Acronym sody
Name small deckishecatonicosachoron,
compound of 10 ex in hi army,
compound of 2 spohi
Coxeter symbol 2{5,3,3}[10{3,3,5}]
Circumradius (1+sqrt(5))/2 = 1.618034
Vertex figure
 ©
General of army hi  
Embedding
                               o2x    f2f  F2o    x2F    F2o  f2f    o2x                               
                                                                                                       
                                                                                                       
                                                                                                       
                   o2f         F2F    B2o         f2B         B2o    F2F         o2f                   
                                                                                                       
            o2o    f2F         B2f    F2B        oC2Co        F2B    B2f         f2F    o2o            
                                                                                                       
                                                                                                       
                                                                                                       
       f2o  F2f   Bx2xB        f2C   Co2FD B2B    D2f    B2B Co2FD   f2C        Bx2xB   F2f  f2o       
                                                                                                       
                                                                                                       
                                                                                                       
                                                                                                       
x2o    F2F  f2B    C2f        FD2DF   B2C  o2T    T2x    o2T  B2C   FD2DF        C2f    f2B  F2F    x2o
                                                                                                       
                                                                                                       
                                                                                                       
f2f    o2B  B2F   FD2Co        C2B   fT2Tf        B2D        fT2Tf   C2B        FD2Co   B2F  o2B    f2f
                                                                                                       
o2F                B2B         T2o                F2T                T2o         B2B                o2F
                                                                                                       
                                                                                                       
                                                                                                       
F2x    B2f oC2Co   f2D         x2T    D2B  T2F    C2C    T2F  D2B    x2T         f2D   oC2Co B2f    F2x
                                                                                                       
                                                                                                       
                                                                                                       
o2F                B2B         T2o                F2T                T2o         B2B                o2F
                                                                                                       
f2f    o2B  B2F   FD2Co        C2B   fT2Tf        B2D        fT2Tf   C2B        FD2Co   B2F  o2B    f2f
                                                                                                       
                                                                                                       
                                                                                                       
x2o    F2F  f2B    C2f        FD2DF   B2C  o2T    T2x    o2T  B2C   FD2DF        C2f    f2B  F2F    x2o
                                                                                                       
                                                                                                       
                                                                                                       
                                                                                                       
       f2o  F2f   Bx2xB        f2C   Co2FD B2B    D2f    B2B Co2FD   f2C        Bx2xB   F2f  f2o       
                                                                                                       
                                                                                                       
                                                                                                       
            o2o    f2F         B2f    F2B        oC2Co        F2B    B2f         f2F    o2o            
                                                                                                       
                   o2f         F2F    B2o         f2B         B2o    F2F         o2f                   
                                                                                                       
                                                                                                       
                                                                                                       
                               o2x    f2f  F2o    x2F    F2o  f2f    o2x                               
hull = hi
(F=ff=f+x, V=2f, B=fff=F+f=2f+x,
C=B+x=2F=2f, D=C+v=3f+x, T=FF=3f+2x)
                                      f2f                     f2f                                      
                                                                                                       
                                                                                                       
                                                                                                       
                               F2F                                   F2F                               
                                                                                                       
            o2o                                  oC2Co                                  o2o            
                                                                                                       
                                                                                                       
                                                                                                       
                  Bx2xB                    B2B           B2B                    Bx2xB                  
                                                                                                       
                                                                                                       
                                                                                                       
                                                                                                       
       F2F                    FD2DF                                 FD2DF                    F2F       
                                                                                                       
                                                                                                       
                                                                                                       
f2f                                  fT2Tf                   fT2Tf                                  f2f
                                                                                                       
                   B2B                                                           B2B                   
                                                                                                       
                                                                                                       
                                                                                                       
           oC2Co                                  C2C                                  oC2Co           
                                                                                                       
                                                                                                       
                                                                                                       
                   B2B                                                           B2B                   
                                                                                                       
f2f                                  fT2Tf                   fT2Tf                                  f2f
                                                                                                       
                                                                                                       
                                                                                                       
       F2F                    FD2DF                                 FD2DF                    F2F       
                                                                                                       
                                                                                                       
                                                                                                       
                                                                                                       
                  Bx2xB                    B2B           B2B                    Bx2xB                  
                                                                                                       
                                                                                                       
                                                                                                       
            o2o                                  oC2Co                                  o2o            
                                                                                                       
                               F2F                                   F2F                               
                                                                                                       
                                                                                                       
                                                                                                       
