The Truncated 24-cell


The truncated 24-cell is a uniform polychoron bounded by 24 cubes and 24 truncated octahedra. It is derived from the 24-cell by truncating its vertices to a depth of 1/3 its edge length.

Truncated 24-cell,
perspective projection

The truncated 24-cell is actually the same as the cantitruncated 16-cell, due to the fact that the 24-cell is just the rectified 16-cell.

Structure

We will explore the truncated 24-cell by means of its parallel projection into 3D. The following image shows its cube-first projection, with all cells except the closest cube to the 4D viewpoint hidden.

Cube-first
parallel projection of the truncated 24-cell into 3D, all cells omitted except
closest cube to viewpoint

This projection has an envelope in the shape of a rhombic dodecahedron with its six axial vertices truncated. The central cube is joined to 6 truncated octahedra, shown in pairs below:

Parallel
projection of the truncated 24-cell, with two more truncated octahedra
shownTwo other truncated
octahedraThird pair of truncated
octahedra

The next image shows all of them together:

Truncated
24-cell, with all 6 truncated octahedra shown

There are 8 obvious gaps into which 8 cubes may fit:

Truncated 24-cell,
with all 8 more cubes shown

We have now reached the limb of the truncated 24-cell. The cells shown so far constitute the “northern hemisphere” cells, lying on the near half of the truncated 24-cell. At the “equator”, there are 6 cubes and 12 truncated octahedra, the former of which is shown below:

Truncated 24-cell,
with 6 limb cubes shown

For reference, we also show the first cube in the center. The 6 cubes on the “equator” appear as flat squares, because they are being viewed at from a 90° angle.

The 12 truncated octahedra at the “equator” project to the hexagonal faces of the projection envelope:

Truncated 24-cell,
with 12 limb truncated octahedra shown

These truncated octahedra appear as flat hexagons because they are being viewed from a 90° angle.

The cells on the far side of the truncated 24-cell (the “southern hemisphere”) have an arrangement that exactly mirrors the near side (the “northern hemisphere”). So in total, in the northern hemisphere we have 1+8=9 cubes, and 6 truncated octahedra. The southern hemisphere also has 9 cubes and 6 truncated octahedra, giving us 18 cubes and 12 truncated octahedra. Adding the equatorial cells, 6 cubes and 12 truncated octahedra, gives us 24 cubes and 24 truncated octahedra in total.

Perspective Projection

The following image shows the perspective projection of the truncated 24-cell:

Truncated 24-cell,
with 12 limb truncated octahedra shown

It is almost the same as the previous parallel projection, except that the limb cells are hidden from view, and the projection envelope shows some foreshortening.

Cantitruncated 16-cell

The truncated 24-cell is the same as the cantitruncated 16-cell, because the 24-cell happens to be the same as the rectified 16-cell. The following parallel projections show the tesseractic symmetry of the truncated 24-cell, when viewed as the cantitruncated 16-cell:

Cantitruncated
16-cell, with nearest truncated octahedron shown

This projection is centered on a truncated octahedral cell. This nearest cell to the 4D viewpoint is shown above in yellow. Its square faces are joined to 6 cubical cells, as shown below:

Cantitruncated
16-cell, with 6 cubes added

These cubes appear slightly flattened, because they are being viewed from a 45° angle; in 4D, they are perfectly regular cubes.

The hexagonal faces of the truncated octahedral cell are joined to 8 other truncated octahedra. We show them in 2 sets of 4 below:

Cantitruncated
16-cell, with 4 more truncated octahedra added Cantitruncated 16-cell, with the other 4
truncated octahedra

Here they are all together:

Cantitruncated
16-cell, with all 8 truncated octahedra

These cells are, of course, perfectly uniform truncated octahedra; they appear a bit squished because they are being viewed from an angle.

These are all the cells in the “Northern Hemisphere” of the cantitruncated 16-cell. Next, we have the equatorial cells, which consist of 6 more truncated octahedra:

Cantitruncated
16-cell, with 6 equatorial truncated octahedra

And 12 cubes:

Cantitruncated
16-cell, with 12 equatorial cubes added

These truncated octahedra and cubes have been foreshortened into octagons and rectangles, because they are being viewed from a 90° angle. In 4D, they are perfectly uniform truncated octahedra and perfectly regular cubes.

Past this point, the cells repeat in reverse, forming the “Southern Hemisphere” of the cantitruncated 16-cell. In summary, we have 1+8=9 truncated octahedra in the northern hemisphere, 6 truncated octahedra on the equator, and another 9 truncated octahedra in the southern hemisphere, making a total of 24 truncated octahedra. There are 6 cubes in each hemisphere and 12 cubes on the equator, also totalling 24 cubes. Thus, we see that the truncated 24-cell and the cantitruncated 16-cell are one and the same.

Coordinates

The coordinates of the truncated 24-cell centered on the origin with an edge length of 2 are all permutations of coordinates and changes of sign of:

Properties

Since the 24-cell is self-dual, the 24 cubes and 24 truncated octahedra of the truncated 24-cell correspond with the cells and vertices of a 24-cell, or, equivalently, with the cells of a 24-cell and the cells of its dual, respectively. With the truncated 24-cell, these two sets of 24 cells have different shapes; the bitruncated 24-cell, on the other hand, has the same shape for both sets of cells, being bounded by 48 truncated octahedra.


Last updated 29 May 2014.

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