Two Geometric Problems with Elegant Solutions?
A question about some problems with elegant solutions was asked recently:
http://answers.yahoo.com/question/index;_ylt=An0DEmptUVt2Fo28ts.g.NLty6IX;_ylv=3?qid=20080816225809AA6Wxwm&show=7#profile-info-a779e940bae6439d1ae9cb7b6c96b882aa
Encouraging further discussion on this fascinating subject, started by Ana, I decided to repost 2 problems despite the fact I know the solutions - they fully deserve it, I'm sure Y!A Math Community will like them very much. Enjoy!
1) Let P is an arbitrary point in space. How many rays, starting at P exist, with the property: the angle between any 2 of them is the same? Prove that the maximal number is 4.
2) Consider all octahedrons, circumscribed around the unit sphere. Prove that the regular octahedron (one of the Platonian Solids) IS NOT the polyhedron with the minimal volume among them.
The most elegant (or most neatly presented in my opinion) answer will be chosen as best, I am not going to put this into vote. Additional Details immediately follow.
1) Answer:
1-dimensional space (linear case): 1+1 = 2 rays;
angle between them arccos(-1/1) = π;
2-dimensional space (planar case): 2+1 = 3 rays;
angle between any 2 of them arccos(-1/2) = 2π/3 ("Mercedes" configuration at P);
3-dimensional space: 3+1 = 4 rays, connecting the centroid of the regular tetrahedron with its vertexes;
angle between any 2 of them arccos(-1/3) ≈ 110°;
Why 4 and not more? That is the question.
2) What polygon with given number of sides (e.g. 8), circumscribed around the unit circle, has the smallest possible area? The answer is well-known and almost obvious - the regular one (e.g. the regular octagon among all octagons). Quite surprisingly the analogy between 2D and 3D cases is misleading here - if a regular polyhedron with given number of faces (e.g. 8) exists, it is not necessarily optimal in similar sense!
Good start by A Cave! Everybody feel free to submit here more problems with elegant solutions instead of answering if You wish!
Zeta, You are on the right track about 2). Keep trying to find! Look here:
http://farm4.static.flickr.com/3004/2802617542_794c3a85db_o.gif
Using the relationship for every circumscribed polyhedron:
3 * Volume = Surface_area * Inscribed_sphere_radius
can be shown that Problem 2) is equivalent to the noteworthy Isoperimetrical Problem for Polyhedrons: what polyhedron with given number F of faces and a given constant surface area has the greatest volume? Unlike the planar case (with perimeter and area involved) where the optimal solution is always a regular polygon, the latter always better than any irregular one with the same number of sides, in the 3D case we can have irregular polyhedrons (#2 and #3 on the picture for F=8), isoperimetrically equivalent (and even better are possible!) to the regular ones! It turns out that only Platonian Solids with trihedral vertexes yield optimal solutions for F=4, F=6 and F=12.
It's time for the solutions that I know.
1) Mark the points exactly as in A Cave's solution. What A Cave has written below can be explained with other words like this: the marked points are vertexes of a regular polyhedron (Platonic Solid). If the rays were 5, we would have an example of a regular polyhedron with 5 vertexes, but such polyhedron DOES NOT EXIST!
http://en.wikipedia.org/wiki/Platonic_solid
If the rays were even more, removing some of them we would reduce to the 5-rays configuration - impossible, as we have just seen already.
2) This is one of my favorite visual ("Look and See") proofs. Circumscribe a regular octahedron around the unit sphere, then follow this link and watch what happens:
http://www.flickr.com/photos/24472398@N06/2795345189/sizes/o/
Operation 3 DECREMENTS THE VOLUME from 4√3 ≈ 6.928 to 16√3(3√2-4) ≈ 6.72!
Here is some more information about this irregular octahedron, isoperimetically better then the regular one:
http://en.wikipedia.org/wiki/Tetragonal_trapezohedron
It is still not the optimal octahedron - M. Goldberg has found an example of the same topological type as #3 (on the first picture): 4 pentagonal and 4 quadrilateral faces, volume ≈ 6.7009. As far as I know, this very difficult isoperimetrical problem for polyhedrons (especially the case F=8) is still awaiting its final solution.