                                      f2f                     f2f                                      
vertex-inscribed non-uniform 2 ex compound, infact
5 such components then build this uniform compound.
scaling factor = fq = sqrt[3+sqrt(5)] = 2.288246
(vertex figure highlighted)
                    o2o                    
                                           
          o2x f2o   x2f   f2o o2x          
                                           
                                           
    x2o   f2f o2F   F2x   o2F f2f   x2o    
                                           
    o2f   F2o       f2F       F2o   o2f    
                                           
                                           
o2o f2x   x2F F2f  Vo2oV  F2f x2F   f2x o2o
                                           
                                           
    o2f   F2o       f2F       F2o   o2f    
                                           
    x2o   f2f o2F   F2x   o2F f2f   x2o    
                                           
                                           
          o2x f2o   x2f   f2o o2x          
                                           
                    o2o                    
single ex component
(display just slanted by 360°/8=45°)
Confer
blend-component:
ex  
compounds:
spohi  

This compound occurs in different forms depending on whether the vertices (coincident by 2) will be counted separately (type A), or will be identified (type C), or whether incident vertices, coplanar triangles, and corealmic tets will be united each (type E). All these would use 10 hexacosachora for compound components. – Two of those cases could be considered on base of the 5 hexacosachora compound (spohi) instead, i.e. using just 2 such components. That then gives rise to (type B) with no identifications, (type D) with identification of vertices only. – As (type E) already cross-connects the tets of either spohi, therefore there will be no according type using those for bases here.

Case E is interesting in that it points out that the altogether 6000 tets here form 2 different corealmic cell clusters. One of those is the well-known compound of 10 tets within an encasing doe, i.e. "e". In fact, each of the 120 does of the encasing hi of sody itself gives rise to that cell compound. But even the remainder of 4800 tet forms non-uniform axial compounds in the sense of mutually rotated pairs of trigonal pyramids with a single common tip each. The corresponding base, a polygonal compound of 2 triangles, with x3f as encasing hull, then connects those 2 cell clusters.


Incidence matrix

(Type A) - no identifications

1200    12 |    30 |   20 ||  1  verf: ike
-----+------+-------+------++---
   2 | 7200 |     5 |    5 ||  1
-----+------+-------+------++---
   3 |    3 | 12000 |    2 ||  1
-----+------+-------+------++---
   4 |    6 |     4 | 6000 ||  1  tet
-----+------+-------+------++---
 120 |  720 |  1200 |  600 || 10  ex

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

1200    12 |    30 |   20 || 1  verf: ike
-----+------+-------+------++--
   2 | 7200 |     5 |    5 || 1
-----+------+-------+------++--
   3 |    3 | 12000 |    2 || 1
-----+------+-------+------++--
   4 |    6 |     4 | 6000 || 1  tet
-----+------+-------+------++--
 600 | 3600 |  6000 | 3000 || 2  spohi

(Type C) - identifying incident vertices only

600    24 |    60 |   40 ||  2  verf: siddo
----+------+-------+------++---
  2 | 7200 |     5 |    5 ||  1
----+------+-------+------++---
  3 |    3 | 12000 |    2 ||  1
----+------+-------+------++---
  4 |    6 |     4 | 6000 ||  1  tet
----+------+-------+------++---
120 |  720 |  1200 |  600 || 10  ex

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

600    24 |    60 |   40 || 2  verf: siddo
----+------+-------+------++--
  2 | 7200 |     5 |    5 || 1
----+------+-------+------++--
  3 |    3 | 12000 |    2 || 1
----+------+-------+------++--
  4 |    6 |     4 | 6000 || 1  tet
----+------+-------+------++--
600 | 3600 |  6000 | 3000 || 2  spohi

(Type E) - identifying vertices, coplanar triangles, and corealmic tets

600    24 |   24   36 |   4   28 ||  2  verf: siddo
----+------+-----------+----------++---
  2 | 7200 |    2    3 |   1    4 ||  1
----+------+-----------+----------++---
  6 |    6 | 2400    * |   1    1 ||  2  red+blue
  3 |    3 |    * 7200 |   0    2 ||  1  yellow+green/orange+magenta
----+------+-----------+----------++---
 20 |   60 |   20    0 | 120    * || 10  red+blue e (10-tet comp.)
  7 |   12 |    1    6 |   * 2400 ||  2  yellow/orange + green/magenta 2-trig.pyr comp.
----+------+-----------+----------++---
120 |  720 |  960  240 | 120  480 || 10  red+yellow/blue+green ex

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