Acronym ...
Name q x4o,
variation of cube,
vertex figure of rectified cubic honeycomb
Circumradius 1
Lace city
in approx. ASCII-art
x x
   
x x
q q
q q
o q o
     
o q o
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dihedral angles
  • between x4o and q x:   90°
  • between q x and q x (at q):   90°
Confer
general prisms:
n-p  
uniform variant:
cube  
variations:
f x4o   u x4o   w x4o  

using edge sizes x = 1 and q = sqrt(2) = 1.414214


Incidence matrix according to Dynkin symbol

q x4o

. . . | 8 | 1 2 | 2 1
------+---+-----+----
q . . | 2 | 4 * | 2 0
. x . | 2 | * 8 | 1 1
------+---+-----+----
q x . | 4 | 2 2 | 4 *
. x4o | 4 | 0 4 | * 2

q x x

. . . | 8 | 1 1 1 | 1 1 1
------+---+-------+------
q . . | 2 | 4 * * | 1 1 0
. x . | 2 | * 4 * | 1 0 1
. . x | 2 | * * 4 | 0 1 1
------+---+-------+------
q x . | 4 | 2 2 0 | 2 * *
q . x | 4 | 2 0 2 | * 2 *
. x x | 4 | 0 2 2 | * * 2

xx4oo&#q   → height = sqrt(2) = 1.414214
({4} || {4})

o.4o.    | 4 * | 2 1 0 | 1 2 0
.o4.o    | * 4 | 0 1 2 | 0 2 1
---------+-----+-------+------
x. ..    | 2 0 | 4 * * | 1 1 0
oo4oo&#q | 1 1 | * 4 * | 0 2 0
.x ..    | 0 2 | * * 4 | 0 1 1
---------+-----+-------+------
x.4o.    | 4 0 | 4 0 0 | 1 * *
xx ..&#q | 2 2 | 1 2 1 | * 4 *
.x4.o    | 0 4 | 0 0 4 | * * 1

xx xx&#q   → height = sqrt(2) = 1.414214
({4} || {4})

o. o.    | 4 * | 1 1 1 0 0 | 1 1 1 0
.o .o    | * 4 | 0 0 1 1 1 | 0 1 1 1
---------+-----+-----------+--------
x. ..    | 2 0 | 2 * * * * | 1 1 0 0
.. x.    | 2 0 | * 2 * * * | 1 0 1 0
oo oo&#q | 1 1 | * * 4 * * | 0 1 1 0
.x ..    | 0 2 | * * * 2 * | 0 1 0 1
.. .x    | 0 2 | * * * * 2 | 0 0 1 1
---------+-----+-----------+--------
x. x.    | 4 0 | 2 2 0 0 0 | 1 * * *
xx ..&#q | 2 2 | 1 0 2 1 0 | * 2 * *
.. xx&#q | 2 2 | 0 1 2 0 1 | * * 2 *
.x .x    | 0 4 | 0 0 0 2 2 | * * * 1

qq xx&#x   → height = 1
((q,x)-{4} || (q,x)-{4})

o. o.    | 4 * | 1 1 1 0 0 | 1 1 1 0
.o .o    | * 4 | 0 0 1 1 1 | 0 1 1 1
---------+-----+-----------+--------
q. ..    | 2 0 | 2 * * * * | 1 1 0 0
.. x.    | 2 0 | * 2 * * * | 1 0 1 0
oo oo&#x | 1 1 | * * 4 * * | 0 1 1 0
.q ..    | 0 2 | * * * 2 * | 0 1 0 1
.. .x    | 0 2 | * * * * 2 | 0 0 1 1
---------+-----+-----------+--------
q. x.    | 4 0 | 2 2 0 0 0 | 1 * * *
qq ..&#x | 2 2 | 1 0 2 1 0 | * 2 * *
.. xx&#x | 2 2 | 0 1 2 0 1 | * * 2 *
.q .x    | 0 4 | 0 0 0 2 2 | * * * 1

qqq oqo&#xt   → both heights = 1/sqrt(2) = 0.707107
(q-line || pseudo q-{4} || q-line)

o.. o..     | 2 * * | 1 2 0 0 0 | 2 1 0
.o. .o.     | * 4 * | 0 1 1 1 0 | 1 1 1
..o ..o     | * * 2 | 0 0 0 2 1 | 0 1 2
------------+-------+-----------+------
q.. ...     | 2 0 0 | 1 * * * * | 2 0 0
oo. oo.&#x  | 1 1 0 | * 4 * * * | 1 1 0
.q. ...     | 0 2 0 | * * 2 * * | 1 0 1
.oo .oo&#x  | 0 1 1 | * * * 4 * | 0 1 1
..q ...     | 0 0 2 | * * * * 1 | 0 0 2
------------+-------+-----------+------
qq. ...&#x  | 2 2 0 | 1 2 1 0 0 | 2 * *
... oqo&#xt | 1 2 1 | 0 2 0 2 0 | * 2 *
.qq ...&#x  | 0 2 2 | 0 0 1 2 1 | * * 2
or
o.. o..      & | 4 * | 1 2 0 | 2 1
.o. .o.        | * 4 | 0 2 1 | 2 1
---------------+-----+-------+----
q.. ...      & | 2 0 | 2 * * | 2 0
oo. oo.&#x   & | 1 1 | * 8 * | 1 1
.q. ...        | 0 2 | * * 2 | 2 0
---------------+-----+-------+----
qq. ...&#x   & | 2 2 | 1 2 1 | 4 *
... oqo&#xt    | 2 2 | 0 4 0 | * 2

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