Acronym	tat
Name	truncated tesseract
	©
Cross sections	©
Circumradius	sqrt[(5+3 sqrt(2))/2] = 2.149726
Inradius wrt. tet	(3+2 sqrt(2))/sqrt(8) = 2.060660
Inradius wrt. tic	[1+sqrt(2)]/2 = 1.207107
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 o3o3x4x o3o3x . tet first o3o . x edge first o . x4x {8} first . o3x4x tic first 2 o3o3w . o3x . w x . o4w . o3o4w 3 o3x3w . o3w . w w . o4w . o3o4w 4 o3w3x . o3W . x W . x4x . o3x4x opposite tic 5 x3w3o . x3w . w w . o4w   6 w3x3o . w3x . w x . o4w 7 w3o3o . W3o . x o . x4x opposite {8} 8 x3o3o . opposite tet w3o . w   9   x3o . w 10 o3o . x opposite edge (W=qw=u+q=x+w)
Lace city in approx. ASCII-art	©   x4x w4o w4o x4x -- o3x4x (tic) w4o w4o -- o3o4w (w-cube) w4o w4o -- o3o4w (w-cube) x4x w4o w4o x4x -- o3x4x (tic)
	x3o w3o w3x x3w o3w o3x o3o W3o o3W o3o o3o W3o o3W o3o x3o w3o w3x x3w o3w o3x
Coordinates	((1+sqrt(2))/2, (1+sqrt(2))/2, (1+sqrt(2))/2, 1/2)   & all permutations, all changes of sign
Volume	(101+72 sqrt(2))/6 = 33.803896
Surface	(168+116 sqrt(2))/3 = 110.682924
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: tet tic tat 16 8 )
Dihedral angles	at {3} between tet and tic:   120° at {8} between tic and tic:   90°
Face vector	64, 128, 88, 24
Confer	blends: otbott   decompositions: rit || tat   related isogonals: o3o3q4x   general polytopal classes: Wythoffian polychora   partial Stott expansions   analogs: truncated hypercube tCn
External links

As abstract polytope tat is isomorphic to quitit, thereby replacing the octagons by octagrams, resp. replacing tic by quith.

Note that tat can be thought of as the external blend of 1 rit + 16 tepes + 8 coatics. This decomposition is described as the degenerate segmentoteron oo3oo3xx4ox&#x. – Alternatively, although subdimensioanlly degenerate, tat can be decomposed into 1 sidpith + 16 hexes + 32 tepes + 24 squicufs + 8 cubatics according to xo3oo3ox4xx&#x.

Incidence matrix according to Dynkin symbol

o3o3x4x

. . . . | 64 |  3  1 |  3  3 |  1 3
--------+----+-------+-------+-----
. . x . |  2 | 96  * |  2  1 |  1 2
. . . x |  2 |  * 32 |  0  3 |  0 3
--------+----+-------+-------+-----
. o3x . |  3 |  3  0 | 64  * |  1 1
. . x4x |  8 |  4  4 |  * 24 |  0 2
--------+----+-------+-------+-----
o3o3x . ♦  4 |  6  0 |  4  0 | 16 *
. o3x4x ♦ 24 | 24 12 |  8  6 |  * 8

snubbed forms: o3o3β4x, o3o3x4s, o3o3β4β

o3o3/2x4x

. .   . . | 64 |  3  1 |  3  3 |  1 3
----------+----+-------+-------+-----
. .   x . |  2 | 96  * |  2  1 |  1 2
. .   . x |  2 |  * 32 |  0  3 |  0 3
----------+----+-------+-------+-----
. o3/2x . |  3 |  3  0 | 64  * |  1 1
. .   x4x |  8 |  4  4 |  * 24 |  0 2
----------+----+-------+-------+-----
o3o3/2x . ♦  4 |  6  0 |  4  0 | 16 *
. o3/2x4x ♦ 24 | 24 12 |  8  6 |  * 8

o3/2o3x4x

.   . . . | 64 |  3  1 |  3  3 |  1 3
----------+----+-------+-------+-----
.   . x . |  2 | 96  * |  2  1 |  1 2
.   . . x |  2 |  * 32 |  0  3 |  0 3
----------+----+-------+-------+-----
.   o3x . |  3 |  3  0 | 64  * |  1 1
.   . x4x |  8 |  4  4 |  * 24 |  0 2
----------+----+-------+-------+-----
o3/2o3x . ♦  4 |  6  0 |  4  0 | 16 *
.   o3x4x ♦ 24 | 24 12 |  8  6 |  * 8

