Acronym traf, K-4.6
Name triangular antifastegium,
triangle oct wedge,
{3} || gyro trip,
{3} || oct,
(vertex-) diminished rap
Segmentochoron display
Lace city
in approx. ASCII-art
```   x3o

x3o   o3x
```
```x o   o x

x x   o o
```
Lace hyper city
in approx. ASCII-art
 ``` o o o ``` ``` x x x ```
Dihedral angles
• at {4} between squippy and trip:   arccos(-sqrt[1/6]) = 114.094843°
• at {3} between oct and tet:   arccos(-1/4) = 104.477512°
• at {3} between squippy and tet:   arccos(-1/4) = 104.477512°
• at {3} between oct and oct:   arccos(1/4) = 75.522488°
• at {3} between oct and squippy:   arccos(1/4) = 75.522488°
• at {3} between oct and trip:   arccos(sqrt[3/8]) = 52.238756°
Confer
segmentochora family:
n-af   n/d-af
uniform relative:
rap
related segmentochora:
bidrap
variations:
xo xo3ox&#q
general polytopal classes:
antifastegia   segmentochora   lace simplices
External

Incidence matrix according to Dynkin symbol

```xoo3oxx&#x   → height(1,2) = height(1,3) = sqrt(2/3) = 0.816497
height(2,3) = 1
( {3} || (dual {3} || dual {3}) )

o..3o..    | 3 * * ♦ 2 2 2 0 0 0 | 1 2 1 2 1 2 0 0 0 | 1 1 2 1 0
.o.3.o.    | * 3 * | 0 2 0 2 1 0 | 0 1 2 0 0 2 1 2 0 | 1 0 1 2 1
..o3..o    | * * 3 | 0 0 2 0 1 2 | 0 0 0 1 2 2 0 2 1 | 0 1 1 2 1
-----------+-------+-------------+-------------------+----------
x.. ...    | 2 0 0 | 3 * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
oo.3oo.&#x | 1 1 0 | * 6 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
o.o3o.o&#x | 1 0 1 | * * 6 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
... .x.    | 0 2 0 | * * * 3 * * | 0 0 1 0 0 0 1 1 0 | 1 0 0 1 1
.oo3.oo&#x | 0 1 1 | * * * * 3 * | 0 0 0 0 0 2 0 2 0 | 0 0 1 2 1
... ..x    | 0 0 2 | * * * * * 3 | 0 0 0 0 1 0 0 1 1 | 0 1 0 1 1
-----------+-------+-------------+-------------------+----------
x..3o..    | 3 0 0 | 3 0 0 0 0 0 | 1 * * * * * * * * | 1 1 0 0 0
xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0
... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * * 3 * * * * * * | 1 0 0 1 0
x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 3 * * * * * | 0 1 1 0 0
... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 3 * * * * | 0 1 0 1 0
ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 6 * * * | 0 0 1 1 0
.o.3.x.    | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 1 * * | 1 0 0 0 1
... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * * * * 3 * | 0 0 0 1 1
..o3..x    | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * 1 | 0 1 0 0 1
-----------+-------+-------------+-------------------+----------
xo.3ox.&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 | 1 * * * *
x.o3o.x&#x ♦ 3 0 3 | 3 0 6 0 0 3 | 1 0 0 3 3 0 0 0 1 | * 1 * * *
xoo ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 | * * 3 * *
... oxx&#x ♦ 1 2 2 | 0 2 2 1 2 1 | 0 0 1 0 1 2 0 1 0 | * * * 3 *
.oo3.xx&#x ♦ 0 3 3 | 0 0 0 3 3 3 | 0 0 0 0 0 0 1 3 1 | * * * * 1
```

