Acronym hexaico, hex || ico
Name hexadecachoron-first rotunda of rectified triacontiditeron,
hexadecachoron atop icositetrachoron,
rectified hexadecachoron-pyramid
Circumradius 1
Lace city
in approx. ASCII-art 
o3o4o  x3o4o  o3o4o		-- x3o3o4o (hex)
                   
                   
                   
x3o4o  o3x4o  x3o4o		-- o3x3o4o (ico)

    x3o3o   o3o3x    		-- x3o3o4o (hex)
                     
                     
                     
o3x3o   x3o3x   o3x3o		-- o3x3o4o (ico)
Coordinates (1/sqrt(2), 1/sqrt(2), 0, 0, 0)   & all permutations, all changes of sign, but last coord non-negative
Face vector 32, 168, 248, 136, 26
Confer
uniform relative:
rat  
related segmentotera:
co || hex  
ambification pre-image:
hexpy  
general polytopal classes:
segmentotera   fundamental lace prisms   lace simplices  

It is quite remarkable that hexpy allows for some of the same operations as regulars do: this polyteron is nothing but the rectification operation being applied to it.


Incidence matrix according to Dynkin symbol

xo3ox3oo4oo&#x   → height = 1/sqrt(2) = 0.707107
(hex || ico)

o.3o.3o.4o.    | 8  * ♦  6  6  0 | 12  6 12  0  0 |  8 12  8  0 0 | 1  8 1 0
.o3.o3.o4.o    | * 24 ♦  0  2  8 |  0  1  8  4  8 |  0  4  8  4 2 | 0  4 2 1
---------------+------+----------+----------------+---------------+---------
x. .. .. ..    | 2  0 | 24  *  * ♦  4  1  0  0  0 |  4  4  0  0 0 | 1  4 0 0
oo3oo3oo4oo&#x | 1  1 |  * 48  * |  0  1  4  0  0 |  0  4  4  0 0 | 0  4 1 0
.. .x .. ..    | 0  2 |  *  * 96 |  0  0  1  1  2 |  0  1  2  2 1 | 0  2 1 1
---------------+------+----------+----------------+---------------+---------
x.3o. .. ..    | 3  0 |  3  0  0 | 32  *  *  *  * |  2  1  0  0 0 | 1  2 0 0
xo .. .. ..&#x | 2  1 |  1  2  0 |  * 24  *  *  * |  0  4  0  0 0 | 0  4 0 0
.. ox .. ..&#x | 1  2 |  0  2  1 |  *  * 96  *  * |  0  1  2  0 0 | 0  2 1 0
.o3.x .. ..    | 0  3 |  0  0  3 |  *  *  * 32  * |  0  1  0  2 0 | 0  2 0 1
.. .x3.o ..    | 0  3 |  0  0  3 |  *  *  *  * 64 |  0  0  1  1 8 | 0  1 1 1
---------------+------+----------+----------------+---------------+---------
x.3o.3o. ..    ♦ 4  0 |  6  0  0 |  4  0  0  0  0 | 16  *  *  * * | 1  1 0 0
xo3ox .. ..&#x ♦ 3  3 |  3  6  3 |  1  3  3  1  0 |  * 32  *  * * | 0  2 0 0
.. ox3oo ..&#x ♦ 1  3 |  0  3  3 |  0  0  3  0  1 |  *  * 64  * * | 0  1 1 0
.o3.x3.o ..    ♦ 0  6 |  0  0 12 |  0  0  0  4  4 |  *  *  * 16 * | 0  1 0 1
.. .x3.o4.o    ♦ 0  6 |  0  0 12 |  0  0  0  0  8 |  *  *  *  * 8 | 0  0 1 1
---------------+------+----------+----------------+---------------+---------
x.3o.3o.4o.    ♦ 8  0 | 24  0  0 | 32  0  0  0  0 | 16  0  0  0 0 | 1  * * *
xo3ox3oo ..&#x ♦ 4  6 |  6 12 12 |  4  6 12  4  4 |  1  4  4  1 0 | * 16 * *
.. ox3oo4oo&#x ♦ 1  6 |  0  6 12 |  0  0 12  0  8 |  0  0  8  0 1 | *  * 8 *
.o3.x3.o4.o    ♦ 0 24 |  0  0 96 |  0  0  0 32 64 |  0  0  0 16 8 | *  * * 1


