Acronym ...
Name 2hi (?)
Circumradius sqrt[7+3 sqrt(5)] = 3.702459
Inradius (7+3 sqrt(5))/4 = 3.427051
General of army hi
Colonel of regiment hi
Confer
non-Grünbaumian master:
hi  

Looks like a compound of 2 hi with vertices and edges identified, while faces and cells coincide by pairs.

As abstract polytope 2hi is isomorphic to 2gogishi, thereby replacing pentagons by pentagrams resp. replacing doe by gissid.


Incidence matrix according to Dynkin symbol

x5o3o3o3/2*b

. . . .      | 600     4 |   12 |   4   4
-------------+-----+------+------+--------
x . . .      |   2 | 1200 |    6 |   3   3
-------------+-----+------+------+--------
x5o . .      |   5 |    5 | 1440 |   1   1
-------------+-----+------+------+--------
x5o3o .        20 |   30 |   12 | 120   *
x5o . o3/2*b   20 |   30 |   12 |   * 120

x5o3o3/2o3*b

. . .   .    | 600     4 |   12 |   4   4
-------------+-----+------+------+--------
x . .   .    |   2 | 1200 |    6 |   3   3
-------------+-----+------+------+--------
x5o .   .    |   5 |    5 | 1440 |   1   1
-------------+-----+------+------+--------
x5o3o   .      20 |   30 |   12 | 120   *
x5o .   o3*b   20 |   30 |   12 |   * 120
or
. . .   .       | 600     4 |   12 |   8
----------------+-----+------+------+----
x . .   .       |   2 | 1200 |    6 |   6
----------------+-----+------+------+----
x5o .   .       |   5 |    5 | 1440 |   2
----------------+-----+------+------+----
x5o3o   .     &   20 |   30 |   12 | 240

x5o3/2o3/2o3/2*b

. .   .   .      | 600     4 |   12 |   4   4
-----------------+-----+------+------+--------
x .   .   .      |   2 | 1200 |    6 |   3   3
-----------------+-----+------+------+--------
x5o   .   .      |   5 |    5 | 1440 |   1   1
-----------------+-----+------+------+--------
x5o3/2o   .        20 |   30 |   12 | 120   *
x5o   .   o3/2*b   20 |   30 |   12 |   * 120
or
. .   .   .         | 600     4 |   12 |   8
--------------------+-----+------+------+----
x .   .   .         |   2 | 1200 |    6 |   6
--------------------+-----+------+------+----
x5o   .   .         |   5 |    5 | 1440 |   2
--------------------+-----+------+------+----
x5o3/2o   .       &   20 |   30 |   12 | 240

x5/4o3o3o3/2*b

.   . . .      | 600     4 |   12 |   4   4
---------------+-----+------+------+--------
x   . . .      |   2 | 1200 |    6 |   3   3
---------------+-----+------+------+--------
x5/4o . .      |   5 |    5 | 1440 |   1   1
---------------+-----+------+------+--------
x5/4o3o .        20 |   30 |   12 | 120   *
x5/4o . o3/2*b   20 |   30 |   12 |   * 120

x5/4o3o3/2o3*b

.   . .   .    | 600     4 |   12 |   4   4
---------------+-----+------+------+--------
x   . .   .    |   2 | 1200 |    6 |   3   3
---------------+-----+------+------+--------
x5/4o .   .    |   5 |    5 | 1440 |   1   1
---------------+-----+------+------+--------
x5/4o3o   .      20 |   30 |   12 | 120   *
x5/4o .   o3*b   20 |   30 |   12 |   * 120
or
.   . .   .       | 600     4 |   12 |   8
------------------+-----+------+------+----
x   . .   .       |   2 | 1200 |    6 |   6
------------------+-----+------+------+----
x5/4o .   .       |   5 |    5 | 1440 |   2
------------------+-----+------+------+----
x5/4o3o   .     &   20 |   30 |   12 | 240

x5/4o3/2o3/2o3/2*b

.   .   .   .      | 600     4 |   12 |   4   4
-------------------+-----+------+------+--------
x   .   .   .      |   2 | 1200 |    6 |   3   3
-------------------+-----+------+------+--------
x5/4o   .   .      |   5 |    5 | 1440 |   1   1
-------------------+-----+------+------+--------
x5/4o3/2o   .        20 |   30 |   12 | 120   *
x5/4o   .   o3/2*b   20 |   30 |   12 |   * 120
or
.   .   .   .         | 600     4 |   12 |   8
----------------------+-----+------+------+----
x   .   .   .         |   2 | 1200 |    6 |   6
----------------------+-----+------+------+----
x5/4o   .   .         |   5 |    5 | 1440 |   2
----------------------+-----+------+------+----
x5/4o3/2o   .       &   20 |   30 |   12 | 240

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