The Bitruncated Tesseract


The bitruncated tesseract is a uniform polychoron constructed by truncating a tesseract halfway to the depth that would yield the dual 16-cell. Equivalently, it can be derived from the 16-cell by truncating at halfway the depth that would yield the dual tesseract. Hence, it is also known as the bitruncated 16-cell. It is bounded by 8 truncated octahedra and 16 truncated tetrahedra.

The following image shows the perspective projection of the bitruncated tesseract, centered on one of the truncated octahedral cells:

Perspective
projection of the bitruncated tesseract, centered on a truncated
octahedron

Structure

We shall explore the structure of the bitruncated tesseract by examining its parallel projection into 3D:

Parallel
projection of the bitruncated tesseract, with nearest truncated octahedron
shown

This image shows its parallel projection centered on a truncated octahedron. The truncated octahedron shown here is the one closest to the 4D viewpoint. Its hexagonal faces are joined to 8 truncated tetrahedra, shown in the next two images:

Parallel
projection of the bitruncated tesseract, now with 4 truncated tetrahedra also
shown Now with the other 4 truncated
tetrahedra shown

These truncated tetrahedra are joined to each other via their triangular faces, and to the truncated octahedra via their hexagonal faces.

The square faces of the central truncated octahedron are joined to 6 other truncated octahedra, shown in the following images:

Parallel
projection of the bitruncated tesseract, with 2 equatorial cells shown Another two equatorial cells Yet another two equatorial
cells

These 6 truncated octahedra lie at a 90° angle to the 4D viewpoint, so they appear to be flat octagons due to foreshortening. In reality, they are perfectly uniform truncated octahedra.

The central truncated octahedron and the 8 truncated tetrahedra may be thought of as lying on the “northern hemisphere” of the bitruncated tesseract. The 6 equatorial cells lie on the “equator”. On the other side of the bitruncated tesseract, the “southern hemisphere”, are another 9 cells that mirror the layout of the northern hemiphere cells. Thus, this makes a total of 1+6+1=8 truncated octahedra, and 8+8=16 truncated tetrahedra.

Other Projections

The following image shows the perspective projection of the bitruncated tesseract, centered on a truncated tetrahedron:

Perspective
projection of the bitruncated tesseract, centered on truncated
tetrahedron

The nearest truncated tetrahedron is shown here in yellow. The 4 truncated tetrahedra surrounding it are shown below:

Perspective
projection of the bitruncated tesseract, centered on truncated tetrahedron,
with 3 of the surrounding truncated tetrahedra shown Now with the 4th adjoining truncated
tetrahedron shown

There are also 4 truncated octahedra surrounding the nearest cell, shown in the following images:

Perspective
projection of the bitruncated tesseract, centered on truncated tetrahedron,
with 1 of the surrounding truncated octahedra shown Now with 2nd adjoining truncated
octahedron 3rd adjoining truncated octahedron 4th adjoining truncated
octahedron

Coordinates

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


Last updated 29 May 2014.

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