oo3ox3oo *b3xo&#x   → height = 1/sqrt(2) = 0.707107
(hex || ico)

o.3o.3o. *b3o.    | 8  * ♦  6  6  0 | 12 12  6  0  0  0 | 4 4  4  4 12 0 0 0 | 1 4 4 1 0
.o3.o3.o *b3.o    | * 24 ♦  0  2  8 |  0  8  1  4  4  4 | 0 0  4  4  4 2 2 2 | 0 2 2 2 1
------------------+------+----------+-------------------+--------------------+----------
.. .. ..    x.    | 2  0 | 24  *  * ♦  4  0  1  0  0  0 | 2 2  0  0  4 0 0 0 | 1 2 2 0 0
oo3oo3oo *b3oo&#x | 1  1 |  * 48  * |  0  4  1  0  0  0 | 0 0  2  2  4 0 0 0 | 0 2 2 1 0
.. .x ..    ..    | 0  2 |  *  * 96 |  0  1  0  1  1  1 | 0 0  1  1  1 1 1 1 | 0 1 1 1 1
------------------+------+----------+-------------------+--------------------+----------
.. o. .. *b3x.    | 3  0 |  3  0  0 | 32  *  *  *  *  * | 1 1  0  0  1 0 0 0 | 1 1 1 0 0
.. ox ..    ..&#x | 1  2 |  0  2  1 |  * 96  *  *  *  * | 0 0  1  1  1 0 0 0 | 0 1 1 1 0
.. .. ..    xo&#x | 2  1 |  1  2  0 |  *  * 24  *  *  * | 0 0  0  0  4 0 0 0 | 0 2 2 0 0
.o3.x ..    ..    | 0  3 |  0  0  3 |  *  *  * 32  *  * | 0 0  1  0  0 1 1 0 | 0 1 0 1 1
.. .x3.o    ..    | 0  3 |  0  0  3 |  *  *  *  * 32  * | 0 0  0  1  0 1 0 1 | 0 0 1 1 1
.. .x .. *b3.o    | 0  3 |  0  0  3 |  *  *  *  *  * 32 | 0 0  0  0  1 0 1 1 | 0 1 1 0 1
------------------+------+----------+-------------------+--------------------+----------
o.3o. .. *b3x.    ♦ 4  0 |  6  0  0 |  4  0  0  0  0  0 | 8 *  *  *  * * * * | 1 1 0 0 0
.. o.3o. *b3x.    ♦ 4  0 |  6  0  0 |  4  0  0  0  0  0 | * 8  *  *  * * * * | 1 0 1 0 0
oo3ox ..    ..&#x ♦ 1  3 |  0  3  3 |  0  3  0  1  0  0 | * * 32  *  * * * * | 0 1 0 1 0
.. ox3oo    ..&#x ♦ 1  3 |  0  3  3 |  0  3  0  0  1  0 | * *  * 32  * * * * | 0 0 1 1 0
.. ox .. *b3xo&#x ♦ 3  3 |  3  6  3 |  1  3  3  0  0  1 | * *  *  * 32 * * * | 0 1 1 0 0
.o3.x3.o    ..    ♦ 0  6 |  0  0 12 |  0  0  0  4  4  0 | * *  *  *  * 8 * * | 0 0 0 1 1
.o3.x .. *b3.o    ♦ 0  6 |  0  0 12 |  0  0  0  4  0  4 | * *  *  *  * * 8 * | 0 1 0 0 1
.. .x3.o *b3.o    ♦ 0  6 |  0  0 12 |  0  0  0  0  4  4 | * *  *  *  * * * 8 | 0 0 1 0 1
------------------+------+----------+-------------------+--------------------+----------
o.3o.3o. *b3x.    ♦ 8  0 | 24  0  0 | 32  0  0  0  0  0 | 8 8  0  0  0 0 0 0 | 1 * * * *
oo3ox .. *b3xo&#x ♦ 4  6 |  6 12 12 |  4 12  6  4  0  4 | 1 0  4  0  4 0 1 0 | * 8 * * *
.. ox3oo *b3xo&#x ♦ 4  6 |  6 12 12 |  4 12  6  0  4  4 | 0 1  0  4  4 0 0 1 | * * 8 * *
oo3ox3oo    ..&#x ♦ 1  6 |  0  6 12 |  0 12  0  4  4  0 | 0 0  4  4  0 1 0 0 | * * * 8 *
.o3.x3.o *b3.o    ♦ 0 24 |  0  0 96 |  0  0  0 32 32 32 | 0 0  0  0  0 8 8 8 | * * * * 1


