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 -- 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
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°
Confer
blends:
otbott
decompositions:
rit || tat
general polytopal classes:
partial Stott expansions
External

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
```