o3/2o3/2x4x

.   .   . . | 64 |  3  1 |  3  3 |  1 3
------------+----+-------+-------+-----
.   .   x . |  2 | 96  * |  2  1 |  1 2
.   .   . x |  2 |  * 32 |  0  3 |  0 3
------------+----+-------+-------+-----
.   o3/2x . |  3 |  3  0 | 64  * |  1 1
.   .   x4x |  8 |  4  4 |  * 24 |  0 2
------------+----+-------+-------+-----
o3/2o3/2x . ♦  4 |  6  0 |  4  0 | 16 *
.   o3/2x4x ♦ 24 | 24 12 |  8  6 |  * 8

oooo3xoox4xwwx&#xt   → outer heights = 1/sqrt(2) = 0.707107
                       inner height = 1
(tic || pseudo w-cube || pseudo w-cube || tic)

o...3o...4o...     | 24 * *  * |  2  1  1 0  0  0  0 | 1 2  2  1  0 0 0 | 1 1 2 0 0
.o..3.o..4.o..     |  * 8 *  * |  0  0  3 1  0  0  0 | 0 0  3  3  0 0 0 | 0 1 3 0 0
..o.3..o.4..o.     |  * * 8  * |  0  0  0 1  3  0  0 | 0 0  0  3  3 0 0 | 0 0 3 1 0
...o3...o4...o     |  * * * 24 |  0  0  0 0  1  2  1 | 0 0  0  1  2 1 2 | 0 0 2 1 1
-------------------+-----------+---------------------+------------------+----------
.... x... ....     |  2 0 0  0 | 24  *  * *  *  *  * | 1 1  1  0  0 0 0 | 1 1 1 0 0
.... .... x...     |  2 0 0  0 |  * 12  * *  *  *  * | 0 2  0  1  0 0 0 | 1 0 2 0 0
oo..3oo..4oo..&#x  |  1 1 0  0 |  *  * 24 *  *  *  * | 0 0  2  1  0 0 0 | 0 1 2 0 0
.oo.3.oo.4.oo.&#x  |  0 1 1  0 |  *  *  * 8  *  *  * | 0 0  0  3  0 0 0 | 0 0 3 0 0
..oo3..oo4..oo&#x  |  0 0 1  1 |  *  *  * * 24  *  * | 0 0  0  1  2 0 0 | 0 0 2 1 0
.... ...x ....     |  0 0 0  2 |  *  *  * *  * 24  * | 0 0  0  0  1 1 1 | 0 0 1 1 1
.... .... ...x     |  0 0 0  2 |  *  *  * *  *  * 12 | 0 0  0  1  0 0 2 | 0 0 2 0 1
-------------------+-----------+---------------------+------------------+----------
o...3x... ....     |  3 0 0  0 |  3  0  0 0  0  0  0 | 8 *  *  *  * * * | 1 1 0 0 0
.... x...4x...     |  8 0 0  0 |  4  4  0 0  0  0  0 | * 6  *  *  * * * | 1 0 1 0 0
.... xo.. ....&#x  |  2 1 0  0 |  1  0  2 0  0  0  0 | * * 24  *  * * * | 0 1 1 0 0
.... .... xwwx&#xt |  2 2 2  2 |  0  1  2 2  2  0  1 | * *  * 12  * * * | 0 0 2 0 0
.... ..ox ....&#x  |  0 0 1  2 |  0  0  0 0  2  1  0 | * *  *  * 24 * * | 0 0 1 1 0
...o3...x ....     |  0 0 0  3 |  0  0  0 0  0  3  0 | * *  *  *  * 8 * | 0 0 0 1 1
.... ...x4...x     |  0 0 0  8 |  0  0  0 0  0  4  4 | * *  *  *  * * 6 | 0 0 1 0 1
-------------------+-----------+---------------------+------------------+----------
o...3x...4x...     ♦ 24 0 0  0 | 24 12  0 0  0  0  0 | 8 6  0  0  0 0 0 | 1 * * * *
oo..3xo.. ....&#x  ♦  3 1 0  0 |  3  0  3 0  0  0  0 | 1 0  3  0  0 0 0 | * 8 * * *
.... xoox4xwwx&#xt ♦  8 4 4  8 |  4  4  8 4  8  4  4 | 0 1  4  4  4 0 1 | * * 6 * *
..oo3..ox ....&#x  ♦  0 0 1  3 |  0  0  0 0  3  3  0 | 0 0  0  0  3 1 0 | * * * 8 *
...o3...x4...x     ♦  0 0 0 24 |  0  0  0 0  0 24 12 | 0 0  0  0  0 8 6 | * * * * 1

