The Rectified Tesseract


The rectified tesseract is a uniform polytope in the tesseract family. It is constructed by truncating the tesseract at the midpoints of its edges, and its surface consists of 16 tetrahedra and 8 cuboctahedra.

The rectified
tesseract

Structure

The structure of the rectified tesseract is relatively simple. It retains the 8 cells of the tesseract it is derived from, except that these cells been rectified into cuboctahedra. The vertices of the original tesseract have been truncated, leaving tetrahedra in their place.

The following image shows the parallel projection of the rectified tesseract, centered on one of its cuboctahedral cells:

Parallel projection
of the rectified tesseract, showing nearest cuboctahedron

For clarity, we show only the nearest cuboctahedron to the 4D viewpoint. The next image shows the 8 tetrahedra that are joined to this cuboctahedron:

Parallel projection
of the rectified tesseract, showing 8 tetrahedra

Here's the cuboctahedron together with the tetrahedra:

Parallel projection
of the rectified tesseract, showing 8 tetrahedra with cuboctahedron

As can be clearly seen, the projection envelope of the rectified tesseract is a cube. The faces of this cube correspond to the projection images of 6 more cuboctahedral cells, shown in pairs below:

Parallel projection
of the rectified tesseract, showing 1st pair of equatorial cuboctahedra Parallel projection of the rectified
tesseract, showing 2nd pair of equatorial cuboctahedra Parallel projection of the rectified
tesseract, showing 3rd pair of equatorial cuboctahedra

These cuboctahedra appear to be flattened into squares; that's because they lie at a 90° angle to the 4D viewpoint. In 4D, they are perfectly uniform cuboctahedra.

On the far side of these cuboctahedra are another 8 tetrahedra and the antipodal cuboctahedron to the first cuboctahedron we saw. The arrangement of these cells is identical to the arrangement of the previous 9 cells on the near side. The following sequence of images shows how all the cells of the rectified tesseract are laid out in the parallel projection:

Nearest cuboctahedron
8 tetrahedra
1st pair of cuboctahedra 2nd pair of cuboctahedra 3rd pair of cuboctahedra
Another 8 tetrahedra
Antipodal cuboctahedron

This layout shows clearly that there are 8 cuboctahedral cells in total, along with 16 tetrahedral cells.

Animations

The following animation shows the rectified tesseract rotating in the WX plane:

Rectified tesseract
rotating in WX plane

For clarity, we've omitted the cuboctahedral cells, showing only the 16 tetrahedral cells, which are alternately colored so that the animation is easier to follow.

Coordinates

The coordinates of the rectified tesseract are all permutations of coordinates and changes of sign of:

These coordinates yield a rectified tesseract of edge length 2, centered on the origin.


Last updated 29 May 2014.

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