# The Joined 24-cell

The joined 24-cell is a Catalan polychoron bounded by 96 triangular bipyramids, 288 isosceles triangles, 240 edges (144 short, 96 long), and 48 vertices. It is the dual of the rectified 24-cell.

The cells are transitive, and are triangular bipyramids with an edge length ratio of √5 : 3, or approximately 0.745 : 1. Thus, they are slightly taller than the dual of the uniform triangular prism, which is also a triangular bipyramid, but with edge length ratio of 2 : 3 (approx. 0.666 : 1). The following image shows an individual cell:

## Structure

We will explore the structure of the joined 24-cell using its parallel projections into 3D, centered on an axial vertex.

### The First Layer

The following image shows this nearest vertex to the 4D viewpoint:

For clarity, we render the rest of the polytope in a light transparent color.

Surrounding this nearest vertex are 12 cells, shown in groups of 3 in the following images:

### The Second Layer

Surrounding these cells are another layer of 24 cells, shown below in groups of 6:

These cells look somewhat distorted because they lie at an angle with the 4D viewpoint and also in an oblique orientation. However, this is merely an artifact of the projection; they are congruent to the cells we saw earlier.

These are all the cells that lie on the near side of the polychoron, the side that faces the 4D viewpoint.

### The Equator

Following this, we come to limb, or equator, of the polytope. These are the cells that lie at a 90° angle to the 4D viewpoint.

The following image shows 8 of them:

Due to their 90° angle to the 4D viewpoint, these cells have been foreshortened into kite-shaped polygons. In 4D, of course, they are full-bodied and congruent with the other cells.

For the sake of clarity, we render the other cells that we have seen before in a light transparent color.

The following images show the rest of the equatorial cells:

There are 24 such cells in total.

The projection envelope is a deltoidal icositetrahedron, albeit with different proportions.

Past this point, we come to the far side of the joined 24-cell, where the layers of cells repeat in reverse order from the cells on the near side.

### Summary

In summary, there are 12+24=36 cells on the near side, 24 cells on the equator, and 24+12=36 cells on the far side, for a total of 36+24+36=96 cells.

## Coordinates

The Cartesian coordinates of the joined 24-cell are all permutations of coordinate and changes of sign of:

• (0, 0, 0, 6)
• (0, 0, 4, 4)
• (3, 3, 3, 3)

These coordinates correspond with a dual rectified 24-cell of edge length √2/12.

Last updated 10 Apr 2024.