Simplex - Cartesian Coordinates For Regular n-dimensional Simplex in Rn

Cartesian Coordinates For Regular n-dimensional Simplex in Rn

The coordinates of the vertices of a regular n-dimensional simplex can be obtained from these two properties,

1. For a regular simplex, the distances of its vertices to its center are equal.
2. The angle subtended by any two vertices of an n-dimensional simplex through its center is

These can be used as follows. Let vectors (v₀, v₁, ..., v_n) represent the vertices of an n-simplex center the origin, all unit vectors so a distance 1 from the origin, satisfying the first property. The second property means the dot product between any pair of the vectors is . This can be used to calculate positions for them.

For example in three dimensions the vectors (v₀, v₁, v₂, v₃) are the vertices of a 3-simplex or tetrahedron. Write these as

Choose the first vector v₀ to have all but the first component zero, so by the first property it must be (1, 0, 0) and the vectors become

By the second property the dot product of v₀ with all other vectors is -1⁄₃, so each of their x components must equal this, and the vectors become

Next choose v₁ to have all but the first two elements zero. The second element is the only unknown. It can be calculated from the first property using the Pythagorean theorem (choose any of the two square roots), and so the second vector can be completed:

The second property can be used to calculate the remaining y components, by taking the dot product of v₁ with each and solving to give

From which the z components can be calculated, using the Pythagorean theorem again to satisfy the first property, the two possible square roots giving the two results

This process can be carried out in any dimension, using n + 1 vectors, applying the first and second properties alternately to determine all the values.