Re: [math-fun] n-dimensional geometry puzzle
Here's my solution to the "n-dimensional geometry puzzle" below: The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is C = (1/n, ..., 1/n). The distance from the point C to any of the basis vectors is sqrt(1-1/n). The distance from C to the origin is sqrt(1/n). Since sqrt(1-1/n) >= sqrt(1/n) for all n >= 2, (exercise), sqrt(1-1/n) solves the problem. As far as I know, only Tom Karzes sent in a solution (almost instantaneously). —Dan ----- Puzzle: ------- In n-dimensional space R^n, find the radius R = R(n) of the smallest sphere containing (whether inside or on the surface) the standard basis vectors {e_k} = {(1,0,...,0), ..., (0,...,0,1)} and the origin 0 = (0,...,0). I.e., R(n) = inf {r > 0 | for some c in R^n ||p - c|| <= r for p = 0 and all p = e_k} Apologies if this has been asked already; I don't recall. -----
Sorry if I'm being dense, but I don't see how Dan got On Fri, Jul 6, 2018 at 4:57 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Here's my solution to the "n-dimensional geometry puzzle" below:
The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is
C = (1/n, ..., 1/n).
The distance from the point C to any of the basis vectors is sqrt(1-1/n).
I get sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) = sqrt((n-1)^2/n^2 + n(1/n)^2) = sqrt((n-1)^2+n)/n^2) = sqrt(n^2-n+1)/n, not sqrt(1-1/n). Am I suffering from end-of-workweek brain-fog? Jim
Hi Jim, This part: sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + n(1/n)^2) sqrt(((n-1)^2+n)/n^2) sqrt(n^2-n+1)/n Should be: sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + (n-1)(1/n)^2) sqrt(((n-1)^2+n-1)/n^2) sqrt((n^2-n)/n^2) sqrt(1-1/n) There are n terms total: One instance of (1-1/n)^2, and (n-1) instances of (0-1/n)^2. You were counting n instances of the latter, resulting in n+1 terms total. Tom James Propp writes:
Sorry if I'm being dense, but I don't see how Dan got
On Fri, Jul 6, 2018 at 4:57 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Here's my solution to the "n-dimensional geometry puzzle" below:
The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is
C = (1/n, ..., 1/n).
The distance from the point C to any of the basis vectors is sqrt(1-1/n).
I get sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) = sqrt((n-1)^2/n^2 + n(1/n)^2) = sqrt((n-1)^2+n)/n^2) = sqrt(n^2-n+1)/n, not sqrt(1-1/n).
Am I suffering from end-of-workweek brain-fog?
Jim
Thanks! On Saturday, July 7, 2018, Tom Karzes <karzes@sonic.net> wrote:
Hi Jim,
This part:
sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + n(1/n)^2) sqrt(((n-1)^2+n)/n^2) sqrt(n^2-n+1)/n
Should be:
sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + (n-1)(1/n)^2) sqrt(((n-1)^2+n-1)/n^2) sqrt((n^2-n)/n^2) sqrt(1-1/n)
There are n terms total: One instance of (1-1/n)^2, and (n-1) instances of (0-1/n)^2. You were counting n instances of the latter, resulting in n+1 terms total.
Tom
James Propp writes:
Sorry if I'm being dense, but I don't see how Dan got
On Fri, Jul 6, 2018 at 4:57 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Here's my solution to the "n-dimensional geometry puzzle" below:
The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is
C = (1/n, ..., 1/n).
The distance from the point C to any of the basis vectors is sqrt(1-1/n).
I get sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) = sqrt((n-1)^2/n^2 + n(1/n)^2) = sqrt((n-1)^2+n)/n^2) = sqrt(n^2-n+1)/n, not sqrt(1-1/n).
Am I suffering from end-of-workweek brain-fog?
Jim
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Tom Karzes