```xo3ox ox&#x   → height = sqrt(5/12) = 0.645497
({3} || gyro trip)

o.3o. o.    | 3 * ♦ 2  4 0 0 | 1 4 2 2 0 0 | 2 2 1 0
.o3.o .o    | * 6 | 0  2 2 1 | 0 1 2 2 1 2 | 1 1 2 1
------------+-----+----------+-------------+--------
x. .. ..    | 2 0 | 3  * * * | 1 2 0 0 0 0 | 2 1 0 0
oo3oo oo&#x | 1 1 | * 12 * * | 0 1 1 1 0 0 | 1 1 1 0
.. .x ..    | 0 2 | *  * 6 * | 0 0 1 0 1 1 | 1 0 1 1
.. .. .x    | 0 2 | *  * * 3 | 0 0 0 2 0 2 | 0 1 2 1
------------+-----+----------+-------------+--------
x.3o. ..    | 3 0 | 3  0 0 0 | 1 * * * * * | 2 0 0 0
xo .. ..&#x | 2 1 | 1  2 0 0 | * 6 * * * * | 1 1 0 0
.. ox ..&#x | 1 2 | 0  2 1 0 | * * 6 * * * | 1 0 1 0
.. .. ox&#x | 1 2 | 0  2 0 1 | * * * 6 * * | 0 1 1 0
.o3.x ..    | 0 3 | 0  0 3 0 | * * * * 2 * | 1 0 0 1
.. .x .x    | 0 4 | 0  0 2 2 | * * * * * 3 | 0 0 1 1
------------+-----+----------+-------------+--------
xo3ox ..&#x ♦ 3 3 | 3  6 3 0 | 1 3 3 0 1 0 | 2 * * *
xo .. ox&#x ♦ 2 2 | 1  4 0 1 | 0 2 0 2 0 0 | * 3 * *
.. ox ox&#x ♦ 1 4 | 0  4 2 2 | 0 0 2 2 0 1 | * * 3 *
.o3.x .x    ♦ 0 6 | 0  0 6 3 | 0 0 0 0 2 3 | * * * 1
```

```{3} || oct   → height = sqrt(5/8) = 0.790569

o..3o..    | 3 * * ♦ 2 2 2 0 0 0 | 1 2 1 2 1 2 0 0 0 | 1 1 2 1 0
.o.3.o.    | * 3 * | 0 2 0 2 1 0 | 0 1 2 0 0 2 1 2 0 | 1 0 1 2 1
..o3..o    | * * 3 | 0 0 2 0 1 2 | 0 0 0 1 2 2 0 2 1 | 0 1 1 2 1
-----------+-------+-------------+-------------------+----------
x.. ...    | 2 0 0 | 3 * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
oo.3oo.&#x | 1 1 0 | * 6 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
o.o3o.o&#x | 1 0 1 | * * 6 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
... .x.    | 0 2 0 | * * * 3 * * | 0 0 1 0 0 0 1 1 0 | 1 0 0 1 1
.oo3.oo&#x | 0 1 1 | * * * * 3 * | 0 0 0 0 0 2 0 2 0 | 0 0 1 2 1
... ..x    | 0 0 2 | * * * * * 3 | 0 0 0 0 1 0 0 1 1 | 0 1 0 1 1
-----------+-------+-------------+-------------------+----------
x..3o..    | 3 0 0 | 3 0 0 0 0 0 | 1 * * * * * * * * | 1 1 0 0 0
xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0
... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * * 3 * * * * * * | 1 0 0 1 0
x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 3 * * * * * | 0 1 1 0 0
... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 3 * * * * | 0 1 0 1 0
ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 6 * * * | 0 0 1 1 0
.o.3.x.    | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 1 * * | 1 0 0 0 1
... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * * * * 3 * | 0 0 0 1 1
..o3..x    | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * 1 | 0 1 0 0 1
-----------+-------+-------------+-------------------+----------
xo.3ox.&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 | 1 * * * *
x.o3o.x&#x ♦ 3 0 3 | 3 0 6 0 0 3 | 1 0 0 3 3 0 0 0 1 | * 1 * * *
xoo ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 | * * 3 * *
... oxx&#x ♦ 1 2 2 | 0 2 2 1 2 1 | 0 0 1 0 1 2 0 1 0 | * * * 3 *
.oo3.xx&#x ♦ 0 3 3 | 0 0 0 3 3 3 | 0 0 0 0 0 0 1 3 1 | * * * * 1
```

