Acronym rit
Name rectified tesseract,
rectified octachoron,

Cross sections
` ©`
Circumradius sqrt(3/2) = 1.224745
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 o3o3x4o o3o3x .tet first o3o . overtex first o . x4o{4} first . o3x4oco first 2 o3x3x . o3x . qvertex figure x . o4q . o3o4q 3 x3x3o . o3u . o u . x4o . o3x4oopposite co 4 x3o3o .opposite tet x3x . q x . o4q 5 u3o . o o . x4oopposite {4} 6 x3o . q 7 o3o . oopposite vertex o3o3o *b3o o3o3o    . o3o . *b3o o . o    o . o3o *b3o 1 x3o3x *b3o x3o3x    .co first x3o . *b3otet first x . x    o{4} first . o3x *b3otet first 2a o3o3u    . x3x . *b3o q . o    x . x3x *b3o 2b u3o3o    . o . q    x 3 x3o3x    .opposite co o3x . *b3x x . x    u . x3o *b3x 4a o3o . *b3xopposite tet q . o    x . o3o *b3xopposite tet 4b o . q    x 5 x . x    oopposite {4}
Lace city
in approx. ASCII-art
```  x3o x3x o3x

o3o u3o o3u o3o

x3o x3x o3x
```
 ``` ©   ``` ```x4o o4q x4o o4q o4q x4o o4q x4o ```
Coordinates (1/sqrt(2), 1/sqrt(2), 1/sqrt(2), 0)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
 by cells: co oho tet tut rit 8 0 16 0 hinnit 8 0 0 16 sto 0 8 8 8 firt 0 0 16 16
& others)
Confer
Grünbaumian relatives:
2rit+64tet
segmentochora:
tet || tut   tut || inv tut
External

Incidence matrix according to Dynkin symbol

```o3o3x4o

. . . . | 32 ♦  6 |  6  3 |  2 3
--------+----+----+-------+-----
. . x . |  2 | 96 |  1  2 |  1 2
--------+----+----+-------+-----
. o3x . |  3 |  3 | 64  * |  1 1
. . x4o |  4 |  4 |  * 24 |  0 2
--------+----+----+-------+-----
o3o3x . ♦  4 |  6 |  4  0 | 16 *
. o3x4o ♦ 12 | 24 |  8  6 |  * 8
```

```o3o3x4/3o

. . .   . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
. . x   . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
. o3x   . |  3 |  3 | 64  * |  1 1
. . x4/3o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3o3x   . ♦  4 |  6 |  4  0 | 16 *
. o3x4/3o ♦ 12 | 24 |  8  6 |  * 8
```

```o3o3/2x4o

. .   . . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
. .   x . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
. o3/2x . |  3 |  3 | 64  * |  1 1
. .   x4o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3o3/2x . ♦  4 |  6 |  4  0 | 16 *
. o3/2x4o ♦ 12 | 24 |  8  6 |  * 8
```

```o3o3/2x4/3o

. .   .   . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
. .   x   . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
. o3/2x   . |  3 |  3 | 64  * |  1 1
. .   x4/3o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3o3/2x   . ♦  4 |  6 |  4  0 | 16 *
. o3/2x4/3o ♦ 12 | 24 |  8  6 |  * 8
```

```o3/2o3x4o

.   . . . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
.   . x . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
.   o3x . |  3 |  3 | 64  * |  1 1
.   . x4o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3/2o3x . ♦  4 |  6 |  4  0 | 16 *
.   o3x4o ♦ 12 | 24 |  8  6 |  * 8
```

```o3/2o3x4/3o

.   . .   . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
.   . x   . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
.   o3x   . |  3 |  3 | 64  * |  1 1
.   . x4/3o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3/2o3x   . ♦  4 |  6 |  4  0 | 16 *
.   o3x4/3o ♦ 12 | 24 |  8  6 |  * 8
```

```o3/2o3/2x4o

.   .   . . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
.   .   x . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
.   o3/2x . |  3 |  3 | 64  * |  1 1
.   .   x4o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3/2o3/2x . ♦  4 |  6 |  4  0 | 16 *
.   o3/2x4o ♦ 12 | 24 |  8  6 |  * 8
```

```o3/2o3/2x4/3o

.   .   .   . | 32 ♦  6 |  6  3 |  2 3
--------------+----+----+-------+-----
.   .   x   . |  2 | 96 |  1  2 |  1 2
--------------+----+----+-------+-----
.   o3/2x   . |  3 |  3 | 64  * |  1 1
.   .   x4/3o |  4 |  4 |  * 24 |  0 2
--------------+----+----+-------+-----
o3/2o3/2x   . ♦  4 |  6 |  4  0 | 16 *
.   o3/2x4/3o ♦ 12 | 24 |  8  6 |  * 8
```

