Convenient Cartesian coördinates for the vertices of a regular simplex.
Version of 9 July 2012.
Dave Barber's other pages.

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.