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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 |
Point | |
Line | |
Triangle | |
Tetrahedon | |
Pentalope |
simplex | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | |
4 | 6 | 4 | 1 | 0 | 0 | 0 | 0 | |
5 | 10 | 10 | 5 | 1 | 0 | 0 | 0 | |
6 | 15 | 20 | 15 | 6 | 1 | 0 | 0 | |
7 | 21 | 35 | 35 | 21 | 7 | 1 | 0 | |
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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