```x3o3x *b3o

. . .    . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-----------+----+-------+----------+------
x . .    . |  2 | 48  * |  2  1  0 | 2 1 0
. . x    . |  2 |  * 48 |  0  1  2 | 2 0 1
-----------+----+-------+----------+------
x3o .    . |  3 |  3  0 | 32  *  * | 1 1 0
x . x    . |  4 |  2  2 |  * 24  * | 2 0 0
. o3x    . |  3 |  0  3 |  *  * 32 | 1 0 1
-----------+----+-------+----------+------
x3o3x    . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3o . *b3o ♦  4 |  6  0 |  4  0  0 | * 8 *
. o3x *b3o ♦  4 |  0  6 |  0  0  4 | * * 8
```

```x3o3x *b3/2o

. . .      . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-------------+----+-------+----------+------
x . .      . |  2 | 48  * |  2  1  0 | 2 1 0
. . x      . |  2 |  * 48 |  0  1  2 | 2 0 1
-------------+----+-------+----------+------
x3o .      . |  3 |  3  0 | 32  *  * | 1 1 0
x . x      . |  4 |  2  2 |  * 24  * | 2 0 0
. o3x      . |  3 |  0  3 |  *  * 32 | 1 0 1
-------------+----+-------+----------+------
x3o3x      . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3o . *b3/2o ♦  4 |  6  0 |  4  0  0 | * 8 *
. o3x *b3/2o ♦  4 |  0  6 |  0  0  4 | * * 8
```

```x3/2o3/2x *b3o

.   .   .    . | 32 ♦  3  3 |  3  3  3 | 3 1 1
---------------+----+-------+----------+------
x   .   .    . |  2 | 48  * |  2  1  0 | 2 1 0
.   .   x    . |  2 |  * 48 |  0  1  2 | 2 0 1
---------------+----+-------+----------+------
x3/2o   .    . |  3 |  3  0 | 32  *  * | 1 1 0
x   .   x    . |  4 |  2  2 |  * 24  * | 2 0 0
.   o3/2x    . |  3 |  0  3 |  *  * 32 | 1 0 1
---------------+----+-------+----------+------
x3/2o3/2x    . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3/2o   . *b3o ♦  4 |  6  0 |  4  0  0 | * 8 *
.   o3/2x *b3o ♦  4 |  0  6 |  0  0  4 | * * 8
```

```x3/2o3/2x *b3/2o

.   .   .      . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-----------------+----+-------+----------+------
x   .   .      . |  2 | 48  * |  2  1  0 | 2 1 0
.   .   x      . |  2 |  * 48 |  0  1  2 | 2 0 1
-----------------+----+-------+----------+------
x3/2o   .      . |  3 |  3  0 | 32  *  * | 1 1 0
x   .   x      . |  4 |  2  2 |  * 24  * | 2 0 0
.   o3/2x      . |  3 |  0  3 |  *  * 32 | 1 0 1
-----------------+----+-------+----------+------
x3/2o3/2x      . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3/2o   . *b3/2o ♦  4 |  6  0 |  4  0  0 | * 8 *
.   o3/2x *b3/2o ♦  4 |  0  6 |  0  0  4 | * * 8
```

```s4x3o3o

demi( . . . . ) | 32 ♦  3  3 |  3  3  3 | 3 1 1
----------------+----+-------+----------+------
demi( . x . . ) |  2 | 48  * |  1  2  0 | 2 1 0
sefa( s4x . . ) |  2 |  * 48 |  1  0  2 | 2 0 1
----------------+----+-------+----------+------
s4x . .   ♦  4 |  2  2 | 24  *  * | 2 0 0
demi( . x3o . ) |  3 |  3  0 |  * 32  * | 1 1 0
sefa( s4x3o . ) |  3 |  0  3 |  *  * 32 | 1 0 1
----------------+----+-------+----------+------
s4x3o .   ♦ 12 | 12 12 |  6  4  4 | 8 * *
demi( . x3o3o ) ♦  4 |  6  0 |  0  4  0 | * 8 *
sefa( s4x3o3o ) ♦  4 |  0  6 |  0  0  4 | * * 8
```

