Acronym ...
Name xo3oo3ox&#q,
longer variant of tetrahedral antiprism,
vertex figure of scad
Circumradius sqrt(3)/2 = 0.866025
Lace city
in approx. ASCII-art 
 x3o o3o
        
        
        
o3o o3x
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Face vector 8, 24, 32, 16
Confer
uniform variant:
hex  

Incidence matrix according to Dynkin symbol

xo3oo3ox&#q   → height = sqrt(3/2) = 1.224745
(tet || dual tet – with q-lacings)

o.3o.3o.    | 4 * ♦ 3  3 0 | 3  6  3 0 | 1 3 3 1 0
.o3.o3.o    | * 4 ♦ 0  3 3 | 0  3  6 3 | 0 1 3 3 1
------------+-----+--------+-----------+----------
x. .. ..    | 2 0 | 6  * * | 2  2  0 0 | 1 2 1 0 0
oo3oo3oo&#q | 1 1 | * 12 * | 0  2  2 0 | 0 1 2 1 0
.. .. .x    | 0 2 | *  * 6 | 0  0  2 2 | 0 0 1 2 1
------------+-----+--------+-----------+----------
x.3o. ..    | 3 0 | 3  0 0 | 4  *  * * | 1 1 0 0 0
xo .. ..&#q | 2 1 | 1  2 0 | * 12  * * | 0 1 1 0 0
.. .. ox&#q | 1 2 | 0  2 1 | *  * 12 * | 0 0 1 1 0
.. .o3.x    | 0 3 | 0  0 3 | *  *  * 4 | 0 0 0 1 1
------------+-----+--------+-----------+----------
x.3o.3o.    ♦ 4 0 | 6  0 0 | 4  0  0 0 | 1 * * * *
xo3oo ..&#q ♦ 3 1 | 3  3 0 | 1  3  0 0 | * 4 * * *
xo .. ox&#q ♦ 2 2 | 1  4 1 | 0  2  2 0 | * * 6 * *
.. oo3ox&#q ♦ 1 3 | 0  3 3 | 0  0  3 1 | * * * 4 *
.o3.o3.x    ♦ 0 4 | 0  0 6 | 0  0  0 4 | * * * * 1

or
o.3o.3o.    & | 8 ♦  3  3 | 3  9 | 1 4 3
--------------+---+-------+------+------
x. .. ..    & | 2 | 12  * | 2  2 | 1 2 1
oo3oo3oo&#q   | 2 |  * 12 | 0  4 | 0 2 2
--------------+---+-------+------+------
x.3o. ..    & | 3 |  3  0 | 8  * | 1 1 0
xo .. ..&#q & | 3 |  1  2 | * 24 | 0 1 1
--------------+---+-------+------+------
x.3o.3o.    & ♦ 4 |  6  0 | 4  0 | 2 * *
xo3oo ..&#q & ♦ 4 |  3  3 | 1  3 | * 8 *
xo .. ox&#q   ♦ 4 |  2  4 | 0  4 | * * 6


oxo3ooo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..4o..    | 1 * * ♦ 6  0 0 | 12 0  0 | 8 0
.o.3.o.4.o.    | * 6 * ♦ 1  4 1 |  4 4  4 | 4 4
..o3..o4..o    | * * 1 ♦ 0  0 6 |  0 0 12 | 0 8
---------------+-------+--------+---------+----
oo.3oo.4oo.&#x | 1 1 0 | 6  * * |  4 0  0 | 4 0
.x. ... ...    | 0 2 0 | * 12 * |  1 2  1 | 2 2
.oo3.oo4.oo&#x | 0 1 1 | *  * 6 |  0 0  4 | 0 4
---------------+-------+--------+---------+----
ox. ... ...&#x | 1 2 0 | 2  1 0 | 12 *  * | 2 0
.x.3.o. ...    | 0 3 0 | 0  3 0 |  * 8  * | 1 1
.xo ... ...&#x | 0 2 1 | 0  1 2 |  * * 12 | 0 2
---------------+-------+--------+---------+----
ox.3oo. ...&#x ♦ 1 3 0 | 3  3 0 |  3 1  0 | 8 *
.xo3.oo ...&#x ♦ 0 3 1 | 0  3 3 |  0 1  3 | * 8

