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The n-Dimensional 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 n-dimensional 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 n-dimensional simplex.

n-dimensional 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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