```s4o3o3x

demi( . . . . ) | 32 ♦  3  3 |  3  3  3 | 1 1 3
----------------+----+-------+----------+------
s4o . .   ♦  2 | 48  * |  2  1  0 | 1 0 2
demi( . . . x ) |  2 |  * 48 |  0  1  2 | 0 1 2
----------------+----+-------+----------+------
sefa( s4o3o . ) |  3 |  3  0 | 32  *  * | 1 0 1
sefa( s4o . x ) |  4 |  2  2 |  * 24  * | 0 0 2
demi( . . o3x ) |  3 |  0  3 |  *  * 32 | 0 1 1
----------------+----+-------+----------+------
s4o3o .   ♦  4 |  6  0 |  4  0  0 | 8 * *
demi( . o3o3x ) ♦  4 |  0  6 |  0  0  4 | * 8 *
sefa( s4o3o3x ) ♦ 12 | 12 12 |  4  6  4 | * * 8
```

```xxoo3oxxo3ooxx&#xt   → all heights = 1/sqrt(2) = 0.707107
(tet || pseudo tut || pseudo inv tut || inv tet)

o...3o...3o...     | 4  *  * * ♦ 3  3 0  0  0  0 0  0 0 | 3 3  3 0  0  0  0 0  0 0 0 | 1 3 1 0 0 0 0
.o..3.o..3.o..     | * 12  * * ♦ 0  1 1  2  2  0 0  0 0 | 0 1  2 1  2  2  1 0  0 0 0 | 0 2 1 1 1 0 0
..o.3..o.3..o.     | *  * 12 * ♦ 0  0 0  0  2  2 1  1 0 | 0 0  0 0  1  2  2 1  2 1 0 | 0 1 0 1 2 1 0
...o3...o3...o     | *  *  * 4 ♦ 0  0 0  0  0  0 0  3 3 | 0 0  0 0  0  0  0 0  3 3 3 | 0 0 0 0 3 1 1
-------------------+-----------+------------------------+----------------------------+--------------
x... .... ....     | 2  0  0 0 | 6  * *  *  *  * *  * * | 2 1  0 0  0  0  0 0  0 0 0 | 1 2 0 0 0 0 0
oo..3oo..3oo..&#x  | 1  1  0 0 | * 12 *  *  *  * *  * * | 0 1  2 0  0  0  0 0  0 0 0 | 0 2 1 0 0 0 0
.x.. .... ....     | 0  2  0 0 | *  * 6  *  *  * *  * * | 0 1  0 0  2  0  0 0  0 0 0 | 0 2 0 1 0 0 0
.... .x.. ....     | 0  2  0 0 | *  * * 12  *  * *  * * | 0 0  1 1  0  1  0 0  0 0 0 | 0 1 1 0 1 0 0
.oo.3.oo.3.oo.&#x  | 0  1  1 0 | *  * *  * 24  * *  * * | 0 0  0 0  1  1  1 0  0 0 0 | 0 1 0 1 1 0 0
.... ..x. ....     | 0  0  2 0 | *  * *  *  * 12 *  * * | 0 0  0 0  0  1  0 1  1 0 0 | 0 1 0 0 1 1 0
.... .... ..x.     | 0  0  2 0 | *  * *  *  *  * 6  * * | 0 0  0 0  0  0  2 0  0 1 0 | 0 0 0 1 2 0 0
..oo3..oo3..oo&#x  | 0  0  1 1 | *  * *  *  *  * * 12 * | 0 0  0 0  0  0  0 0  2 1 0 | 0 0 0 0 2 1 0
.... .... ...x     | 0  0  0 2 | *  * *  *  *  * *  * 6 | 0 0  0 0  0  0  0 0  0 1 2 | 0 0 0 0 2 0 1
-------------------+-----------+------------------------+----------------------------+--------------
x...3o... ....     | 3  0  0 0 | 3  0 0  0  0  0 0  0 0 | 4 *  * *  *  *  * *  * * * | 1 1 0 0 0 0 0
xx.. .... ....&#x  | 2  2  0 0 | 1  2 1  0  0  0 0  0 0 | * 6  * *  *  *  * *  * * * | 0 2 0 0 0 0 0
.... ox.. ....&#x  | 1  2  0 0 | 0  2 0  1  0  0 0  0 0 | * * 12 *  *  *  * *  * * * | 0 1 1 0 0 0 0
.... .x..3.o..     | 0  3  0 0 | 0  0 0  3  0  0 0  0 0 | * *  * 4  *  *  * *  * * * | 0 0 1 0 1 0 0
.xo. .... ....&#x  | 0  2  1 0 | 0  0 1  0  2  0 0  0 0 | * *  * * 12  *  * *  * * * | 0 1 0 1 0 0 0
.... .xx. ....&#x  | 0  2  2 0 | 0  0 0  1  2  1 0  0 0 | * *  * *  * 12  * *  * * * | 0 1 0 0 1 0 0
.... .... .ox.&#x  | 0  1  2 0 | 0  0 0  0  2  0 1  0 0 | * *  * *  *  * 12 *  * * * | 0 0 0 1 1 0 0
..o.3..x. ....     | 0  0  3 0 | 0  0 0  0  0  3 0  0 0 | * *  * *  *  *  * 4  * * * | 0 1 0 0 0 1 0
.... ..xo ....&#x  | 0  0  2 1 | 0  0 0  0  0  1 0  2 0 | * *  * *  *  *  * * 12 * * | 0 0 0 0 1 1 0
.... .... ..xx&#x  | 0  0  2 2 | 0  0 0  0  0  0 1  2 1 | * *  * *  *  *  * *  * 6 * | 0 0 0 0 2 0 0
.... ...o3...x     | 0  0  0 3 | 0  0 0  0  0  0 0  0 3 | * *  * *  *  *  * *  * * 4 | 0 0 0 0 1 0 1
-------------------+-----------+------------------------+----------------------------+--------------
x...3o...3o...     ♦ 4  0  0 0 | 6  0 0  0  0  0 0  0 0 | 4 0  0 0  0  0  0 0  0 0 0 | 1 * * * * * *
xxo.3oxx. ....&#xt ♦ 3  6  3 0 | 3  6 3  3  6  3 0  0 0 | 1 3  3 0  3  3  0 1  0 0 0 | * 4 * * * * *
.... ox..3oo..&#x  ♦ 1  3  0 0 | 0  3 0  3  0  0 0  0 0 | 0 0  3 1  0  0  0 0  0 0 0 | * * 4 * * * *
.xo. .... .ox.&#x  ♦ 0  2  2 0 | 0  0 1  0  4  0 1  0 0 | 0 0  0 0  2  0  2 0  0 0 0 | * * * 6 * * *
.... .xxo3.oxx&#xt ♦ 0  3  6 3 | 0  0 0  3  6  3 3  6 3 | 0 0  0 1  0  3  3 0  3 3 1 | * * * * 4 * *
..oo3..xo ....&#x  ♦ 0  0  3 1 | 0  0 0  0  0  3 0  3 0 | 0 0  0 0  0  0  0 1  3 0 0 | * * * * * 4 *
...o3...o3...x     ♦ 0  0  0 4 | 0  0 0  0  0  0 0  0 6 | 0 0  0 0  0  0  0 0  0 0 4 | * * * * * * 1

