The n-Dimensional Simplexes and Pascal's Triangle
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|
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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