Geometric fun

I was doing some geometric figuring a few days ago.  Zero dimensions:  a point is a point.  One dimension:  a segment has two end points.  Two dimensions:  a square has four sides, with eight points coming together in twos to make four corners.  Three dimensions:  a cube has six faces, 24 segments coming together in twos to make twelve edges, and 24 points coming together in threes to make eight corners.

If you put all this together you can see a pattern:

Points      1     2     2*4/2=4     4*6/3=8     

Segments        1     4                4*6/2=12

Squares                  1                6

Cubes                                       1

Total         1     3     9                27

Horizontally, the number of points are exponents of two. But if you look at the diagonal lines upper left to lower right, you'll see a constant one, then a progression of even numbers, then the number to the left multiplied by the even number below and divided by two, then the same but divided by three, then four and five and so on.  And notice the total for each column:  exponents of three!

It follows that you'd get the following numbers for a tesseract and further dimensions. (Can't visualize it, but I can calculate it!)

8*8/4=16     16*10/5=32     32*12/6=64

12*8/3=32    32*10/4=80    80*12/5=192

6*8/2=24     24*10/3=80     80*12/4=240

8                    8*10/2=40     40*12/3=160

1                    10                   10*12/2=60

*                     1                    12

*                      *                    1

Total: 81          243               729

But then I thought about triangles, tetrahedra (four faces) and tetrahedral tesseracts!  It's the same for zero and one dimension, but then it gets different.

2*3/2=3     3*4/3=4     4*5/4=5

3                3*4/2=6     6*5/3=10

1                4                 4*5/2=10

*                1                 5

*                                   1

Total     7     15            31

This time, instead of 1, 2, 4, 8... the top row is 1, 2, 3, 4... Instead of 2,4,6... the first diagonal line is 2,3,4...  And the total is exponents of two minus one!

I suppose I could also do this for pentagons and docedahedra (twelve faces) and dodecahedral tesseracts and beyond, and even for the other Pythagorean solids: triangles and octahedra (eight faces), or triangles and icosahedrea (20 faces). This stuff fascinates me.