or
o..3o..4o..    & | 2 * ♦  6  0 | 12 0 |  8
.o.3.o.4.o.      | * 6 ♦  2  4 |  8 4 |  8
-----------------+-----+-------+------+---
oo.3oo.4oo.&#x & | 1 1 | 12  * |  4 0 |  4
.x. ... ...      | 0 2 |  * 12 |  2 2 |  4
-----------------+-----+-------+------+---
ox. ... ...&#x & | 1 2 |  2  1 | 24 * |  2
.x.3.o. ...      | 0 3 |  0  3 |  * 8 |  2
-----------------+-----+-------+------+---
ox.3oo. ...&#x & ♦ 1 3 |  3  3 |  3 1 | 16


ooo3oxo3ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..3o..    | 1 * * ♦ 6  0 0 | 12 0 0  0 | 4 4 0 0
.o.3.o.3.o.    | * 6 * ♦ 1  4 1 |  4 2 2  4 | 2 2 2 2
..o3..o3..o    | * * 1 ♦ 0  0 6 |  0 0 0 12 | 0 0 4 4
---------------+-------+--------+-----------+--------
oo.3oo.3oo.&#x | 1 1 0 | 6  * * |  4 0 0  0 | 2 2 0 0
... .x. ...    | 0 2 0 | * 12 * |  1 1 1  1 | 1 1 1 1
.oo3.oo3.oo&#x | 0 1 1 | *  * 6 |  0 0 0  4 | 0 0 2 2
---------------+-------+--------+-----------+--------
... ox. ...&#x | 1 2 0 | 2  1 0 | 12 * *  * | 1 1 0 0
.o.3.x. ...    | 0 3 0 | 0  3 0 |  * 4 *  * | 1 0 1 0
... .x.3.o.    | 0 3 0 | 0  3 0 |  * * 4  * | 0 1 0 1
... .xo ...&#x | 0 2 1 | 0  1 2 |  * * * 12 | 0 0 1 1
---------------+-------+--------+-----------+--------
oo.3ox. ...&#x ♦ 1 3 0 | 3  3 0 |  3 1 0  0 | 4 * * *
... ox.3oo.&#x ♦ 1 3 0 | 3  3 0 |  3 0 1  0 | * 4 * *
.oo3.xo ...&#x ♦ 0 3 1 | 0  3 3 |  0 1 0  3 | * * 4 *
... .xo3.oo&#x ♦ 0 3 1 | 0  3 3 |  0 0 1  3 | * * * 4

or
o..3o..3o..    & | 2 * ♦  6  0 | 12 0 0 | 4 4
.o.3.o.3.o.      | * 6 ♦  2  4 |  8 2 2 | 4 4
-----------------+-----+-------+--------+----
oo.3oo.3oo.&#x & | 1 1 | 12  * |  4 0 0 | 2 2
... .x. ...      | 0 2 |  * 12 |  2 1 1 | 2 2
-----------------+-----+-------+--------+----
... ox. ...&#x & | 1 2 |  2  1 | 24 * * | 1 1
.o.3.x. ...      | 0 3 |  0  3 |  * 4 * | 2 0
... .x.3.o.      | 0 3 |  0  3 |  * * 4 | 0 2
-----------------+-----+-------+--------+----
oo.3ox. ...&#x & ♦ 1 3 |  3  3 |  3 1 0 | 8 *
... ox.3oo.&#x & ♦ 1 3 |  3  3 |  3 0 1 | * 8


xox oxo4ooo&#xt   → both heights = 1/2
(line || perp pseudo {4} || line)

