Truncated tesseract

Tesseract	Truncated tesseract	Rectified tesseract	Bitruncated tesseract
Schlegel diagrams centered on [4,3] (cells visible at [3,3])
16-cell	Truncated 16-cell	Rectified 16-cell (24-cell)	Bitruncated tesseract
Schlegel diagrams centered on [3,3] (cells visible at [4,3])

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

Truncated tesseract
Schlegel diagram (tetrahedron cells visible)
Type	Uniform 4-polytope
Schläfli symbol	t{4,3,3}
Coxeter diagrams
Cells	24	8 3.8.8 16 3.3.3
Faces	88	64 {3} 24 {8}
Edges	128
Vertices	64
Vertex figure	( )v{3}
Dual	Tetrakis 16-cell
Symmetry group	B₄, [4,3,3], order 384
Properties	convex
Uniform index	12 13 14

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)^[1]

Construction

The truncated tesseract may be constructed by truncating the vertices of the tesseract at $1/({\sqrt {2}}+2)$ 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:

\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

orthographic projections
Coxeter plane	B₄	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	F₄	A₃
Graph
Dihedral symmetry	[12/3]	[4]

A polyhedral net	Truncated tesseract projected onto the 3-sphere with a stereographic projection into 3-space.

Related polytopes

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

Truncated hypercubes
Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Bitruncated tesseract

Bitruncated tesseract
Two Schlegel diagrams, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden.
Type	Uniform 4-polytope
Schläfli symbol	2t{4,3,3} 2t{3,3^1,1} h_2,3{4,3,3}
Coxeter diagrams	=
Cells	24	8 4.6.6 16 3.6.6
Faces	120	32 {3} 24 {4} 64 {6}
Edges	192
Vertices	96
Vertex figure	Digonal disphenoid
Symmetry group	B₄, [3,3,4], order 384 D₄, [3^1,1,1], order 192
Properties	convex, vertex-transitive
Uniform index	15 16 17

The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation operation applied to the tesseract. It can also be called a runcicantic tesseract with half the vertices of a runcicantellated tesseract with a construction.