Sorry, I should have said: Then "bend" the n rays ... (as opposed to n-1 rays). Tom
Here's an obvious followup question: In n dimensions, is the maximum number of mutually-obtuse rays with a common origin always n+1?
Here's a suggested construction that I believe generates canonical, symmetrical solutions: For n dimensions, start with the n-1 dimension solution that has n rays. Add the new dimension perpendicular to the previous n-1 dimensions. Then "bend" the n-1 rays all in the same direction in the new dimension, and add a new ray in the opposite direction along the new dimension. If the previous rays are all bent by the correct amount, then I believe all rays, including the new one, will be isomorphic to each other (and all angles will be the same).
Starting with one dimension, the solution involves 2 rays at 180 degrees to each other. To extend that to 2 dimensions, bend the two rays to an angle of 120 degrees, then add the new ray in the opposite direction. For 3 dimensions, bend those three rays, then add a fourth in the opposite direction, resulting in tetrahedral symmetry. And so on.
Tom
On 8/30/10, rcs@xmission.com <rcs@xmission.com> wrote:
Forwarded from SBG ...
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Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer.
Steve Gray
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