The Truncated Tesseract


The truncated tesseract is a relatively simple uniform polychoron bounded by 8 truncated cubes and 16 tetrahedra. It is an analogue of the 3D truncated cube, with its tetrahedra being the analogue of the truncated cube's triangular faces, and its truncated cubes the analogue of the truncated cube's octagonal faces.

Perspective
projection of the truncated tesseract into 3D

Structure

We will explore the structure of the truncated tesseract by using its parallel projection into 3D, centered on a truncated octahedral cell:

Parallel projection
of the truncated tesseract into 3D

As can be seen, its projection envelope is a cube. Nine cells are visible from this viewpoint: 1 truncated cube (in yellow), and 8 tetrahedra (in cyan). The truncated cube, which we may regard as lying on the “north pole” of the truncated tesseract, is shown on its own below:

Parallel projection
of the truncated tesseract, showing the nearest truncated cube

The eight gaps are, of course, occupied by 8 tetrahedra:

Parallel projection
of the truncated tesseract, showing surrounding tetrahedra

The six square faces of the projection envelope are, in fact, the images of six truncated cubes, shown below in pairs:

Parallel
projection of the truncated tesseract, showing a pair of equatorial truncated
cubes Parallel projection of the
truncated tesseract, showing another pair of equatorial truncated cubes Parallel projection of the truncated
tesseract, showing the third pair of equatorial truncated cubes

They appear flattened into squares in the projection because they are being viewed from a 90° angle. In reality, they are perfectly uniform truncated cubes. They lie on the “equator” of the truncated tesseract.

On the far side of the truncated tesseract, another 8 tetrahedra are joined to these 6 equatorial cells, as well as the last truncated cube at the “south pole”. These cells have exactly the same arrangement as the “north pole“ truncated cube and the 8 tetrahedra we have seen above.

In total, there are the “north pole” truncated cube, 6 truncated cubes on the equator, and the “south pole” truncated cube, for a total of 8 truncated cubes. There are a total of 8+8=16 tetrahedra.

Perspective Projections

This image shows the perspective projection of the truncated tesseract from another viewpoint, centered on one of the 16 tetrahedra:

Perspective
projection of the truncated tesseract, centered on tetrahedron

For the sake of clarity, we've omitted the cells that lie on the far side of the truncated tesseract. The nearest tetrahedron to the 4D viewpoint lies at the center of the projection envelope, as shown below:

Perspective
projection of the truncated tesseract, showing nearest tetrahedron

Surrounding it are 4 truncated cubes:

Perspective
projection of the truncated tesseract, showing one of the truncated cubes Perspective projection of the truncated
tesseract, showing another truncated cube Perspective projection of the truncated tesseract, showing yet another
truncated cube Perspective
projection of the truncated tesseract, showing last truncated cube

These truncated cubes are somewhat foreshortened because they are being viewed from an angle. In 4D, of course, they are perfectly uniform.

There are another 4 tetrahedra visible from this viewpoint besides the nearest tetrahedron:

Perspective
projection of the truncated tesseract, showing 5 visible tetrahedra

There are 6 pairs of triangular faces that lie on the projection envelope; these belong to 6 tetrahedra that lie on the far side of the projection, and so are not visible here. The pairs of triangular faces that are visible actually belong to the 4 truncated cubes we saw, but we show them in isolation below to indicate where those 6 hidden tetrahedra connect with the truncated cubes.

Perspective
projection of the truncated tesseract, showing 6 pairs of triangular
faces

Finally, the last 4 triangular faces on the projection envelope are joined to 4 tetrahedra on the far side of the truncated tesseract that are in dual orientation to the 4 visible tetrahedra we saw. Again, they belong to the 4 truncated cubes, but we show them below to indicate where they lie on the far side:

Perspective
projection of the truncated tesseract, showing last 4 triangular faces

They are surrounded by another 4 truncated cubes, also in dual orientation, which in turn surrounds the antipodal tetrahedron.

Animation

The following animation shows the truncated tesseract rotating in the XW plane.

Rotating truncated
tesseract

We turned off visibility clipping so that you can trace the full path of each cell as it rotates relative to the 4D viewpoint.

Coordinates

The coordinates of the truncated tesseract centered at the origin with edge length 2 are all permutations of coordinates and changes of sign of:


Last updated 24 Feb 2013.

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