The table below shows Cartesian coördinates for some regular simplices in various numbers of dimensions.

Each (one-dimensional) edge is of length one; beyond that the location and orientation of each figure was selected to make as obvious a pattern of numbers as possible. Of course, the coördinates may be scaled, rotated and translated as desired.

For clarity, define:

a	= 1 ÷ √ ( 2 ×	2 )
b	= 1 ÷ √ ( 3 ×	4 )
c	= 1 ÷ √ ( 4 ×	6 )
d	= 1 ÷ √ ( 5 ×	8 )
e	= 1 ÷ √ ( 6 ×	10 )
f	= 1 ÷ √ ( 7 ×	12 )

Then it is simple to write the coördinates:

In 2 dimensions the 3 vertices, which form an equilateral triangle, are:	( 0, 0 ) ( 2a, 0 ) ( a, 3b )
In 3 dimensions the 4 vertices, which form a regular tetrahedron, are:	( 0, 0, 0 ) ( 2a, 0, 0 ) ( a, 3b, 0 ) ( a, b, 4c )
In 4 dimensions the 5 vertices are:	( 0, 0, 0, 0 ) ( 2a, 0, 0, 0 ) ( a, 3b, 0, 0 ) ( a, b, 4c, 0 ) ( a, b, c, 5d )
In 5 dimensions the 6 vertices are:	( 0, 0, 0, 0, 0 ) ( 2a, 0, 0, 0, 0 ) ( a, 3b, 0, 0, 0 ) ( a, b, 4c, 0, 0 ) ( a, b, c, 5d, 0 ) ( a, b, c, d, 6e )
In 6 dimensions the 7 vertices are:	( 0, 0, 0, 0, 0, 0 ) ( 2a, 0, 0, 0, 0, 0 ) ( a, 3b, 0, 0, 0, 0 ) ( a, b, 4c, 0, 0, 0 ) ( a, b, c, 5d, 0, 0 ) ( a, b, c, d, 6e, 0 ) ( a, b, c, d, e, 7f )
Higher dimensionalities generalize in the obvious fashion.