Re: HyperCube (Diamond Explained)

Frank Newhouse wrote:
> 
> On Wed, 11 Jun 1997 18:48:45 +0100, salesman
> <sal...@g-grids.demon.co.uk> wrote:
> 
> >>I think these figures are right (I never tried to counts edges and faces
> >>of a hypercube before!)
> >>
> >>                            Dimensions   Vertices   Edges   Faces
> >>Point       0           1?
> >>Segment     1           2         1
> >>Square      2           4         4       1
> >>Cube        3           8        12       6
> >>H-cube      4          16        30      18
> 
> Your explanations and drawings are quite good, but -sorry to tell you-
> your figures are wrong
> 
> The tesseract, or hypercube, has
> -4 dimensions,
> -16 vertices -OK so far-,
> - 32 edges, not 30 (not too hard to see; you have 12 edges in the cube
> you begin with, another 12 edges in the cube you end with, and eight
> edges connecting each vertex in the first cube with its homologue in
> the second cube)
> -24 faces, not 18 (again, 6 faces in cube 1 plus 6 faces in cube 2,
> plus 12 faces connecting each edge of cube 1 with its homologue in
> cube 2)
> -and last but not least, 8 cubical "faces" (cube 1, cube 2, and 6
> cubes connecting each face of cube 1 with its homologue in cube 2)

Making the above table:

      Dimension  Points  Lines Squares Cubes H-Cubes H-H-Cubes    
                          
Point       0       1
Line        1       2      1
Square      2       4      4      1
Cube        3       8     12      6      1
H-cube      4      16     32     24      8       1     
H-H-cube    5      32     80     80      40      10      1

etc.

the number of Y-dimensional objects in an X-dimensional object
is 2 x number of Y dimensional object in an X-1 dimensional
object + number of Y-1 dimensional objects in an X-1 dimensional
objects.

Is theis a bit clear? No? reread! Still not clear? No? Bummer! :-)

It sounds a bit complicated, but once you get it it's quite simple
example: number of lines in cube: 2 x number of lines in squares
+ number of points in square == 2*4 + 4 = 12

Greetz,
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