o.. o..4o..    | 2 * * ♦ 1 4 1 0 0 0 | 4 4 4 0 0 | 4 4 0
.o. .o.4.o.    | * 4 * ♦ 0 2 0 2 2 0 | 1 4 2 1 4 | 2 4 2
..o ..o4..o    | * * 2 ♦ 0 0 1 0 4 1 | 0 0 4 4 4 | 0 4 4
---------------+-------+-------------+-----------+------
x.. ... ...    | 2 0 0 | 1 * * * * * | 4 0 0 0 0 | 4 0 0
oo. oo.4oo.&#x | 1 1 0 | * 8 * * * * | 1 2 1 0 0 | 2 2 0
o.o o.o4o.o&#x | 1 0 1 | * * 2 * * * | 0 0 4 0 0 | 0 4 0
... .x. ...    | 0 2 0 | * * * 4 * * | 0 2 0 0 2 | 1 2 1
.oo .oo4.oo&#x | 0 1 1 | * * * * 8 * | 0 0 1 1 2 | 0 2 2
..x ... ...    | 0 0 2 | * * * * * 1 | 0 0 0 4 0 | 0 0 4
---------------+-------+-------------+-----------+------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 4 * * * * | 2 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 8 * * * | 1 1 0
ooo ooo4ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 8 * * | 0 2 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 2 1 | * * * 4 * | 0 0 2
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * 8 | 0 1 1
---------------+-------+-------------+-----------+------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 | 2 2 0 0 0 | 4 * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 1 2 0 1 | * 8 *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 4 1 | 0 0 0 2 2 | * * 4

or
o.. o..4o..    & | 4 * ♦ 1  4 1 0 | 4  4 4 | 4 4
.o. .o.4.o.      | * 4 ♦ 0  4 0 2 | 2  8 2 | 4 4
-----------------+-----+----------+--------+----
x.. ... ...    & | 2 0 | 2  * * * | 4  0 0 | 4 0
oo. oo.4oo.&#x & | 1 1 | * 16 * * | 1  2 1 | 2 2
o.o o.o4o.o&#x   | 2 0 | *  * 2 * | 0  0 4 | 0 4
... .x. ...      | 0 2 | *  * * 4 | 0  4 0 | 2 2
-----------------+-----+----------+--------+----
xo. ... ...&#x & | 2 1 | 1  2 0 0 | 8  * * | 2 0
... ox. ...&#x & | 1 2 | 0  2 0 1 | * 16 * | 1 1
ooo ooo4ooo&#x   | 2 1 | 0  2 1 0 | *  * 8 | 0 2
-----------------+-----+----------+--------+----
xo. ox. ...&#x & ♦ 2 2 | 1  4 0 1 | 2  2 0 | 8 *
... oxo ...&#x   ♦ 2 2 | 0  4 1 1 | 0  2 2 | * 8


xox oxo oxo&#xt   → both heights = 1/2
(line || perp pseudo {4} || line)