```tet || squippy   → height = sqrt(5/8) = 0.790569

2 * * * | 1 2 2 0 0 0 0 0 0 | 2 2 1 2 1 0 0 0 0 0 0 | 1 2 1 1 0 0  top edge of tet
* 2 * * ♦ 0 2 0 1 2 1 0 0 0 | 1 0 2 2 0 1 1 2 0 0 0 | 1 1 0 2 1 0  bottom edge of tet
* * 4 * | 0 0 1 0 1 0 1 1 1 | 0 1 0 1 1 0 1 1 1 1 1 | 0 1 1 1 1 1  top base of squippy
* * * 1 ♦ 0 0 0 0 0 2 0 0 4 | 0 0 0 0 0 1 0 4 0 2 2 | 0 0 0 2 2 1  bottom tip of squippy
--------+-------------------+-----------------------+------------
2 0 0 0 | 1 * * * * * * * * | 2 2 0 0 0 0 0 0 0 0 0 | 1 2 1 0 0 0
1 1 0 0 | * 4 * * * * * * * | 1 0 1 1 0 0 0 0 0 0 0 | 1 1 0 1 0 0
1 0 1 0 | * * 4 * * * * * * | 0 1 0 1 1 0 0 0 0 0 0 | 0 1 1 1 0 0
0 2 0 0 | * * * 1 * * * * * | 0 0 2 0 0 1 0 0 0 0 0 | 1 0 0 2 0 0
0 1 1 0 | * * * * 4 * * * * | 0 0 0 1 0 0 1 1 0 0 0 | 0 1 0 1 1 0
0 1 0 1 | * * * * * 2 * * * | 0 0 0 0 0 1 0 2 0 0 0 | 0 0 0 2 1 0
0 0 2 0 | * * * * * * 2 * * | 0 1 0 0 0 0 1 0 1 1 0 | 0 1 1 0 1 1
0 0 2 0 | * * * * * * * 2 * | 0 0 0 0 1 0 0 0 1 0 1 | 0 0 1 1 0 1
0 0 1 1 | * * * * * * * * 4 | 0 0 0 0 0 0 0 1 0 1 1 | 0 0 0 1 1 1
--------+-------------------+-----------------------+------------
2 1 0 0 | 1 2 0 0 0 0 0 0 0 | 2 * * * * * * * * * * | 1 1 0 0 0 0
2 0 2 0 | 1 0 2 0 0 0 1 0 0 | * 2 * * * * * * * * * | 0 1 1 0 0 0
1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 2 * * * * * * * * | 1 0 0 1 0 0
1 1 1 0 | 0 1 1 0 1 0 0 0 0 | * * * 4 * * * * * * * | 0 1 0 1 0 0
1 0 2 0 | 0 0 2 0 0 0 0 1 0 | * * * * 2 * * * * * * | 0 0 1 1 0 0
0 2 0 1 | 0 0 0 1 0 2 0 0 0 | * * * * * 1 * * * * * | 0 0 0 2 0 0  gyrated {3}
0 1 2 0 | 0 0 0 0 2 0 1 0 0 | * * * * * * 2 * * * * | 0 1 0 0 1 0
0 1 1 1 | 0 0 0 0 1 1 0 0 1 | * * * * * * * 4 * * * | 0 0 0 1 1 0
0 0 4 0 | 0 0 0 0 0 0 2 2 0 | * * * * * * * * 1 * * | 0 0 1 0 0 1
0 0 2 1 | 0 0 0 0 0 0 1 0 2 | * * * * * * * * * 2 * | 0 0 0 0 1 1
0 0 2 1 | 0 0 0 0 0 0 0 1 2 | * * * * * * * * * * 2 | 0 0 0 1 0 1
--------+-------------------+-----------------------+------------
2 2 0 0 | 1 4 0 1 0 0 0 0 0 | 2 0 2 0 0 0 0 0 0 0 0 | 1 * * * * *  tet
2 1 2 0 | 1 2 2 0 2 0 1 0 0 | 1 1 0 2 0 0 1 0 0 0 0 | * 2 * * * *  squippy
2 0 4 0 | 1 0 4 0 0 0 2 2 0 | 0 2 0 0 2 0 0 0 1 0 0 | * * 1 * * *  trip
1 2 2 1 | 0 2 2 1 2 2 0 1 2 | 0 0 1 2 1 1 0 2 0 0 1 | * * * 2 * *  oct
0 1 2 1 | 0 0 0 0 2 1 1 0 2 | 0 0 0 0 0 0 1 2 0 1 0 | * * * * 2 *  tet
0 0 4 1 | 0 0 0 0 0 0 2 2 4 | 0 0 0 0 0 0 0 0 1 2 2 | * * * * * 1  squippy
```

