Truncated tesseract

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell.

Truncated tesseract

The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra.

Alternate names

Truncated tesseract (Norman W. Johnson)

Truncated tesseract (Acronym: tat) (George Olshevsky and Jonathan Bowers)

Construction

The truncated tesseract may be constructed by truncating the vertices of the tesseract at

1

/

(

2

+

2

)

{\displaystyle 1/({\sqrt {2}}+2)}

of the edge length. A regular tetrahedron is formed at each truncated vertex.

The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:

(

±

1

,

±

(

1

+

2

)

,

±

(

1

+

2

)

,

±

(

1

+

2

)

)

{\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}

Projections

In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:

The projection envelope is a cube.

Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope.

The other 6 truncated cubes project onto the square faces of the envelope.

The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.

Images

Related polytopes

The truncated tesseract, is third in a sequence of truncated hypercubes:

Bitruncated tesseract

The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation operation applied to the tesseract.