The Bitruncated 24-cell


The Bitruncated 24-cell is a uniform polychoron bounded by 48 cells in the shape of truncated cubes. It is both vertex-transitive and cell-transitive. It can be obtained by truncating the 24-cell at a depth halfway to obtaining the dual 24-cell.

Perspective
projection of the bitruncated 24-cell

It is also known as “octagonny” (due to Wendy Krieger) because of its connection with octagonal numbers (Z4).

Structure

The following cell-first parallel projection of the bitruncated 24-cell shows the nearest cell to the 4D viewpoint.

Cell-first
projection of bitruncated 24-cell into 3D, with nearest cell shown.

The octagonal faces of this cell are joined to 6 other cells, shown in the following image:

Cell-first
projection of bitruncated 24-cell into 3D, with 6 cells surrounding nearest
cell shown.

These cells are arranged in a cubical arrangement.

The triangular faces of the nearest cell are joined to 8 other cells. The following image shows 8 cells surrounding the nearest cell.

Cell-first
projection of bitruncated 24-cell into 3D, with 8 cells surrounding
nearest cell shown.

These cells are arranged in an octahedral arrangement.

The 15 cells we've seen so far lie on the “northern hemisphere” of the bitruncated 24-cell. The next image shows 6 of the 18 cells that lie on the “equator”.

Cell-first
projection of bitruncated 24-cell into 3D, with 6 equatorial cells
shown.

These equatorial cells appear to be flat squares because they are being viewed from a 90° angle. In reality, they are perfectly uniform truncated cubes.

Together with the “north pole” and “south pole” cells, these cells correspond with the 8 cells of a tesseract that the bitruncated 24-cell can be inscribed into. The next image shows these cells (minus the “south pole” cell, which lies behind the central cell in the fourth direction and is obscured from view here.

Cell-first
projection of bitruncated 24-cell into 3D, with 6 equatorial cells
shown and nearest cell shown.

The next image shows the other 12 equatorial cells. These correspond with the 12 edges of a cube, or the faces of a rhombic dodecahedron.

Cell-first
projection of bitruncated 24-cell into 3D, with the other 12 equatorial cells
shown.

Again, these cells also appear to be flat because they are being viewed at from a 90° angle. In 4D, they are perfectly uniform truncated cubes.

After these cells, we have the “southern hemisphere” cells, which mirror the arrangement of the northern hemisphere cells. Hence, we have a total of 1+6+8=15 northern hemisphere cells, 15 southern hemisphere cells, and 6+12=18 equatorial cells, for a total of 48 cells.

Perspective Projections

The following image shows a perspective projection of the bitruncated 24-cell.

Cell-first
perspective projection of bitruncated 24-cell into 3D.

The nearest cell is rendered in blue, the cells sharing an octagonal face with it in red, and the cells sharing a triangular face with it in green. The equatorial cells are not visible because they make an angle larger than 90° with the viewpoint.

Coordinates

The coordinates of the vertices of the bitruncated 24-cell are all permutations of coordinates and changes of sign of:

Properties

The bitruncated 24-cell is one of the two non-prismatic, non-regular uniform polychora that are both cell-transitive and vertex-transitive. The regular polychora have this property, as well as the bitruncated 5-cell and the n,n-duoprisms.


Last updated 29 May 2014.

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