©

Similar to bilbiro, likewise gibil biro can be obtained by means of an expanded kaleido-faceting, but here by starting with gike instead.

This polyhedron is related also to gid and qrid, because in fact its surface could be decomposed into 4 regions (2 pentagrams plus 2 triangles around an "inner" vertex each, respectively the lunes of a square and the 2 attached triangles of the "outer" envelope), each of which either belong to the one or the other polyhedron.

As abstract polytope gibil biro is isomorphic to bilbiro, thereby replacing pentagrams by pentagons.

Incidence matrix according to Dynkin symbol

xVoVx xo(-v)ox&#xt    → outer heights = (1+sqrt(5))/4 = 0.809017
(V=vv=x-v)              inner heights = -1/2
({4} || pseudo V-line || pseudo ortho v-line || pseudo V-line || {4})

o.... o.( .)..     | 4 * * * * | 1 1 1 1 0 0 0 0 0 | 1 1 1 1 0 0 0
.o... .o( .)..     | * 2 * * * | 0 0 2 0 1 0 0 0 0 | 0 1 0 2 0 0 0
..o.. ..( o)..     | * * 2 * * | 0 0 0 2 0 2 0 0 0 | 0 0 1 2 1 0 0
...o. ..( .)o.     | * * * 2 * | 0 0 0 0 1 0 2 0 0 | 0 0 0 2 0 1 0
....o ..( .).o     | * * * * 4 | 0 0 0 0 0 1 1 1 1 | 0 0 0 1 1 1 1
-------------------+-----------+-------------------+--------------
x.... ..( .)..     | 2 0 0 0 0 | 2 * * * * * * * * | 1 0 1 0 0 0 0
..... x.( .)..     | 2 0 0 0 0 | * 2 * * * * * * * | 1 1 0 0 0 0 0
oo... oo( .)..&#x  | 1 1 0 0 0 | * * 4 * * * * * * | 0 1 0 1 0 0 0
o.o.. o.( o)..&#x  | 1 0 1 0 0 | * * * 4 * * * * * | 0 0 1 1 0 0 0
.o.o. .o( .)o.&#x  | 0 1 0 1 0 | * * * * 2 * * * * | 0 0 0 2 0 0 0
..o.o ..( o).o&#x  | 0 0 1 0 1 | * * * * * 4 * * * | 0 0 0 1 1 0 0
...oo ..( .)oo&#x  | 0 0 0 1 1 | * * * * * * 4 * * | 0 0 0 1 0 1 0
....x ..( .)..     | 0 0 0 0 2 | * * * * * * * 2 * | 0 0 0 0 1 0 1
..... ..( .).x     | 0 0 0 0 2 | * * * * * * * * 2 | 0 0 0 0 0 1 1
-------------------+-----------+-------------------+--------------
x.... x.( .)..     | 4 0 0 0 0 | 2 2 0 0 0 0 0 0 0 | 1 * * * * * *
..... xo( .)..&#x  | 2 1 0 0 0 | 0 1 2 0 0 0 0 0 0 | * 2 * * * * *
x.o.. ..( .)..&#x  | 2 0 1 0 0 | 1 0 0 2 0 0 0 0 0 | * * 2 * * * *
ooooo oo( o)oo&#xt | 1 1 1 1 1 | 0 0 1 1 1 1 1 0 0 | * * * 4 * * *
..o.x ..( .)..&#x  | 0 0 1 0 2 | 0 0 0 0 0 2 0 1 0 | * * * * 2 * *
..... ..( .)ox&#x  | 0 0 0 1 2 | 0 0 0 0 0 0 2 0 1 | * * * * * 2 *
....x ..( .).x     | 0 0 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * 1

or
o.... o.( .)..     & | 8 * * | 1 1 1 1 0 | 1 1 1 1  [3,4,3/2,5/3]
.o... .o( .)..     & | * 4 * | 0 0 2 0 1 | 0 1 0 2  [3,(5/2)2]
..o.. ..( o)..       | * * 2 | 0 0 0 4 0 | 0 0 2 2  [(3,5/2)2]
---------------------+-------+-----------+--------
x.... ..( .)..     & | 2 0 0 | 4 * * * * | 1 0 1 0
..... x.( .)..     & | 2 0 0 | * 4 * * * | 1 1 0 0
oo... oo( .)..&#x  & | 1 1 0 | * * 8 * * | 0 1 0 1
o.o.. o.( o)..&#x  & | 1 0 1 | * * * 8 * | 0 0 1 1
.o.o. .o( .)o.&#x    | 0 2 0 | * * * * 2 | 0 0 0 2
---------------------+-------+-----------+--------
x.... x.( .)..     & | 4 0 0 | 2 2 0 0 0 | 2 * * *
..... xo( .)..&#x  & | 2 1 0 | 0 1 2 0 0 | * 4 * *
x.o.. ..( .)..&#x  & | 2 0 1 | 1 0 0 2 0 | * * 4 *
ooooo oo( o)oo&#xt   | 2 2 1 | 0 0 2 2 1 | * * * 4

oxVxo ov(-x)vo&#xt   → outer heights = (1+sqrt(5))/4 = 0.809017
(V=vv=x-v)              inner height = -1/2
(pt || pseudo (x,v)-{4} || pseudo (V,x)-{4} || pseudo (x,v)-{4} || pt)

