensional space, Rn=1/2
and the central sphere is tangent to the surface of the unit cube. For
any higher dimension the sphere protrudes outside of the unit cube.
The final vote was divided as follows
a) Lim = 0 4 votes
b) Lim = PI/8 5 votes
c) Lim = infinity 6 votes
d) Lim = 1/2 3 votes
Thus one out of three had the right intuition (Which is better
than I would have thought).
Following are some of the comments I received along with the
answers. They are interesting as they show some insight into
peoples thought processes. The last comment includes another
question on insight, this time in only four dimensions (which
is hard enough as is.) Beware that the question is closely followed
by the solution.
1)
My intuition was the answer is 1/2. I have calculated
the correct answer and found it surprising . I think
this is more a reflection of poor intuition about
dimension tending to infinity rather than poor intuition
about dimension n for n large.
2)
To my opinion b) is the right solution.
a) should not be right because it seems to me that the sequence Rn is increasing
(consider R1=0).
c) is also wrong because Rn <= 1.
and d) means that the inner circle would reach the boudaries of the surrounding
square (cube, ...) what seems impossible in my intuition.
3)
Thanks for an entertaining problem. I'm afraid my intuition is in
need of brushing up however; I incorrectly guessed (a) 0.
I suspect that infinite-dimensional intuition (or an asymptotic
version thereof, as in your example) is hard to come by. (Who can
visualize Hilbert space, even one of countably infinite
dimensionality?)
You might try posing questions on Euclidean geomentry in four
dimensions, as an alternative. (To what extent does mathematicians'
spatial perception [the ability to rotate figures, etc.] carry over to
four dimensions?) I think I'd do a bit better. At least, I hope so.
4)
In response to your survey:
First, I thought the answer is 1/2.
But that is because I misunderstood the problem.
(I thought that ignoring the little cubes
meant also ignoring the little spheres
inscribed in the little cubes.)
Then, I thought that the answer is pi/8.
Then, I could not make up my mind between 1/2 and pi/8.
Then, it occurred to me that the answer might be infinity
because the length of the diagonal gets long.
I was about to guess infinity when it occurred to me
that if the radius is greater than 1/2,
then the sphere sticks out of the cube.
Therefore, I am now totally confused.
But there is no law against the sphere
sticking out of the cube provided
it does not stick into the 2-to-the-n-th spheres,
so I guess infinity anyway.
5)
The question was too easy
In particular you could `see' the answer algebraically rather than
geometrically, since the distance of a corner to the center tends to
infinity, while the distance between adjacent corners is constant.
If you REALLY want to test higher dimensional intuition try asking the
following.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
QUESTION:
~~~~~~~~~
If you slice a cube along a plane perpendicular to an interior diagonal
the intersection of the cube with the plane is shaped like a regular hexagon.
Now take a 4-dimensional hypercube, and slice it along a three dimensional
hyperplane which perpendicularly bisects an interior diagonal. The three
dimensional solid which results is a ...... ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Ask people to try to do it in their heads without writing anything, and ask
them to indicate how long it took.
(answer below)
octahedron
Proof: It is a regular platonic solid. It touches all `faces' of the hypercube
and thus has 8 faces. QED.
Thanks for your attention
David Klein
KLEIN@HUJIVMS
Disclaimer: These opinions aren't even my own, never mind anybody else's.
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