The Pentatope


The pentatope is the 4D equivalent of the tetrahedron. It consists of 5 regular tetrahedra joined at their faces, folded into 4D to form a 4D volume. There are 3 tetrahedra surrounding every edge. It is also known as the 5-cell because it is made of 5 tetrahedral cells. Another name for it is the 4D simplex, so called because it is the simplest possible polychoron that encloses a non-zero 4D volume. It is the shape of Pento's pyramid in The Legend of the Pyramid.

Edge-first projection of
pentatope

Cell-first projection

The cell-first perspective projection of the pentatope into 3D is a tetrahedron, which is the nearest cell to the 4D viewpoint.

Cell-first projection of
pentatope

The other 4 cells are on the far side of the pentatope, and are not shown here.

Vertex-first projection

The vertex-first projection of the pentatope also has a tetrahedral envelope. This time, four of the cells are visible.

Vertex-first projection
of pentatope

The vertex at the center of this image, where the four internal edges meet, is actually the apex of the pentatope pointing at us from the 4th direction. It is the nearest vertex to the 4D viewpoint.

The following images show the layout of these four cells in the projected image:

Vertex-first projection
of pentatope, first cell shown Vertex-first projection of pentatope, second
cell shown Vertex-first projection of pentatope, third
cell shown Vertex-first projection of pentatope, fourth
cell shown

The fifth cell is not visible here, as it lies on the far side of the pentatope. It covers the entire tetrahedral volume of the projection.

Note that although the cells appear here as slightly flattened tetrahedra, this is only because they are being viewed from an angle. In actuality, they are perfectly regular tetrahedra.

Face-first projection

The next image shows the pentatope viewed at face-first.

Face-first projection of
pentatope

This projection has a trigonal bipyramid as its envelope. There are two cells visible here, forming the upper and lower halves of the bipyramid, respectively. The following images show each of these two cells.

trigonal bipyramid
projection of pentatope, first cell shown trigonal bipyramid projection of
pentatope, second cell shown

The other 3 cells are not visible from this viewpoint, because they lie on the far side of the pentatope.

These cells appear somewhat deformed from a regular tetrahedron, because they are all being viewed from an angle.

Edge-first projection

The edge-first projection of the pentatope also has a trigonal bipyramidal envelope.

Edge-first projection of
pentatope

The vertical edge in this image is the closest edge to the 4D viewpoint. Three tetrahedral cells meet at this edge, as shown in the following images.

Edge-first projection
of pentatope Edge-first projection of pentatope Edge-first projection of pentatope

Coordinates

The coordinates of an origin-centered 5-cell with edge length 2 are:

Simpler coordinates can be obtained in 5D as all permutations of coordinates of:

The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection.


Last updated 13 Sep 2011.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid XHTML 1.1! Proud to be Microsoft-free