3 dimensional shapes

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Suppose an icosahedron is centered at the origin and three of its twelve vertices are at
(0, 0, 1), (0, 0,−1), and (a, 0, b). What are the values of a and b?

It is hard for me to picture this, if the center is at (0,0,0), I think you can rotate the icosahedron such that one vertex is at the very top and one is at the very bottom, like a north and south poles, but are these points (0,0,1) and (0,0,-1)? If so, how do you find a and b from that.

Any help is appreciated.
 
Are we assuming the icosahedron is regular? Because I don't see how any conclusion could otherwise be made...?

Thank you.

Eliz.
 
Hi there, yes the icosahedron is regular, made up of equilateral triangles.

I found a picture here http://en.wikipedia.org/wiki/Image:Icos ... angles.png

I photoshopped it to try and see if I could find the missing vertex and I found one, this is how I labeled it:

ico.jpg


I labeled one vertex blue.
I noticed that there is at least 2 vertices that could be (a,0,b) , with the other one located at (-a,0,-b)

I think the purple plane in the picture represents y=0

I was also trying to picture the shape from a top view looking down, this would give you a pentagon, made up of 5 equilateral triangles, so from this I was thinking that you could find one angle and vector. But the vector from the origin to the blue vertex is confusing for me to even picture
 
problem

Okay, i think the key is to try to imagine how it looks like when looking above the y-axis. I think you will see that you will have a hexagon, with those vertices on x,z coordinates.
And if i havent made a mistake, i find a=(squareroot 3)/2, and b=1/2.
 
Hi there, I was trying to imaging it from a top view perspective like you said but if you have a vertex pointed straight up on an icosahedron then you will see a pentagon, not a hexagon. Each side of the pentagon has equal sides. If you look at it from the side, then you see a hezagon, but it does not appear to be regular.

Can you explain more on how you derived your results?

thanks
 
problem

Sorry, I could be wrong.
But look at the picture you have.
Don't you see a hexagon there?
But, I could have looking in wrong direction.
 
The net looks like
Code: 
      1     2     3 *    5
     /\    /\    /  *    /\
    /  \  /  \  /   *   /  \
   /____\/____\/__  * \/____\
  /\    /\    /\    * /\    /
 /  \  /  \  /  \   *   \  /
/____\/____\/____\/_* ___\/
\    /\    /\    /\ *    /
 \  /  \  /  \  /   *   /
  \/    \/    \/    * \/
made up of 20 equilateral triangles so you should be seeing a pentagon when you look down the z axis (line (0,0,1),(0,0,0),(0,0,-1)

PS. Whoops, I can see the hexagon too but the sides are slanted in the z direction. The sides of the pentagon are co-plainer.
 
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