Acronym tat
Name truncated tesseract

Cross sections
` ©`
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 o3o3x4x o3o3x .tet first o3o . xedge first o . x4x{8} first . o3x4xtic first 2 o3o3w . o3x . w x . o4w . o3o4w 3 o3x3w . o3w . w w . o4w . o3o4w 4 o3w3x . o3W . x W . x4x . o3x4xopposite tic 5 x3w3o . x3w . w w . o4w 6 w3x3o . w3x . w x . o4w 7 w3o3o . W3o . x o . x4xopposite {8} 8 x3o3o .opposite tet w3o . w 9 x3o . w 10 o3o . xopposite edge
(W=qw=u+q=x+w)
Lace city
in approx. ASCII-art
 ``` ©   ``` ```x4x w4o w4o x4x w4o w4o w4o w4o x4x w4o w4o x4x ```
```   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
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
 by cells: tet tic tat 16 8
)
Confer
blends:
otbott
External

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

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
```

```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

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
```