Regular Simplices

A regular *n*-simplex
is
figure that consists of *n+1* vertices all of which are the same
distance from one another. An *n*-simplex can only sit in a space
of dimension *n* or more. A regular 2-simplex is
an equilateral triangle and a regular 3-simplex is a regular tetrahedron.
Regular simplices make a nice benchmark for projection algorithms. Since
their vertices are equal distant, we know what to expect in the projected
figure. The projected vertices should be about as equal distant to one
another as possible in two dimensions.

A regular 4-simplex is a 4-simplex whose edges all have the same length. A regular 4-simplex is one of the simplest four dimensional objects and it is useful for illustrating the projection of objects whose dimension is too high for us to see directly.

On the left the vertices of a regular 4-simplex is shown projected into a plane using the Sammon map. This 4-simplex actually sits inside of a five dimensional space and it has five fold rotational symmetry about its center (which is not a vertex in this case). This symmetry is preserved in the projected image. Each of the five vertices are equal distant to one another in the five dimensional space. These distances cannot be preserved in a two dimensional projection. The best that can be done is to have the vertices form a regular pentagon.

The edges connecting the vertices have been included on the right figure. There are ten edges in the 4-simplex. In the projected image these edges form a pentagram inscribed inside of a pentagon. The edges which are projected to the pentagram portion do not really cross in the five dimensional space but it is not possible to project the edges of a 4-simplex into plane without producing at least one extraneous crossing point. In this case because the five fold symmetry is preserved in the projection we get five extraneous crossing points.

A regular 4-simplex has ten equilateral triangular faces in the five dimensional space. Each equilateral triangle is projected to one of two types of triangles in the plane. One type of triangle has two edges in the pentagon portion of the projected image and the remaining edge is in the pentagram portion of the projected image. The second type of triangle has two edges in the pentagram portion of the projected image and the remaining edge is in the pentagon portion of the projected image. The edges in the projected image are slightly curved to strike a balance between the two types of triangles. This keeps the lengths of the edges of the triangles close in value and the angles of the triangles close in value. This helps provide a better overall picture of the shape of the triangular faces of a regular 4-simplex than simply connecting the vertices with straight lines in the projected image.

Shown below on the left is a 20-simplex; to its right is a pair of nested 20-simplices, one "inside" the other. Notice that the Sammon algorithm has done it best to maintain equal distances between the vertices within each simplex, and, in the figure to the right, to maintain equal distances between the corresponding vertices of the nested simplices.