Acronym bitefa
Name bitegmic hecatontetracontatetrachoron,
compound of 12 ico in spic army,
compound of 2 kitefa
Coxeter symbol 2{;3,4,3;}[12{3,4,3}]
Circumradius 1
Vertex figure
 ©
General of army spic  
Embedding
                o4o         o4x         o4o                
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o        
                                                           
                                                           
                                                           
o4o                         o4w                         o4o
                                                           
        x4x         w4o             w4o         x4x        
                                                           
                                                           
                                                           
o4x             o4w         q4q         o4w             o4x
                                                           
                                                           
                                                           
        x4x         w4o             w4o         x4x        
                                                           
o4o                         o4w                         o4o
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o        
                                                           
                                                           
                                                           
                o4o         o4x         o4o                
hull = spic
                o4o                                        
                                                           
                                                           
                                                           
                                    x4x                    
                                                           
                                                           
                                                           
                                                        o4o
                                                           
        x4x                                                
                                                           
                                                           
                                                           
                            q4q                            
                                                           
                                                           
                                                           
                                                x4x        
                                                           
o4o                                                        
                                                           
                                                           
                                                           
                    x4x                                    
                                                           
                                                           
                                                           
                                        o4o                
vertex-inscribed non-uniform 2 ico compound, infact
6 such components then build this uniform compound.
scaling factor = sqrt[2+sqrt(2)] = 1.847759
(vertex figure highlighted)
o4o x4o o4o
           
x4o o4q x4o
           
o4o x4o o4o
single ico component
(display just slanted by 360°/16=22.5°)
Confer
blend-component:
ico  
compounds:
kitefa  
©

This compound occurs in different forms depending on whether the vertices (coincident by 2) will be counted separately (type A), or will be identified while the corealmic cells will still be counted separately (type C), or whether both would be unified (type E). All these would use 12 icositetrachora for compound components. – Two of those cases could be considered on base of the 6 icositetrachora compound (kitefa) instead, i.e. using just 2 such components. That then gives rise to (type B) with no identifications and (type D) with identification of vertices only. – As (type E) already cross-connects the octs of either kitefa, therefore there will be no according type using those for bases here. In fact, there are non-uniform compounds of 4 octs being used here. There is one central one plus three more, one each per axis, rotated half way. That whole cell is being displayed at the right.

How could this internal structure be understood? – The main idea to decipher the structure is to divide the vertex figure compound into 2 separate square prisms, which would just use a 2-square compound for base. Those bases for sure correspond to the 8-fold vertices of the right 4-oct compound. The 4 lateral faces of one of those cubes of the vertex figure then correspond to the only 4-fold vertices of the right compound cell. The 4 lateral faces of the other cube correspond to the remaining (normal) octs. The set of the triangles thus is to be divided too into subsets according to the adjoining cell types. Finally this requires a distinction of the edges as well, corresponding to the vertices of the 2 different cubes of the vertex figure.


Incidence matrix

(Type A) - no identifications

288     8 |   12 |   6 ||  1  verf: cube
----+------+------+-----++---
  2 | 1152 |    3 |   3 ||  1
----+------+------+-----++---
  3 |    3 | 1152 |   2 ||  1
----+------+------+-----++---
  6 |   12 |    8 | 288 ||  1  oct
----+------+------+-----++---
 24 |   96 |   96 |  24 || 12  ico

(Type B) - no identifications, but as 2 kitefa comp.

288     8 |   12 |   6 ||  1  verf: cube
----+------+------+-----++---
  2 | 1152 |    3 |   3 ||  1
----+------+------+-----++---
  3 |    3 | 1152 |   2 ||  1
----+------+------+-----++---
  6 |   12 |    8 | 288 ||  1  oct
----+------+------+-----++---
144 |  576 |  576 | 144 || 2  kitefa

(Type C) - identifying incident vertices only 

144 |   16 |   24 |  12 ||  2  verf: twirled face-first 2-cube comp. w/o identifications
----+------+------+-----++---
  2 | 1152 |    3 |   3 ||  1
----+------+------+-----++---
  3 |    3 | 1152 |   2 ||  1
----+------+------+-----++---
  6 |   12 |    8 | 288 ||  1  oct
----+------+------+-----++---
 24 |   96 |   96 |  24 || 12  ico

(Type D) - identifying incident vertices only, but as 2 kitefa comp.

144 |   16 |   24 |  12 || 2  verf: twirled face-first 2-cube comp. w/o identifications
----+------+------+-----++--
  2 | 1152 |    3 |   3 || 1
----+------+------+-----++--
  3 |    3 | 1152 |   2 || 1
----+------+------+-----++--
  6 |   12 |    8 | 288 || 1  oct
----+------+------+-----++--
144 |  576 |  576 | 144 || 2  kitefa

(Type E) - identifying vertices and corealmic cells

144 |   8   8 |  12   8   4 |  6  4 ||  2  verf: twirled face-first 2-cube comp. with coplanar faces unified as compound
----+---------+-------------+-------++---
  2 | 576   * |   3   0   0 |  3  0 ||  1
  2 |   * 576 |   0   2   1 |  1  2 ||  1
----+---------+-------------+-------++---
  3 |   3   0 | 576   *   * |  2  0 ||  1  faces of non-central oct of 4-oct
  3 |   0   3 |   * 384   * |  1  1 ||  1  faces of central oct of 4-oct
  3 |   0   3 |   *   * 192 |  0  2 ||  1
----+---------+-------------+-------++---
 18 |  36  12 |  24   8   0 | 48  * ||  4  4-oct compound
  6 |   0  12 |   0   4   4 |  * 96 ||  1  oct (in tet sym)
----+---------+-------------+-------++---
 24 |  48  48 |  48  32  16 |  4  8 || 12  ico

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