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The nDimensional Simplexes and Pascal's Triangle
"Number" is not resticted to the 3 dimensions of our physical world but spans all dimensions from zero to infinity. In any dimension there is a shape which constitutes the simplest configuration.
The simplest possible shape in any dimension is called an ndimensional simplex where n denotes the number of dimensions. For example:
in 0 dimensional space the point is the simplest shape.
in 1 dimensional space the line is the simplest shape.
in 2 dimensional space the triangle is the simplest shape.
in 3 dimensional space the tetrahedon is the simplest shape.
...and so on to infinity. Of course we cannot use a different name for the simplest shape in every dimension so we use the shorthand of
ndimensional simplex.
ndimensional simplex  English Name 
0  Point 
1  Line 
2  Triangle 
3  Tetrahedon 
4  Pentalope 
simplex  0  1  2
 3  4  5  6  7

0  1  0  0  0  0  0  0  0

1  2  1  0  0  0  0  0  0

2  3  3  1  0  0  0  0  0

3  4  6  4  1  0  0  0  0

4  5  10  10  5  1  0  0  0

5  6  15  20  15  6  1  0  0

6  7  21  35  35  21  7  1  0

7  8  28  56  70  56  28  8  1

This table is a slightly modified Pascal's triangle illustrating a remarkable connection between simplices and simplex figurate numbers.
The features of a given simplex correspond exactly with a particular level of Pascal's triangle which in turn corresponds with a particular numbers' partition structure.
These partition structures can be represented with binary figures. Base 2 is the simplest possible numerical system of notation. So there is a direct relation between the simplest shape, the simplest base and ways in which a number can be partitioned.
The number of ways in which 5 can be partitioned corresponds with the Tetrahedron.
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