Acronym bilbiro, J91 Name bilunabirotunda ` © ©   ©` Vertex figures [(3,5)2], [3,4,3,5], [3,52] Lace cityin approx. ASCII-art ``` x f f o o f f x ``` ``` o o x x f x x o o ``` ``` x x F where: o o F=ff=x+f F x x ``` Coordinates (F/2, 1/2, 0)         & all changes of sign (1/2, f/2, 1/2)       & all changes of sign (0, 0, f/2)             & all changes of sign where f = (1+sqrt(5))/2 = 1.618034, F = (3+sqrt(5))/2 = 2.618034 General of army (is itself convex) Colonel of regiment (is itself locally convex) Dihedral angles between {3} and {4} (lunal):   arccos(-(1+sqrt(5))/sqrt(12)) = 159.094843° between {3} and {5} (rotundal):   arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632° between {3} and {4} (across rim):   arccos(-sqrt[(3-sqrt(5))/6]) = 110.905157° between {3} and {5} (across rim):   arccos(-sqrt[(5-2 sqrt(5))/15]) = 100.812317° between {5} and {5}:   arccos(1/sqrt(5)) = 63.434949° Confer uniform relative: ike   id   srid   related Johnson solids: pocuro   pero   pecu   thawro   general polytopal classes: Johnson solids   expanded kaleido-facetings Externallinks

The right pic shows how bilbiro can be obtained by means of an expanded kaleido-faceting from ike. And this relation too is why bilbiro occures not too seldomly as a cell within CRFs.

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

Incidence matrix according to Dynkin symbol

```xfofx oxfxo&#xt   → outer heights = (1+sqrt(5))/4 = 0.809017
inner heights = 1/2
(line || pseudo (f,x)-{4} || pseudo ortho f-line || pseudo (f,x)-{4} || line)

o.... o....     | 2 * * * * | 1 2 0 0 0 0 0 0 0 | 1 2 0 0 0 0
.o... .o...     | * 4 * * * | 0 1 1 1 1 0 0 0 0 | 1 1 1 1 0 0
..o.. ..o..     | * * 2 * * | 0 0 0 2 0 2 0 0 0 | 0 1 0 2 1 0
...o. ...o.     | * * * 4 * | 0 0 0 0 1 1 1 1 0 | 0 0 1 1 1 1
....o ....o     | * * * * 2 | 0 0 0 0 0 0 0 2 1 | 0 0 0 0 2 1
----------------+-----------+-------------------+------------
x.... .....     | 2 0 0 0 0 | 1 * * * * * * * * | 0 2 0 0 0 0
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.. .oo..&#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
..oo. ..oo.&#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
....x .....     | 0 0 0 0 2 | * * * * * * * * 1 | 0 0 0 0 2 0
----------------+-----------+-------------------+------------
..... ox...&#x  | 1 2 0 0 0 | 0 2 1 0 0 0 0 0 0 | 2 * * * * *
xfo.. .....&#xt | 2 2 1 0 0 | 1 2 0 2 0 0 0 0 0 | * 2 * * * *
..... .x.x.&#x  | 0 2 0 2 0 | 0 0 1 0 2 0 1 0 0 | * * 2 * * *
.ooo. .ooo.&#xt | 0 1 1 1 0 | 0 0 0 1 1 1 0 0 0 | * * * 4 * *
..ofx .....&#xt | 0 0 1 2 2 | 0 0 0 0 0 2 0 2 1 | * * * * 2 *
..... ...xo&#x  | 0 0 0 2 1 | 0 0 0 0 0 0 1 2 0 | * * * * * 2
```
```or
o.... o....      & | 4 * * | 1 2 0 0 0 | 1 2 0 0  [3,52]
.o... .o...      & | * 8 * | 0 1 1 1 1 | 1 1 1 1  [3,4,3,5]
..o.. ..o..        | * * 2 | 0 0 0 4 0 | 0 2 0 2  [(3,5)2]
-------------------+-------+-----------+--------
x.... .....      & | 2 0 0 | 2 * * * * | 0 2 0 0
oo... oo...&#x   & | 1 1 0 | * 8 * * * | 1 1 0 0
..... .x...      & | 0 2 0 | * * 4 * * | 1 0 1 0
.oo.. .oo..&#x   & | 0 1 1 | * * * 8 * | 0 1 0 1
.o.o. .o.o.&#x     | 0 2 0 | * * * * 4 | 0 0 1 1
-------------------+-------+-----------+--------
..... ox...&#x   & | 1 2 0 | 0 2 1 0 0 | 4 * * *
xfo.. .....&#xt  & | 2 2 1 | 1 2 0 2 0 | * 4 * *
..... .x.x.&#x     | 0 4 0 | 0 0 2 0 2 | * * 2 *
.ooo. .ooo.&#xt    | 0 2 1 | 0 0 0 2 1 | * * * 4
```

```xFoFx xofox&#xt    → outer heights = (sqrt(5)-1)/4 = 0.309017
(F=ff=x+f)           inner heights = 1/2
({4} || pseudo F-line || pseudo ortho f-line || pseudo F-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 ooooo&#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  [3,4,3,5]
.o... .o...     & | * 4 * | 0 0 2 0 1 | 0 1 0 2  [3,52]
..o.. ..o..       | * * 2 | 0 0 0 4 0 | 0 0 2 2  [(3,5)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 ooooo&#xt   | 2 2 1 | 0 0 2 2 1 | * * * 4
```

```oxFxo ofxfo&#xt   → outer heights = (sqrt(5)-1)/4 = 0.309017
(F=ff=x+f)           inner heights = 1/2
(pt || pseudo (x,f)-{4} || pseudo (F,x)-{4} || pseudo (x,f)-{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.. .oo..&#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. ..oo.&#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 * * * * *
..... ofx..&#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. .ooo.&#xt | 0 1 1 1 0 | 0 0 1 1 0 1 0 0 | * * * 4 * *
..... ..xfo&#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]
.o... .o...     & | * 8 * | 1 1 1 1 0 | 1 1 1 1  [3,4,3,5]
..o.. ..o..       | * * 4 | 0 0 2 0 1 | 0 2 0 1  [3,52]
------------------+-------+-----------+--------
oo... oo...&#x  & | 1 1 0 | 8 * * * * | 1 1 0 0
.x... .....     & | 0 2 0 | * 4 * * * | 1 0 1 0
.oo.. .oo..&#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 * * *
..... ofx..&#xt & | 1 2 2 | 2 0 2 0 1 | * 4 * *
.x.x. .....&#x    | 0 4 0 | 0 2 0 2 0 | * * 2 *
.ooo. .ooo.&#xt   | 0 2 1 | 0 0 2 1 0 | * * * 4
```

```fxo ofx oxF&#zxt   → both heights = 0
(F=ff=x+f)
(pseudo (f,o,o)-line || pseudo (x,f,x)-cube || pseudo (o,x,F)-{4})

o.. o.. o..     | 2 * * | 4 0 0 0 0 | 2 2 0 0  [(3,5)2]
.o. .o. .o.     | * 8 * | 1 1 1 1 0 | 1 1 1 1  [3,4,3,5]
..o ..o ..o     | * * 4 | 0 0 0 2 1 | 2 0 0 1  [3,52]
----------------+-------+-----------+--------
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 .oo .oo&#x  | 0 1 1 | * * * 8 * | 1 0 0 1
... ..x ...     | 0 0 2 | * * * * 2 | 2 0 0 0
----------------+-------+-----------+--------
... ofx ...&#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
```

 © 2004-2021 top of page