rect( xoo3ooo3oxo&#x )

mids(  x.. ... ...    ) & | 12  * * ♦  4  4  0  2  0  0  0 | 2  2  1  0 2  4 0  2  4 0  4  0  0 | 1 2 1 2  2 2 2 0  4 0 0  2 0 | 1 1 2 1 2 0
mids(  oo.3oo.3oo.&#x )   |  * 12 * ♦  0  4  4  0  0  2  0 | 0  4  0  1 0  2 2  4  0 0  4  4  0 | 0 2 2 0  4 2 0 0  2 2 0  4 0 | 1 0 2 2 2 0
mids(  o.o3o.o3o.o&#x ) & |  *  * 8 ♦  0  0  0  3  3  3  3 | 0  0  3  3 0  0 0  0  3 3  6  3  9 | 0 0 0 3  9 0 1 1  3 1 4  3 3 | 0 1 4 3 1 1
--------------------------+---------+----------------------+------------------------------------+------------------------------+------------
verf(  x..3o.. ...    ) & |  2  0 0 | 24  *  *  *  *  *  * | 1  0  0  0 1  1 0  0  1 0  0  0  0 | 1 1 0 1  0 1 1 0  1 0 0  0 0 | 1 1 1 0 1 0
verfA( xo. ... ...&#x ) & |  1  1 0 |  * 48  *  *  *  *  * | 0  1  0  0 0  1 0  1  0 0  1  0  0 | 0 1 1 0  1 1 0 0  1 0 0  1 0 | 1 0 1 1 1 0
verfB( xo. ... ...&#x ) & |  0  2 0 |  *  * 24  *  *  *  * | 0  1  0  0 0  0 1  1  0 0  0  1  0 | 0 1 1 0  1 1 0 0  0 1 0  1 0 | 1 0 1 1 1 0
verfA( x.o ... ...&#x ) & |  1  0 1 |  *  *  * 24  *  *  * | 0  0  1  0 0  0 0  0  2 0  2  0  0 | 0 0 0 2  2 0 1 0  2 0 0  1 0 | 0 1 2 1 1 0
verfB( x.o ... ...&#x ) & |  0  0 2 |  *  *  *  * 12  *  * ♦ 0  0  1  0 0  0 0  0  0 2  0  0  2 | 0 0 0 2  2 0 0 1  0 0 2  0 1 | 0 1 2 1 0 1
verfA( ooo3ooo3ooo&#x )   |  0  1 1 |  *  *  *  *  * 24  * | 0  0  0  1 0  0 0  0  0 0  2  2  0 | 0 0 0 0  4 0 0 0  1 1 0  2 0 | 0 0 2 2 1 0
verfB( ooo3ooo3ooo&#x )   |  0  0 2 |  *  *  *  *  *  * 12 ♦ 0  0  0  1 0  0 0  0  0 0  0  0  4 | 0 0 0 0  4 0 0 0  0 0 2  0 2 | 0 0 2 2 0 1
--------------------------+---------+----------------------+------------------------------------+------------------------------+------------
rect(  x..3o.. ...    ) & |  3  0 0 |  3  0  0  0  0  0  0 | 8  *  *  * *  * *  *  * *  *  *  * | 1 1 0 1  0 0 0 0  0 0 0  0 0 | 1 1 1 0 0 0
rect(  xo. ... ...&#x ) & |  1  2 0 |  0  2  1  0  0  0  0 | * 24  *  * *  * *  *  * *  *  *  * | 0 1 1 0  1 0 0 0  0 0 0  0 0 | 1 0 1 1 0 0
rect(  x.o ... ...&#x ) & |  1  0 2 |  0  0  0  2  1  0  0 | *  * 12  * *  * *  *  * *  *  *  * | 0 0 0 2  2 0 0 0  0 0 0  0 0 | 0 1 2 1 0 0
