array9

 just putting this down as a project to think about.

if you have n points in space, how many potential flat surfaces have you got?

Any three points define a single surface.  So the answer is n.(n-1).(n-2) / 3.2

So n=3  you have one surface defined.

     n=4 you have 4.3.2 / (3.2) = 4

     n=5 you have 5.4.3 / 6 = 10

     n=6 you have 6.5.4 / 6 = 20

     n=7 you have 7.6.5 / 6 = 35

     n=8 you have 8.7.6 / 6 = 56

     n=9 you have 9.8.7 / 6 = 84

The thing that intrigues me is that this does not depend on the dimensionality of the space.  The dimensionality must come in somewhere surely.  And I am currently guessing that it is more about how many you can 'see partially' if the surfaces are opaque. So in three dimensions

n   most visible   out of       comment

4           4                  4        go inside tetrahedron

             3                  4        go outside

5           6+             10        go inside ( eg square pyramid )

             6                10        go outside (eg triangle with two triangular pyramids) 

I cannot cope with the difficulty of imagining these spaces at the moment, and the figures above may be wrong.

I am also intrigued by the thought of inside and outside.  Where did that come from?

martin

# posted by martinW @ 3:54 pm