Acronym ...
Name (1,0,8,12,0)-diminished vertex-first-rotunda of ex,
(subsymmetrical) icosiena-diminished vertex-first rotunda of hexacosachoron
Circumradius (1+sqrt(5))/2 = 1.618034
Lace city
in approx. ASCII-art
        x3o   o3f f3o   o3x        
                                   
         demi      demi            
     o3f  (f3x)     (x3f)  f3o     
                                   
                                       (F=ff)
                                   
                                   
o3x   x3f F3o   f3f   o3F f3x   x3o
          o2x f2o   x2f   f2o o2x          
                                           
                                           
    x2o       o2F   F2x   o2F       x2o    
                                               (F=ff)
                                               (V=2f)
                                           
                                           
o2o f2x   x2F F2f  Vo2oV  F2f x2F   f2x o2o
Confer
segmentochora:
ike || doe   doe || id  
other CRFs:
rotunda of ex   pt || ike || doe || id   ike || doe || id   ike || tet-dim-doe || id   ike || f-ike || id  
uniform relative:
ex  

The medial section here is the cubically diminished dodecahedron. From that section the whole polychoron inherits its chirality.

Note that there are different known icosiena-diminishings of rotunda of ex. This (1,0,8,12,0)-diminishing is an axially subsymmetrical one (showing up pyritohedral around-symmetry only). But there is also the (1,0,20,0,0)-diminished vertex-first-rotunda of ex (ike || f-ike || id), which has full icosahedral around-symmetry.


Incidence matrix

ike || pseudo cube-dim-doe || id   → height(1,2) = 1/2, height(2,3) = (1+sqrt(5))/4 = 0.809017