```xoxoxo&#xr   → all consecutive pairwise heights = sqrt(3)/2 = 0.866025
all alternating pairwise heights = 1
Note: these lengths show that this cycle is not flat, rather it is wobbling up and down!

o.....     & | 6 * | 1  2 2 0 | 2 2 2 1 1 0 | 2 1 1 1
.o....     & | * 3 ♦ 0  4 0 2 | 2 2 0 0 4 1 | 1 2 0 2
-------------+-----+----------+-------------+--------
x.....     & | 2 0 | 3  * * * | 2 0 2 0 0 0 | 2 0 1 1
oo....&#x  & | 1 1 | * 12 * * | 1 1 0 0 1 0 | 1 1 0 1
o.o...&#x  & | 2 0 | *  * 6 * | 0 1 1 1 0 0 | 1 1 1 0
.o.o..&#x  & | 0 2 | *  * * 3 | 0 0 0 0 2 1 | 0 2 0 1
-------------+-----+----------+-------------+--------
xo....&#x  & | 2 1 | 1  2 0 0 | 6 * * * * * | 1 0 0 1
ooo...&#x  & | 2 1 | 0  2 1 0 | * 6 * * * * | 1 1 0 0
x.x...&#x  & | 4 0 | 2  0 2 0 | * * 3 * * * | 1 0 1 0
o.o.o.&#x    | 3 0 | 0  0 3 0 | * * * 2 * * | 0 1 1 0
.ooo..&#x  & | 1 2 | 0  2 0 1 | * * * * 6 * | 0 1 0 1
.o.o.o&#x    | 0 3 | 0  0 0 3 | * * * * * 1 | 0 2 0 0
-------------+-----+----------+-------------+--------
xox...&#x  & ♦ 4 1 | 2  4 2 0 | 2 2 1 0 0 0 | 3 * * *
oooooo&#x    ♦ 3 3 | 0  6 3 3 | 0 3 0 1 3 1 | * 2 * *
x.x.x.&#x    ♦ 6 0 | 3  0 6 0 | 0 0 3 2 0 0 | * * 1 *
.oxo..&#x  & ♦ 2 2 | 1  4 0 1 | 2 0 0 0 2 0 | * * * 3
```

```ooxx oxox&#xr   → height(1,2) = height(2,4) = height(3,4) = sqrt(3)/2 = 0.866025
height(1,4) = height(2,3) = 1/sqrt(2) = 0.707107