rect(  ooo3ooo3ooo&#x )   |  0  1 2 |  0  0  0  0  0  2  1 | *  *  * 12 *  * *  *  * *  *  *  * | 0 0 0 0  4 0 0 0  0 0 0  0 0 | 0 0 2 2 0 0
verf(  x..3o..3o..    ) & |  3  0 0 |  3  0  0  0  0  0  0 | *  *  *  * 8  * *  *  * *  *  *  * | 1 0 0 0  0 1 1 0  0 0 0  0 0 | 1 1 0 0 1 0
verfA( xo.3oo. ...&#x ) & |  2  1 0 |  1  2  0  0  0  0  0 | *  *  *  * * 24 *  *  * *  *  *  * | 0 1 0 0  0 1 0 0  1 0 0  0 0 | 1 0 1 0 1 0
verfB( xo.3oo. ...&#x ) & |  0  3 0 |  0  0  3  0  0  0  0 | *  *  *  * *  * 8  *  * *  *  *  * | 0 1 0 0  0 1 0 0  0 1 0  0 0 | 1 0 1 0 1 0
verf(  xo. ... ox.&#x )   |  1  2 0 |  0  2  1  0  0  0  0 | *  *  *  * *  * * 24  * *  *  *  * | 0 0 1 0  0 1 0 0  0 0 0  1 0 | 1 0 0 1 1 0
verfA( x.o3o.o ...&#x ) & |  2  0 1 |  1  0  0  2  0  0  0 | *  *  *  * *  * *  * 24 *  *  *  * | 0 0 0 1  0 0 1 0  1 0 0  0 0 | 0 1 1 0 1 0
verfB( x.o3o.o ...&#x ) & |  0  0 3 |  0  0  0  0  3  0  0 | *  *  *  * *  * *  *  * 8  *  *  * | 0 0 0 1  0 0 0 1  0 0 1  0 0 | 0 1 1 0 0 1
verfA( xoo ... ...&#x ) & |  1  1 1 |  0  1  0  1  0  1  0 | *  *  *  * *  * *  *  * * 48  *  * | 0 0 0 0  1 0 0 0  1 0 0  1 0 | 0 0 1 1 1 0
verfB( xoo ... ...&#x ) & |  0  2 1 |  0  0  1  0  0  2  0 | *  *  *  * *  * *  *  * *  * 24  * | 0 0 0 0  1 0 0 0  0 1 0  1 0 | 0 0 1 1 1 0
verfC( xoo ... ...&#x ) & |  0  0 3 |  0  0  0  0  1  0  2 | *  *  *  * *  * *  *  * *  *  * 24 | 0 0 0 0  1 0 0 0  0 0 1  0 1 | 0 0 1 1 0 1
--------------------------+---------+----------------------+------------------------------------+------------------------------+------------
rect(  x..3o..3o..    ) & ♦  6  0 0 | 12  0  0  0  0  0  0 | 4  0  0  0 4  0 0  0  0 0  0  0  0 | 2 * * *  * * * *  * * *  * * | 1 1 0 0 0 0
rect(  xo.3oo. ...&#x ) & ♦  3  3 0 |  3  6  3  0  0  0  0 | 1  3  0  0 0  3 1  0  0 0  0  0  0 | * 8 * *  * * * *  * * *  * * | 1 0 1 0 0 0
rect(  xo. ... ox.&#x )   ♦  2  4 0 |  0  8  4  0  0  0  0 | 0  4  0  0 0  0 0  4  0 0  0  0  0 | * * 6 *  * * * *  * * *  * * | 1 0 0 1 0 0
rect(  x.o3o.o ...&#x ) & ♦  3  0 3 |  3  0  0  6  3  0  0 | 1  0  3  0 0  0 0  0  3 1  0  0  0 | * * * 8  * * * *  * * *  * * | 0 1 1 0 0 0
rect(  xoo ... ...&#x ) & ♦  1  2 3 |  0  2  1  2  1  4  2 | 0  1  1  2 0  0 0  0  0 0  2  1  1 | * * * * 24 * * *  * * *  * * | 0 0 1 1 0 0