or
o...3o...4o...     & | 48  * |  2  1  1 0 |  1  2  2  1 | 1  1 2
.o..3.o..4.o..     & |  * 16 |  0  0  3 1 |  0  0  3  3 | 0  1 3
---------------------+-------+------------+-------------+-------
.... x... ....     & |  2  0 | 48  *  * * |  1  1  1  0 | 1  1 1
.... .... x...     & |  2  0 |  * 24  * * |  0  2  0  1 | 1  0 2
oo..3oo..4oo..&#x  & |  1  1 |  *  * 48 * |  0  0  2  1 | 0  1 2
.oo.3.oo.4.oo.&#x    |  0  2 |  *  *  * 8 |  0  0  0  3 | 0  0 3
---------------------+-------+------------+-------------+-------
o...3x... ....     & |  3  0 |  3  0  0 0 | 16  *  *  * | 1  1 0
.... x...4x...     & |  8  0 |  4  4  0 0 |  * 12  *  * | 1  0 1
.... xo.. ....&#x  & |  2  1 |  1  0  2 0 |  *  * 48  * | 0  1 1
.... .... xwwx&#xt   |  4  4 |  0  2  4 2 |  *  *  * 12 | 0  0 2
---------------------+-------+------------+-------------+-------
o...3x...4x...     & ♦ 24  0 | 24 12  0 0 |  8  6  0  0 | 2  * *
oo..3xo.. ....&#x  & ♦  3  1 |  3  0  3 0 |  1  0  3  0 | * 16 *
.... xoox4xwwx&#xt   ♦ 16  8 |  8  8 16 4 |  0  2  8  4 | *  * 6

xwwxoooo3ooxwwxoo3ooooxwwx&#xt   → height(1,2) = height(3,4) = height(5,6) = height(7,8) = 1/2
                       height(2,3) = height(4,5) = height(6,7) = 1/sqrt(2) = 0.707107
(tet || pseudo w-tet || pseudo (w,x)-tut || pseudo (x,w)-tut || pseudo inv (x,w)-tut || pseudo inv (w,x)-tut || pseudo dual w-tet || dual tet)

o.......3o.......3o.......      & | 8 *  *  * |  3 1  0  0  0  0  0 | 3  3  0 0  0  0 | 1 3 0 0
.o......3.o......3.o......      & | * 8  *  * |  0 1  3  0  0  0  0 | 0  3  3 0  0  0 | 0 3 1 0
..o.....3..o.....3..o.....      & | * * 24  * |  0 0  1  2  1  0  0 | 0  1  2 1  2  0 | 0 3 1 0
...o....3...o....3...o....      & | * *  * 24 |  0 0  0  0  1  1  2 | 0  1  0 0  2  3 | 0 3 0 1
----------------------------------+-----------+---------------------+-----------------+--------
x....... ........ ........      & | 2 0  0  0 | 12 *  *  *  *  *  * | 2  1  0 0  0  0 | 1 2 0 0
oo......3oo......3oo......&#x   & | 1 1  0  0 |  * 8  *  *  *  *  * | 0  3  0 0  0  0 | 0 3 0 0
.oo.....3.oo.....3.oo.....&#x   & | 0 1  1  0 |  * * 24  *  *  *  * | 0  1  2 0  0  0 | 0 2 1 0
........ ..x..... ........      & | 0 0  2  0 |  * *  * 24  *  *  * | 0  0  1 1  1  0 | 0 2 1 0
..oo....3..oo....3..oo....&#x   & | 0 0  1  1 |  * *  *  * 24  *  * | 0  1  0 0  2  0 | 0 3 0 0
...x.... ........ ........      & | 0 0  0  2 |  * *  *  *  * 12  * | 0  1  0 0  0  2 | 0 2 0 1
...oo...3...oo...3...oo...&#x     | 0 0  0  2 |  * *  *  *  *  * 24 | 0  0  0 0  1  2 | 0 2 0 1
----------------------------------+-----------+---------------------+-----------------+--------
x.......3o....... ........      & | 3 0  0  0 |  3 0  0  0  0  0  0 | 8  *  * *  *  * | 1 1 0 0
xwwx.... ........ ........&#xt  & | 2 2  2  2 |  1 2  2  0  2  1  0 | * 12  * *  *  * | 0 2 0 0
........ .ox..... ........&#x   & | 0 1  2  0 |  0 0  2  1  0  0  0 | *  * 24 *  *  * | 0 1 1 0
........ ..x.....3..o.....      & | 0 0  3  0 |  0 0  0  3  0  0  0 | *  *  * 8  *  * | 0 1 1 0
........ ..xwwx.. ........&#xt    | 0 0  4  4 |  0 0  0  2  4  0  2 | *  *  * * 12  * | 0 2 0 0
...xo... ........ ........&#x   & | 0 0  0  3 |  0 0  0  0  0  1  2 | *  *  * *  * 24 | 0 1 0 1
----------------------------------+-----------+---------------------+-----------------+--------
x.......3o.......3o.......      & ♦ 4 0  0  0 |  6 0  0  0  0  0  0 | 4  0  0 0  0  0 | 2 * * *
xwwxoo..3ooxwwx.. ........&#xt  & ♦ 3 3  9  9 |  3 3  6  6  9  3  6 | 1  3  3 1  3  3 | * 8 * *
........ .ox.....3.oo.....&#x   & ♦ 0 1  3  0 |  0 0  3  3  0  0  0 | 0  0  3 1  0  0 | * * 8 *
...xo... ........ ...ox...&#x     ♦ 0 0  0  4 |  0 0  0  0  0  2  4 | 0  0  0 0  0  4 | * * * 6

