hi rajinikanth0602

Is this a problem you want help with?  Or an exercise for the rest of us?

And is it exactly five rays or at least five?

Bob

I would say 72°.

Then it seems it is not possible... I might be wrong though...

I assumed that these rays are in 3D space with every ray at the same angle to the other 4.

eg. 6 rays can all be at 90 to each other by taking them along the coordinate axes x,y and z.

eg.  4 rays:  Take the start point as the centroid of a regular tetrahedron with each ray being a ray to one vertex.

If O is the centre of a sphere and A,B,C,D, and E are points on the surface of the sphere, I can imagine these points being equally spaced so that every radius is at the same angle to the other 4.  I would think the angle would be obtuse.  But my imagination isn't providing the positions at the moment. 

rajinikanth0602:  You didn't answer my other question.  Is this a puzzle that you can do and have set as a challenge; or one that you want help with?

Bob

hi Stefy,

6 lines;  Oh yes, I forgot the 180s.  Silly me. 

I don't see what is wrong with the tetrahedron though.

I'm going from the middle of the solid out to the 4 vertices.  As it is symmetrical they must all be at the same angle to each other surely ??

bob

That depends on what you say.

If you need to ask the question, then best to say nothing.


Here's a tetrahedron to show what I meant above.  I claim each red line is at the same angle to the other 3.

Bob

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Hi Bob

Yes, the 4 rays solutions is ok, sorry.

I'm working on whether the picture is a possible solution or not.  ie.  Can this exist?

Bob

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hi Stefy,

In post 12, I did a similar thing or a tetrahedron.  It's hard to show this in 2D.  I am assuming that it is possible to do this so that the diagram looks the same whichever point (A,B,C,D or E) is "at the top".  OR to prove this cannot be done.

If you can prove what you have said, then it only remains to prove that, for a solution,  all rays must come from a single point.  If this is so, then you have proved impossibilty.

LATER EDIT:

5 rays are drawn from a point in space

I re-read the question.  So all five come from one point.  proving impossibility has suddenly got easier.

EVEN LATER EDIT:

Assume it is possible.

Let O be the point the rays come from and OA,OB,OC,OD,OE be the rays.

There is no loss of generality in placing A,B,C,D,E on a sphere, since the rays can always be extended until they cut the sphere without changing the angles between them.

Let A be at the "top" of the sphere.  B,C,D,E must lie in a horizontal plane since these angles are equal: AOB,AOC,AOD,AOE.

I consider three cases: (i) B,C,D,E lie in a plane that cuts O (ii) B,C,D,E lie in a plane below O (iii) B,C,D,E lie in a plane above O.

(i)  angle DOB = 180; angle DOC = 90 =><=

(ii) angle AOB is obtuse.

Let DB and EC cross at F.  angle BFC = 90.

O is the vertex of a square based pyramid OBCDE so BOC < 90  =><=

(iii)  DOA + AOB = DOB  =><=

So it is impossible.

Bob

Bob

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