o.... o.( .)..     | 1 * * * * | 4 0 0 0 0 0 0 0 | 2 2 0 0 0 0
.o... .o( .)..     | * 4 * * * | 1 1 1 1 0 0 0 0 | 1 1 1 1 0 0
..o.. ..( o)..     | * * 4 * * | 0 0 1 0 1 1 0 0 | 0 1 0 1 1 0
...o. ..( .)o.     | * * * 4 * | 0 0 0 1 0 1 1 1 | 0 0 1 1 1 1
....o ..( .).o     | * * * * 1 | 0 0 0 0 0 0 0 4 | 0 0 0 0 2 2
-------------------+-----------+-----------------+------------
oo... oo( .)..&#x  | 1 1 0 0 0 | 4 * * * * * * * | 1 1 0 0 0 0
.x... ..( .)..     | 0 2 0 0 0 | * 2 * * * * * * | 1 0 1 0 0 0
.oo.. .o( o)..&#x  | 0 1 1 0 0 | * * 4 * * * * * | 0 1 0 1 0 0
.o.o. .o( .)o.&#x  | 0 1 0 1 0 | * * * 4 * * * * | 0 0 1 1 0 0
..... ..(-x)..     | 0 0 2 0 0 | * * * * 2 * * * | 0 1 0 0 1 0
..oo. ..( o)o.&#x  | 0 0 1 1 0 | * * * * * 4 * * | 0 0 0 1 1 0
...x. ..( .)..     | 0 0 0 2 0 | * * * * * * 2 * | 0 0 1 0 0 1
...oo ..( .)oo&#x  | 0 0 0 1 1 | * * * * * * * 4 | 0 0 0 0 1 1
-------------------+-----------+-----------------+------------
ox... ..( .)..&#x  | 1 2 0 0 0 | 2 1 0 0 0 0 0 0 | 2 * * * * *
..... ov(-x)..&#xt | 1 2 2 0 0 | 2 0 2 0 1 0 0 0 | * 2 * * * *
.x.x. ..( .)..&#x  | 0 2 0 2 0 | 0 1 0 2 0 0 1 0 | * * 2 * * *
.ooo. .o( o)o.&#xt | 0 1 1 1 0 | 0 0 1 1 0 1 0 0 | * * * 4 * *
..... ..(-x)vo&#xt | 0 0 2 2 1 | 0 0 0 0 1 2 0 2 | * * * * 2 *
...xo ..( .)..&#x  | 0 0 0 2 1 | 0 0 0 0 0 0 1 2 | * * * * * 2

or
o.... o.( .)..     & | 2 * * | 4 0 0 0 0 | 2 2 0 0  [(3,5/2)2]
.o... .o( .)..     & | * 8 * | 1 1 1 1 0 | 1 1 1 1  [3,4,3/2,5/3]
..o.. ..( o)..       | * * 4 | 0 0 2 0 1 | 0 2 0 1  [3,(5/2)2]
---------------------+-------+-----------+--------
oo... oo( .)..&#x  & | 1 1 0 | 8 * * * * | 1 1 0 0
.x... ..( .)..     & | 0 2 0 | * 4 * * * | 1 0 1 0
.oo.. .o( o)..&#x  & | 0 1 1 | * * 8 * * | 0 1 0 1
.o.o. .o( .)o.&#x    | 0 2 0 | * * * 4 * | 0 0 1 1
..... ..(-x)..       | 0 0 2 | * * * * 2 | 0 2 0 0
---------------------+-------+-----------+--------
ox... ..( .)..&#x  & | 1 2 0 | 2 1 0 0 0 | 4 * * *
..... ov(-x)..&#xt & | 1 2 2 | 2 0 2 0 1 | * 4 * *
.x.x. ..( .)..&#x    | 0 4 0 | 0 2 0 2 0 | * * 2 *
.ooo. .o( o)o.&#xt   | 0 2 1 | 0 0 2 1 0 | * * * 4

vxo ov(-x) oxV&#zxt   → both heights = 0
(V=vv=x-v)
(pseudo (v,o,o)-line || pseudo (x,v,x)-cube || pseudo (o,x,V)-{4})

o.. o.( .) o..     | 2 * * | 4 0 0 0 0 | 2 2 0 0  [(3,5/2)2]
.o. .o( .) .o.     | * 8 * | 1 1 1 1 0 | 1 1 1 1  [3,4,3/2,5/3]
..o ..( o) ..o     | * * 4 | 0 0 0 2 1 | 2 0 0 1  [3,(5/2)2]
-------------------+-------+-----------+--------
oo. oo( .) oo.&#x  | 1 1 0 | 8 * * * * | 1 1 0 0
.x. ..( .) ...     | 0 2 0 | * 4 * * * | 0 0 1 1
... ..( .) .x.     | 0 2 0 | * * 4 * * | 0 1 1 0
.oo .o( o) .oo&#x  | 0 1 1 | * * * 8 * | 1 0 0 1
... ..(-x) ...     | 0 0 2 | * * * * 2 | 2 0 0 0
-------------------+-------+-----------+--------
... ov(-x) ...&#xt | 1 2 2 | 2 0 0 2 1 | 4 * * *
... ..( .) ox.&#x  | 1 2 0 | 2 0 1 0 0 | * 4 * *
.x. ..( .) .x.     | 0 4 0 | 0 2 2 0 0 | * * 2 *
.xo ..( .) ...&#x  | 0 2 1 | 0 1 0 2 0 | * * * 4