12  *  * * |  4 1  1  2 0  0  0  0  0  0 | 2  3  4  2  2  1 0  0  0  0 0  0 | 1 2  3 1  1  0 0  top layer vertices
 * 12  * * |  0 0  1  2 1  2  1  0  0  0 | 0  0  2  1  4  2 1  1  2  0 0  0 | 0 2  1 1  3  1 0  medial layer vertices
 *  * 24 * |  0 0  0  0 0  1  0  2  1  1 | 0  0  0  0  2  0 0  1  1  2 1  1 | 0 1  0 0  2  1 1  A-type botom layer vertices
 *  *  * 6 |  0 0  0  0 0  0  2  0  0  4 | 0  0  0  0  0  0 1  0  4  2 0  2 | 0 0  0 0  2  2 1  B-type botom layer vertices
-----------+-----------------------------+----------------------------------+-----------------
 2  0  0 0 | 24 *  *  * *  *  *  *  *  * | 1  1  1  0  0  0 0  0  0  0 0  0 | 1 1  1 0  0  0 0  top layer edges of teddies
 2  0  0 0 |  * 6  *  * *  *  *  *  *  * | 0  2  0  2  0  0 0  0  0  0 0  0 | 1 0  2 1  0  0 0  other top layer edges
 1  1  0 0 |  * * 12  * *  *  *  *  *  * | 0  0  2  0  2  0 0  0  0  0 0  0 | 0 2  1 0  1  0 0  upper lacing edges towards acute mid-layer triangle
 1  1  0 0 |  * *  * 24 *  *  *  *  *  * | 0  0  1  1  1  1 0  0  0  0 0  0 | 0 1  1 1  1  0 0  lacings of dangling tet
 0  2  0 0 |  * *  *  * 6  *  *  *  *  * | 0  0  0  0  0  2 1  0  0  0 0  0 | 0 0  0 1  2  0 0
 0  1  1 0 |  * *  *  * * 24  *  *  *  * | 0  0  0  0  2  0 0  1  1  0 0  0 | 0 1  0 0  2  1 0
 0  1  0 1 |  * *  *  * *  * 12  *  *  * | 0  0  0  0  0  0 1  0  2  0 0  0 | 0 0  0 0  2  1 0
 0  0  2 0 |  * *  *  * *  *  * 24  *  * | 0  0  0  0  1  0 0  0  0  1 1  0 | 0 1  0 0  1  0 1  bottom {5} lower laterals
 0  0  2 0 |  * *  *  * *  *  *  * 12  * | 0  0  0  0  0  0 0  1  0  1 0  1 | 0 0  0 0  1  1 1  bottom {5} base
 0  0  1 1 |  * *  *  * *  *  *  *  * 24 | 0  0  0  0  0  0 0  0  1  1 0  1 | 0 0  0 0  1  1 1  bottom {5} upper laterals
-----------+-----------------------------+----------------------------------+-----------------
 3  0  0 0 |  3 0  0  0 0  0  0  0  0  0 | 8  *  *  *  *  * *  *  *  * *  * | 1 1  0 0  0  0 0  top layer {3} of teddies
 3  0  0 0 |  2 1  0  0 0  0  0  0  0  0 | * 12  *  *  *  * *  *  *  * *  * | 1 0  1 0  0  0 0
 2  1  0 0 |  1 0  1  1 0  0  0  0  0  0 | *  * 24  *  *  * *  *  *  * *  * | 0 1  1 0  0  0 0
 2  1  0 0 |  0 1  0  2 0  0  0  0  0  0 | *  *  * 12  *  * *  *  *  * *  * | 0 0  1 1  0  0 0
 1  2  2 0 |  0 0  1  1 0  2  0  1  0  0 | *  *  *  * 24  * *  *  *  * *  * | 0 1  0 0  1  0 0  lacing {5}
 1  2  0 0 |  0 0  0  2 1  0  0  0  0  0 | *  *  *  *  * 12 *  *  *  * *  * | 0 0  0 1  1  0 0
 0  2  0 1 |  0 0  0  0 1  0  2  0  0  0 | *  *  *  *  *  * 6  *  *  * *  * | 0 0  0 0  2  0 0
 0  1  2 0 |  0 0  0  0 0  2  0  0  1  0 | *  *  *  *  *  * * 12  *  * *  * | 0 0  0 0  1  1 0
 0  1  1 1 |  0 0  0  0 0  1  1  0  0  1 | *  *  *  *  *  * *  * 24  * *  * | 0 0  0 0  1  1 0
 0  0  4 1 |  0 0  0  0 0  0  0  2  1  2 | *  *  *  *  *  * *  *  * 12 *  * | 0 0  0 0  1  0 1  bottom {5}
 0  0  3 0 |  0 0  0  0 0  0  0  3  0  0 | *  *  *  *  *  * *  *  *  * 8  * | 0 1  0 0  0  0 1
 0  0  2 1 |  0 0  0  0 0  0  0  0  1  2 | *  *  *  *  *  * *  *  *  * * 12 | 0 0  0 0  0  1 1
-----------+-----------------------------+----------------------------------+-----------------
12  0  0 0 | 24 6  0  0 0  0  0  0  0  0 | 8 12  0  0  0  0 0  0  0  0 0  0 | 1 *  * *  *  * *  top ike
 3  3  3 0 |  3 0  3  3 0  3  0  3  0  0 | 1  0  3  0  3  0 0  0  0  0 1  0 | * 8  * *  *  * *  vertical teddi
 3  1  0 0 |  2 1  1  2 0  0  0  0  0  0 | 0  1  2  1  0  0 0  0  0  0 0  0 | * * 12 *  *  * *  top-down tet
 2  2  0 0 |  0 1  0  4 1  0  0  0  0  0 | 0  0  0  2  0  2 0  0  0  0 0  0 | * *  * 6  *  * *  dangling tet
 1  3  4 1 |  0 0  1  2 1  4  2  2  1  2 | 0  0  0  0  2  1 1  1  2  1 0  0 | * *  * * 12  * *  lying teddi
 0  1  2 1 |  0 0  0  0 0  2  1  0  1  2 | 0  0  0  0  0  0 0  1  2  0 0  1 | * *  * *  * 12 *  bottom-up tet
 0  0 24 6 |  0 0  0  0 0  0  0 24 12 24 | 0  0  0  0  0  0 0  0  0 12 8 12 | * *  * *  *  * 1  bottom id

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