verf(  xo.3oo.3ox.&#x )   ♦  3  3 0 |  3  6  3  0  0  0  0 | 0  0  0  0 1  3 1  3  0 0  0  0  0 | * * * *  * 8 * *  * * *  * * | 1 0 0 0 1 0
verfA( x.o3o.o3o.o&#x ) & ♦  3  0 1 |  3  0  0  3  0  0  0 | 0  0  0  0 1  0 0  0  3 0  0  0  0 | * * * *  * * 8 *  * * *  * * | 0 1 0 0 1 0
verfB( x.o3o.o3o.o&#x ) & ♦  0  0 4 |  0  0  0  0  6  0  0 | 0  0  0  0 0  0 0  0  0 4  0  0  0 | * * * *  * * * 2  * * *  * * | 0 1 0 0 0 1
verfA( xoo3ooo ...&#x ) & ♦  2  1 1 |  1  2  0  2  0  1  0 | 0  0  0  0 0  1 0  0  1 0  2  0  0 | * * * *  * * * * 24 * *  * * | 0 0 1 0 1 0
verfB( xoo3ooo ...&#x ) & ♦  0  3 1 |  0  0  3  0  0  3  0 | 0  0  0  0 0  0 1  0  0 0  0  3  0 | * * * *  * * * *  * 8 *  * * | 0 0 1 0 1 0
verfC( xoo3ooo ...&#x ) & ♦  0  0 4 |  0  0  0  0  3  0  3 | 0  0  0  0 0  0 0  0  0 1  0  0  3 | * * * *  * * * *  * * 8  * * | 0 0 1 0 0 1
verfA( xoo ... oxo&#x )   ♦  1  2 1 |  0  2  1  1  0  2  0 | 0  0  0  0 0  0 0  1  0 0  2  1  0 | * * * *  * * * *  * * * 24 * | 0 0 0 1 1 0
verfB( xoo ... oxo&#x )   ♦  0  0 4 |  0  0  0  0  2  0  4 | 0  0  0  0 0  0 0  0  0 0  0  0  4 | * * * *  * * * *  * * *  * 6 | 0 0 0 1 0 1
--------------------------+---------+----------------------+------------------------------------+------------------------------+------------
rect(  xo.3oo.3ox.&#x )   ♦ 12 12 0 | 24 48 24  0  0  0  0 | 8 24  0  0 8 24 8 24  0 0  0  0  0 | 2 8 6 0  0 8 0 0  0 0 0  0 0 | 1 * * * * *
rect(  x.o3o.o3o.o&#x ) & ♦  6  0 4 | 12  0  0 12  6  0  0 | 4  0  6  0 4  0 0  0 12 4  0  0  0 | 1 0 0 4  0 0 4 1  0 0 0  0 0 | * 2 * * * *
rect(  xoo3ooo ...&#x ) & ♦  3  3 4 |  3  6  3  6  3  6  3 | 1  3  3  3 0  3 1  0  3 1  6  3  3 | 0 1 0 1  3 0 0 0  3 1 1  0 0 | * * 8 * * *
rect(  xoo ... oxo&#x )   ♦  2  4 4 |  0  8  4  4  2  8  4 | 0  4  2  4 0  0 0  4  0 0  8  4  4 | 0 0 1 0  4 0 0 0  0 0 0  4 1 | * * * 6 * *
verfA( xoo3ooo3oxo&#x )   ♦  3  3 1 |  3  6  3  3  0  3  0 | 0  0  0  0 1  3 1  3  3 0  6  3  0 | 0 0 0 0  0 1 1 0  3 1 0  3 0 | * * * * 8 *
verfB( xoo3ooo3oxo&#x )   ♦  0  0 8 |  0  0  0  0 12  0 12 | 0  0  0  0 0  0 0  0  0 8  0  0 24 | 0 0 0 0  0 0 0 2  0 0 8  0 6 | * * * * * 1

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