2gahi

Acronym	...
Name	2gahi (?)
Circumradius	(1+sqrt(5))/2 = 1.618034
Inradius	(1+sqrt(5))/4 = 0.809017
Coordinates	(τ, 0, 0, 0) & all permutations, all changes of sign (vertex inscribed f/q-hex) (τ/2, τ/2, τ/2, τ/2) & all permutations, all changes of sign (vertex inscribed f-tes) (τ²/2, τ/2, 1/2, 0) & even permutations, all changes of sign (vertex inscribed sadi) where τ = (1+sqrt(5))/2 (a. and b. together define a vertex inscribed f-ico)
General of army	ex
Colonel of regiment	ex
Confer	non-Grünbaumian master: gahi Grünbaumian relatives: fix+gahi+120id gahi+gohi general polytopal classes: Wythoffian polychora

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

As abstract polytope 2gahi is isomorphic to 2gishi, thereby replacing doe by gissid, resp. replacing pentagonal faces by pentagrammal ones, resp. replacing 2gike vertex figures by 2ike ones.

Incidence matrix according to Dynkin symbol

x5o3o5o3/2*b

. . . .      | 120 ♦  12 |   60 |  20  20
-------------+-----+-----+------+--------
x . . .      |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x5o . .      |   5 |   5 | 1440 |   1   1
-------------+-----+-----+------+--------
x5o3o .      ♦  20 |  30 |   12 | 120   *
x5o . o3/2*b ♦  20 |  30 |   12 |   * 120

x5o3o5/4o3*b

. . .   .    | 120 ♦  12 |   60 |  20  20
-------------+-----+-----+------+--------
x . .   .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x5o .   .    |   5 |   5 | 1440 |   1   1
-------------+-----+-----+------+--------
x5o3o   .    ♦  20 |  30 |   12 | 120   *
x5o .   o3*b ♦  20 |  30 |   12 |   * 120

or
. . .   .       | 120 ♦  12 |   60 |  40
----------------+-----+-----+------+----
x . .   .       |   2 | 720 |   10 |  10
----------------+-----+-----+------+----
x5o .   .       |   5 |   5 | 1440 |   2
----------------+-----+-----+------+----
x5o3o   .     & ♦  20 |  30 |   12 | 240

x5o3/2o5/4o3/2*b

. .   .   .      | 120 ♦  12 |   60 |  20  20
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5o   .   o3/2*b ♦  20 |  30 |   12 |   * 120

or
. .   .   .         | 120 ♦  12 |   60 |  40
--------------------+-----+-----+------+----
x .   .   .         |   2 | 720 |   10 |  10
--------------------+-----+-----+------+----
x5o   .   .         |   5 |   5 | 1440 |   2
--------------------+-----+-----+------+----
x5o3/2o   .       & ♦  20 |  30 |   12 | 240

x5/4o3o5o3/2*b

.   . . .      | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/4o . .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/4o3o .      ♦  20 |  30 |   12 | 120   *
x5/4o . o3/2*b ♦  20 |  30 |   12 |   * 120

x5/4o3o5/4o3*b

.   . .   .    | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/4o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/4o3o   .    ♦  20 |  30 |   12 | 120   *
x5/4o .   o3*b ♦  20 |  30 |   12 |   * 120

or
.   . .   .       | 120 ♦  12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/4o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/4o3o   .     & ♦  20 |  30 |   12 | 240

x5/4o3/2o5/4o3/2*b

.   .   .   .      | 120 ♦  12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/4o   .   o3/2*b ♦  20 |  30 |   12 |   * 120

or
.   .   .   .         | 120 ♦  12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/4o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/4o3/2o   .       & ♦  20 |  30 |   12 | 240

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