o.. o.. o..    | 2 * * ♦ 1 4 1 0 0 0 0 | 4 2 2 4 0 0 0 | 2 2 2 2 0 0
.o. .o. .o.    | * 4 * ♦ 0 2 0 1 1 2 0 | 1 2 2 2 1 2 2 | 1 1 2 2 1 1
..o ..o ..o    | * * 2 ♦ 0 0 1 0 0 4 1 | 0 0 0 4 4 2 2 | 0 0 2 2 2 2
---------------+-------+---------------+---------------+------------
x.. ... ...    | 2 0 0 | 1 * * * * * * | 4 0 0 0 0 0 0 | 2 2 0 0 0 0
oo. oo. oo.&#x | 1 1 0 | * 8 * * * * * | 1 1 1 1 0 0 0 | 1 1 1 1 0 0
o.o o.o o.o&#x | 1 0 1 | * * 2 * * * * | 0 0 0 4 0 0 0 | 0 0 2 2 0 0
... .x. ...    | 0 2 0 | * * * 2 * * * | 0 2 0 0 0 2 0 | 1 0 2 0 1 0
... ... .x.    | 0 2 0 | * * * * 2 * * | 0 0 2 0 0 0 2 | 0 1 0 2 0 1
.oo .oo .oo&#x | 0 1 1 | * * * * * 8 * | 0 0 0 1 1 1 1 | 0 0 1 1 1 1
..x ... ...    | 0 0 2 | * * * * * * 1 | 0 0 0 0 4 0 0 | 0 0 0 0 2 2
---------------+-------+---------------+---------------+------------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 0 | * 4 * * * * * | 1 0 1 0 0 0
... ... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * 4 * * * * | 0 1 0 1 0 0
ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * 8 * * * | 0 0 1 1 0 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 0 2 1 | * * * * 4 * * | 0 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * 4 * | 0 0 1 0 1 0
... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * 4 | 0 0 0 1 0 1
---------------+-------+---------------+---------------+------------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 0 | 2 2 0 0 0 0 0 | 2 * * * * *
xo. ... ox.&#x ♦ 2 2 0 | 1 4 0 0 1 0 0 | 2 0 2 0 0 0 0 | * 2 * * * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 0 2 0 | 0 1 0 2 0 1 0 | * * 4 * * *
... ... oxo&#x ♦ 1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 2 0 0 1 | * * * 4 * *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 0 4 1 | 0 0 0 0 2 2 0 | * * * * 2 *
.ox ... .xo&#x ♦ 0 2 2 | 0 0 0 0 1 4 1 | 0 0 0 0 2 0 2 | * * * * * 2

or
o.. o.. o..    & | 4 * ♦ 1  4 1 0 0 | 4 2 2 4 | 2 2 2 2
.o. .o. .o.      | * 4 ♦ 0  4 0 1 1 | 2 4 4 2 | 2 2 2 2
-----------------+-----+------------+---------+--------
x.. ... ...    & | 2 0 | 2  * * * * | 4 0 0 0 | 2 2 0 0
oo. oo. oo.&#x & | 1 1 | * 16 * * * | 1 1 1 1 | 1 1 1 1
o.o o.o o.o&#x   | 2 0 | *  * 2 * * | 0 0 0 4 | 0 0 2 2
... .x. ...      | 0 2 | *  * * 2 * | 0 4 0 0 | 2 0 2 0
... ... .x.      | 0 2 | *  * * * 2 | 0 0 4 0 | 0 2 0 2
-----------------+-----+------------+---------+--------
xo. ... ...&#x & | 2 1 | 1  2 0 0 0 | 8 * * * | 1 1 0 0
... ox. ...&#x & | 1 2 | 0  2 0 1 0 | * 8 * * | 1 0 1 0
... ... ox.&#x & | 1 2 | 0  2 0 0 1 | * * 8 * | 0 1 0 1
ooo ooo ooo&#x   | 2 1 | 0  2 1 0 0 | * * * 8 | 0 0 1 1
-----------------+-----+------------+---------+--------
xo. ox. ...&#x & ♦ 2 2 | 1  4 0 1 0 | 2 2 0 0 | 4 * * *
xo. ... ox.&#x & ♦ 2 2 | 1  4 0 0 1 | 2 0 2 0 | * 4 * *
... oxo ...&#x   ♦ 2 2 | 0  4 1 1 0 | 0 2 0 2 | * * 4 *
... ... oxo&#x   ♦ 2 2 | 0  4 1 0 1 | 0 0 2 2 | * * * 4


xoo3oox oqo&#xt   → both heights = 1/sqrt(6) = 0.408248
({3} || perp q-line || dual {3})