wx3oo3xw *b3oo&#zx   → height = 0
(tegum sum of 2 mutually gyrated (w,x)-rits)

o.3o.3o. *b3o.     | 32  * |  3  1  0 |  3  3  0 | 1 3 0
.o3.o3.o *b3.o     |  * 32 |  0  1  3 |  0  3  3 | 0 3 1
-------------------+-------+----------+----------+------
.. .. x.    ..     |  2  0 | 48  *  * |  2  1  0 | 1 2 0
oo3oo3oo *b3oo&#x  |  1  1 |  * 32  * |  0  3  0 | 0 3 0
.x .. ..    ..     |  0  2 |  *  * 48 |  0  1  2 | 0 2 1
-------------------+-------+----------+----------+------
.. o.3x.    ..     |  3  0 |  3  0  0 | 32  *  * | 1 1 0
wx .. xw    ..&#zx |  4  4 |  2  4  2 |  * 24  * | 0 2 0
.x3.o ..    ..     |  0  3 |  0  0  3 |  *  * 32 | 0 1 1
-------------------+-------+----------+----------+------
.. o.3x. *b3o.     ♦  4  0 |  6  0  0 |  4  0  0 | 8 * *
wx3oo3xw    ..&#zx ♦ 12 12 | 12 12 12 |  4  6  4 | * 8 *
.x3.o .. *b3.o     ♦  0  4 |  0  0  6 |  0  0  4 | * * 8

wx oo3xo4xw&#zx   → height = 0
(tegum sum of (w,x,x)-ticcup and (x,w,w,w)-tes)

o. o.3o.4o.     | 48  * |  2  1  1 0 |  1  2  1  2 | 1 2  1
.o .o3.o4.o     |  * 16 |  0  0  3 1 |  0  0  3  3 | 0 3  1
----------------+-------+------------+-------------+-------
.. .. x. ..     |  2  0 | 48  *  * * |  1  1  0  1 | 1 1  1
.. .. .. x.     |  2  0 |  * 24  * * |  0  2  1  0 | 1 2  0
oo oo3oo4oo&#x  |  1  1 |  *  * 48 * |  0  0  1  2 | 0 2  1
.x .. .. ..     |  0  2 |  *  *  * 8 |  0  0  3  0 | 0 3  0
----------------+-------+------------+-------------+-------
.. o.3x. ..     |  3  0 |  3  0  0 0 | 16  *  *  * | 1 0  1
.. .. x.4x.     |  8  0 |  4  4  0 0 |  * 12  *  * | 1 1  0
wx .. .. xw&#zx |  4  4 |  0  2  4 2 |  *  * 12  * | 0 2  0
.. .. xo ..&#x  |  2  1 |  1  0  2 0 |  *  *  * 48 | 0 1  1
----------------+-------+------------+-------------+-------
.. o.3x.4x.     ♦ 24  0 | 24 12  0 0 |  8  6  0  0 | 2 *  *
wx .. xo4xw&#zx ♦ 16  8 |  8  8 16 4 |  0  2  4  8 | * 6  *
.. oo3xo ..&#x  ♦  3  1 |  3  0  3 0 |  1  0  0  3 | * * 16

ox4wx xo4xw&#zx   → height = 0
(tegum sum of 2 interchanged (w,x,x)-sodips)

o.4o. o.4o.     & | 64 |  1  1  2 | 1  3  2 |  1 3
------------------+----+----------+---------+-----
.. .. x. ..     & |  2 | 32  *  * | 1  2  0 |  1 2
.. .. .. x.     & |  2 |  * 32  * | 1  0  2 |  0 3
oo4oo oo4oo&#x    |  2 |  *  * 64 | 0  2  1 |  1 2
------------------+----+----------+---------+-----
.. .. x.4x.     & |  8 |  4  4  0 | 8  *  * |  0 2
ox .. .. ..&#x  & |  3 |  1  0  2 | * 64  * |  1 1
.. wx .. xw&#zx   |  8 |  0  4  4 | *  * 16 |  0 2
------------------+----+----------+---------+-----
ox .. xo ..&#x    ♦  4 |  2  0  4 | 0  4  0 | 16 *
ox4wx .. xw&#zx & ♦ 24 |  8 12 16 | 2  8  4 |  * 8