Acronym tat
Name truncated tesseract
 
©  
Cross sections
 ©
Circumradius sqrt[(5+3 sqrt(2))/2] = 2.149726
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o4o o3o3o . o3o . o o . o4o . o3o4o
1o3o3x4x 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
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: tet tic
tat 168
)
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
links
hedrondude   wikipedia   WikiChoron   quickfur

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, even so 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

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