or
o...3o...3o...     & | 8  * ♦  3  3  0  0  0 | 3  3  3 0  0  0 | 1 3 1 0
.o..3.o..3.o..     & | * 24 ♦  0  1  1  2  2 | 0  1  2 1  3  2 | 0 3 1 1
---------------------+------+----------------+-----------------+--------
x... .... ....     & | 2  0 | 12  *  *  *  * | 2  1  0 0  0  0 | 1 2 0 0
oo..3oo..3oo..&#x  & | 1  1 |  * 24  *  *  * | 0  1  2 0  0  0 | 0 2 1 0
.x.. .... ....     & | 0  2 |  *  * 12  *  * | 0  1  0 0  2  0 | 0 2 0 1
.... .x.. ....     & | 0  2 |  *  *  * 24  * | 0  0  1 1  0  1 | 0 2 1 0
.oo.3.oo.3.oo.&#x    | 0  2 |  *  *  *  * 24 | 0  0  0 0  2  1 | 0 2 0 1
---------------------+------+----------------+-----------------+--------
x...3o... ....     & | 3  0 |  3  0  0  0  0 | 8  *  * *  *  * | 1 1 0 0
xx.. .... ....&#x  & | 2  2 |  1  2  1  0  0 | * 12  * *  *  * | 0 2 0 0
.... ox.. ....&#x  & | 1  2 |  0  2  0  1  0 | *  * 24 *  *  * | 0 1 1 0
.... .x..3.o..     & | 0  3 |  0  0  0  3  0 | *  *  * 8  *  * | 0 1 1 0
.xo. .... ....&#x  & | 0  3 |  0  0  1  0  2 | *  *  * * 24  * | 0 1 0 1
.... .xx. ....&#x    | 0  4 |  0  0  0  2  2 | *  *  * *  * 12 | 0 2 0 0
---------------------+------+----------------+-----------------+--------
x...3o...3o...     & ♦ 4  0 |  6  0  0  0  0 | 4  0  0 0  0  0 | 2 * * *
xxo.3oxx. ....&#xt & ♦ 3  9 |  3  6  3  6  6 | 1  3  3 1  3  3 | * 8 * *
.... ox..3oo..&#x  & ♦ 1  3 |  0  3  0  3  0 | 0  0  3 1  0  0 | * * 8 *
.xo. .... .ox.&#x    ♦ 0  4 |  0  0  2  0  4 | 0  0  0 0  4  0 | * * * 6
```