o..3o.. o..     | 3 * * ♦ 2 2 2 0 0 | 1 4 2 1  4 0 0 | 2 4 2 0
.o.3.o. .o.     | * 2 * ♦ 0 3 0 3 0 | 0 3 0 0  6 3 0 | 1 3 3 1
..o3..o ..o     | * * 3 ♦ 0 0 2 2 2 | 0 0 1 2  4 4 1 | 0 2 4 2
----------------+-------+-----------+----------------+--------
x.. ... ...     | 2 0 0 | 3 * * * * | 1 2 1 0  0 0 0 | 2 2 0 0
oo.3oo. oo.&#x  | 1 1 0 | * 6 * * * | 0 2 0 0  2 0 0 | 1 2 1 0
o.o3o.o o.o&#x  | 1 0 1 | * * 6 * * | 0 0 1 1  2 0 0 | 0 2 2 0
.oo3.oo .oo&#x  | 0 1 1 | * * * 6 * | 0 0 0 0  2 2 0 | 0 1 2 1
... ..x ...     | 0 0 2 | * * * * 3 | 0 0 0 1  0 2 1 | 0 0 2 2
----------------+-------+-----------+----------------+--------
x..3o.. ...     | 3 0 0 | 3 0 0 0 0 | 1 * * *  * * * | 2 0 0 0
xo. ... ...&#x  | 2 1 0 | 1 2 0 0 0 | * 6 * *  * * * | 1 1 0 0
x.o ... ...&#x  | 2 0 1 | 1 0 2 0 0 | * * 3 *  * * * | 0 2 0 0
... o.x ...&#x  | 1 0 2 | 0 0 2 0 1 | * * * 3  * * * | 0 0 2 0
ooo3ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * * * * 12 * * | 0 1 1 0
... .ox ...&#x  | 0 1 2 | 0 0 0 2 1 | * * * *  * 6 * | 0 0 1 1
..o3..x ...     | 0 0 3 | 0 0 0 0 3 | * * * *  * * 1 | 0 0 0 2
----------------+-------+-----------+----------------+--------
xo.3oo. ...&#x  ♦ 3 1 0 | 3 3 0 0 0 | 1 3 0 0  0 0 0 | 2 * * *
xoo ... ...&#xt ♦ 2 1 1 | 1 2 2 1 0 | 0 1 1 0  2 0 0 | * 6 * *
... oox ...&#xt ♦ 1 1 2 | 0 1 2 2 1 | 0 0 0 1  2 1 0 | * * 6 *
.oo3.ox ...&#x  ♦ 0 1 3 | 0 0 0 3 3 | 0 0 0 0  0 3 1 | * * * 2

or
o..3o.. o..      & | 6 * ♦ 2  2 2 | 1  4 3  4 | 2  6
.o.3.o. .o.        | * 2 ♦ 0  6 0 | 0  6 0  6 | 2  6
-------------------+-----+--------+-----------+-----
x.. ... ...      & | 2 0 | 6  * * | 1  2 1  0 | 2  2
oo.3oo. oo.&#x   & | 1 1 | * 12 * | 0  2 0  2 | 1  3
o.o3o.o o.o&#x     | 2 0 | *  * 6 | 0  0 2  2 | 0  4
-------------------+-----+--------+-----------+-----
x..3o.. ...      & | 3 0 | 3  0 0 | 2  * *  * | 2  0
xo. ... ...&#x   & | 2 1 | 1  2 0 | * 12 *  * | 1  1
x.o ... ...&#x   & | 3 0 | 1  0 2 | *  * 6  * | 0  2
ooo3ooo ooo&#xt    | 2 1 | 0  2 1 | *  * * 12 | 0  2
-------------------+-----+--------+-----------+-----
xo.3oo. ...&#x   & ♦ 3 1 | 3  3 0 | 1  3 0  0 | 4  *
xoo ... ...&#xt  & ♦ 3 1 | 1  3 2 | 0  1 1  2 | * 12


oxoo3ooox&#xr   → all heights = sqrt(2/3) = 0.816497
(pt || ({3} || inv {3}) || pt)

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

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