o... o...     | 1 * * * ♦ 2 4 0 0 0 0 0 0 0 | 1 4 2 2 0 0 0 0 0 0 0 | 2 2 1 0 0 0
.o.. .o..     | * 2 * * ♦ 1 0 1 2 2 0 0 0 0 | 1 2 0 0 1 2 2 1 0 0 0 | 2 1 0 1 1 0
..o. ..o.     | * * 2 * | 0 0 0 2 0 1 2 0 0 | 0 0 0 0 2 1 2 0 2 1 0 | 1 0 0 1 2 1
...o ...o     | * * * 4 | 0 1 0 0 1 0 1 1 1 | 0 1 1 1 0 0 1 1 1 1 1 | 1 1 1 0 1 1
--------------+---------+-------------------+-----------------------+------------
oo.. oo..&#x  | 1 1 0 0 | 2 * * * * * * * * | 1 2 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0
o..o o..o&#x  | 1 0 0 1 | * 4 * * * * * * * | 0 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0 0
.... .x..     | 0 2 0 0 | * * 1 * * * * * * | 1 0 0 0 0 2 0 0 0 0 0 | 2 0 0 1 0 0
.oo. .oo.&#x  | 0 1 1 0 | * * * 4 * * * * * | 0 0 0 0 1 1 1 0 0 0 0 | 1 0 0 1 1 0
.o.o .o.o&#x  | 0 1 0 1 | * * * * 4 * * * * | 0 1 0 0 0 0 1 1 0 0 0 | 1 1 0 0 1 0
..x. ....     | 0 0 2 0 | * * * * * 1 * * * | 0 0 0 0 2 0 0 0 2 0 0 | 0 0 0 1 2 1
..oo ..oo&#x  | 0 0 1 1 | * * * * * * 4 * * | 0 0 0 0 0 0 1 0 1 1 0 | 1 0 0 0 1 1
...x ....     | 0 0 0 2 | * * * * * * * 2 * | 0 0 1 0 0 0 0 1 1 0 1 | 0 1 1 0 1 1
.... ...x     | 0 0 0 2 | * * * * * * * * 2 | 0 0 0 1 0 0 0 0 0 1 1 | 1 0 1 0 0 1
--------------+---------+-------------------+-----------------------+------------
.... ox..&#x  | 1 2 0 0 | 2 0 1 0 0 0 0 0 0 | 1 * * * * * * * * * * | 2 0 0 0 0 0
oo.o oo.o&#x  | 1 1 0 1 | 1 1 0 0 1 0 0 0 0 | * 4 * * * * * * * * * | 1 1 0 0 0 0
o..x ....&#x  | 1 0 0 2 | 0 2 0 0 0 0 0 1 0 | * * 2 * * * * * * * * | 0 1 1 0 0 0
.... o..x&#x  | 1 0 0 2 | 0 2 0 0 0 0 0 0 1 | * * * 2 * * * * * * * | 1 0 1 0 0 0
.ox. ....&#x  | 0 1 2 0 | 0 0 0 2 0 1 0 0 0 | * * * * 2 * * * * * * | 0 0 0 1 1 0
.... .xo.&#x  | 0 2 1 0 | 0 0 1 2 0 0 0 0 0 | * * * * * 2 * * * * * | 1 0 0 1 0 0
.ooo .ooo&#x  | 0 1 1 1 | 0 0 0 1 1 0 1 0 0 | * * * * * * 4 * * * * | 1 0 0 0 1 0
.o.x ....&#x  | 0 1 0 2 | 0 0 0 0 2 0 0 1 0 | * * * * * * * 2 * * * | 0 1 0 0 1 0
..xx ....&#x  | 0 0 2 2 | 0 0 0 0 0 1 2 1 0 | * * * * * * * * 2 * * | 0 0 0 0 1 1
.... ..ox&#x  | 0 0 1 2 | 0 0 0 0 0 0 2 0 1 | * * * * * * * * * 2 * | 1 0 0 0 0 1
...x ...x     | 0 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * 1 | 0 0 1 0 0 1
--------------+---------+-------------------+-----------------------+------------
.... oxox&#xr ♦ 1 2 1 2 | 2 2 1 2 2 0 2 0 1 | 1 2 0 1 0 1 2 0 0 1 0 | 2 * * * * *
oo.x ....&#x  ♦ 1 1 0 2 | 1 2 0 0 2 0 0 1 0 | 0 2 1 0 0 0 0 1 0 0 0 | * 2 * * * *
o..x o..x&#x  ♦ 1 0 0 4 | 0 4 0 0 0 0 0 2 2 | 0 0 2 2 0 0 0 0 0 0 1 | * * 1 * * *
.ox. .xo.&#x  ♦ 0 2 2 0 | 0 0 1 4 0 1 0 0 0 | 0 0 0 0 2 2 0 0 0 0 0 | * * * 1 * *
.oxx ....&#x  ♦ 0 1 2 2 | 0 0 0 2 2 1 2 1 0 | 0 0 0 0 1 0 2 1 1 0 0 | * * * * 2 *
..xx ..ox&#x  ♦ 0 0 2 4 | 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 2 2 1 | * * * * * 1
```