```ooo3xox4oqo&#xt   → both heights = 1/sqrt(2) = 0.707107
(co || pseudo q-cube || co)

o..3o..4o..     | 12 *  * ♦  4  2  0  0 | 2 2  4  1  0 0 0 | 1 2 2 0 0
.o.3.o.4.o.     |  * 8  * ♦  0  3  3  0 | 0 0  3  3  3 0 0 | 0 1 3 1 0
..o3..o4..o     |  * * 12 ♦  0  0  2  4 | 0 0  0  1  4 2 2 | 0 0 2 2 1
----------------+---------+-------------+------------------+----------
... x.. ...     |  2 0  0 | 24  *  *  * | 1 1  1  0  0 0 0 | 1 1 1 0 0
oo.3oo.4oo.&#x  |  1 1  0 |  * 24  *  * | 0 0  2  1  0 0 0 | 0 1 2 0 0
.oo3.oo4.oo&#x  |  0 1  1 |  *  * 24  * | 0 0  0  1  2 0 0 | 0 0 2 1 0
... ..x ...     |  0 0  2 |  *  *  * 24 | 0 0  0  0  1 1 1 | 0 0 1 1 1
----------------+---------+-------------+------------------+----------
o..3x.. ...     |  3 0  0 |  3  0  0  0 | 8 *  *  *  * * * | 1 1 0 0 0
... x..4o..     |  4 0  0 |  4  0  0  0 | * 6  *  *  * * * | 1 0 1 0 0
... xo. ...&#x  |  2 1  0 |  1  2  0  0 | * * 24  *  * * * | 0 1 1 0 0
... ... oqo&#xt |  1 2  1 |  0  2  2  0 | * *  * 12  * * * | 0 0 2 0 0
... .ox ...&#x  |  0 1  2 |  0  0  2  1 | * *  *  * 24 * * | 0 0 1 1 0
..o3..x ...     |  0 0  3 |  0  0  0  3 | * *  *  *  * 8 * | 0 0 0 1 1
... ..x4..o     |  0 0  4 |  0  0  0  4 | * *  *  *  * * 6 | 0 0 1 0 1
----------------+---------+-------------+------------------+----------
o..3x..4o..     ♦ 12 0  0 | 24  0  0  0 | 8 6  0  0  0 0 0 | 1 * * * *
oo.3xo. ...&#x  ♦  3 1  0 |  3  3  0  0 | 1 0  3  0  0 0 0 | * 8 * * *
... xox4oqo&#xt ♦  4 4  4 |  4  8  8  4 | 0 1  4  4  4 0 1 | * * 6 * *
.oo3.ox ...&#x  ♦  0 1  3 |  0  0  3  3 | 0 0  0  0  3 1 0 | * * * 8 *
..o3..x4..o     ♦  0 0 12 |  0  0  0 24 | 0 0  0  0  0 8 6 | * * * * 1

or
o..3o..4o..      & | 24 * ♦  4  2 |  2  2  4  1 | 1  2 2
.o.3.o.4.o.        |  * 8 ♦  0  6 |  0  0  6  3 | 0  2 3
-------------------+------+-------+-------------+-------
... x.. ...      & |  2 0 | 48  * |  1  1  1  0 | 1  1 1
oo.3oo.4oo.&#x   & |  1 1 |  * 48 |  0  0  2  1 | 0  1 2
-------------------+------+-------+-------------+-------
o..3x.. ...      & |  3 0 |  3  0 | 16  *  *  * | 1  1 0
... x..4o..      & |  4 0 |  4  0 |  * 12  *  * | 1  0 1
... xo. ...&#x   & |  2 1 |  1  2 |  *  * 48  * | 0  1 1
... ... oqo&#xt    |  2 2 |  0  4 |  *  *  * 12 | 0  0 2
-------------------+------+-------+-------------+-------
o..3x..4o..      & ♦ 12 0 | 24  0 |  8  6  0  0 | 2  * *
oo.3xo. ...&#x   & ♦  3 1 |  3  3 |  1  0  3  0 | * 16 *
... xox4oqo&#xt    ♦  8 4 |  8 16 |  0  2  8  4 | *  * 6
```

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