The Pentatope

The pentatope 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.

Cell-first/vertex-first projection

The following diagram shows a perspective projection of the pentatope.

The dotted line shows the far edge of the outer tetrahedron. The blue lines are inside the outer tetrahedron in this projection. The center of the projection where these blue lines meet is actually the apex of the pentatope pointing away from us in the 4th direction.

All 5 tetrahedral cells of the pentatope are present in this diagram: the outer tetrahedron, and the 4 “inner” tetrahedra outlined by one triangular face of the outer tetrahedron and 3 of the blue lines each. Although they appear as slightly flattened tetrahedra, this is only because they are being viewed at from an angle. In actuality, they are perfectly regular tetrahedra.

The following diagrams illustrate 3 of these cells.

Note that if a 4D being were to look at an actual pentatope, it would not be able to see all 5 cells at once. Rather, it would either see only the outer tetrahedron (if it looks at the “base” of the pentatope), or the 4 inner cells (if it looks at the apex of the pentatope).

Edge-first/face-first projection

The next diagram shows the pentatope viewed at from another angle.

In this diagram, three of the pentatope's cells are arranged around the central axis indicated by the blue line. The other two cells are the upper and lower halves of the outer shape, which is called a trigonal bipyramid. All the cells appear somewhat deformed from a regular tetrahedron, because they are all being viewed at from an angle. Some of these cells are shown below:

Again, this projection represents two possible views of the pentatope. From one view, the 4D being would see only the upper and lower tetrahedra. From the opposite view, it would see the 3 inner tetrahedra. It cannot see all 5 cells at once unless the pentatope is transparent.