a3b5c

Acronym ...

Name a3b5c,
general variation of great rhombicosidodecahedron

Circumradius sqrt[(f+2)a²+4(f+1)ab+2(2f+1)ac+4(f+1)b²+4(2f+1)bc+3(f+1)c²]/2

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o5o`	`o3o .`	`. o5o`
1	`a3b5c`	`a3b .` (a,b)-{6} first	`. b5c` (b,c)-{10} first
2		`a3B .`	`. C5c`
3		`C3D .`	`. a5A`
4		`b3F .` (layers 4 & 5 interchange for a > c)	`. a5D`
5		`X3A .`	`. C5B`
6		`G3c .` (layers 6 & 7 interchange for b > af)	`. X5b` (layers 6 & 7 interchange for c > af)
7		`B3Z .`	`. b5X`
8		`H3c .` (layers 8 & 9 interchange for a > bf)	`. B5C`
9		`D3Z .`	`. D5a`
10		`F3A .` (layers 10 & 11 interchange for a > cff)	`. A5a`
11		`A3F .`	`. c5C`
12		`Z3D .` (layers 12 & 13 interchange for a > bf)	`. c5b` opposite (c,b)-{10}
13		`c3H .`
14		`Z3B .` (layers 14 & 15 interchange for b > af)
15		`c3G .`
16		`A3X .` (layers 16 & 17 interchange for b > af)
17		`F3b .`
18		`D3C .`
19		`B3a .`
20		`b3a .` opposite (b,a)-{6}

(A=fb+c, B=b+fc, C=a+b, D=f(b+c), X=a+b+fc, Z=fa+fb+c, F=f(a+b+c), G=a+ffb+fc, H=f(a+fb+c))

Lace tower
and approx. ASCII-art

                     b   a   b                            
                   o---o---o---o                     - a3b
               c/  B  c| a |c  B  \c                     height = cy
             o---------o---o---------o               - a3B
          b/    D    b/  C  \b    D    \b                height = bfy
         o-----------o-------o-----------o           - C3D
       c/    A    c/ a\  b  /a \c    A    \c             height = cfy  resp. height = afy
       o---------o     o---o     o---------o         - b3F + X3A [comp. only if c=a,   else 2 layers: heigth = |c-a|fy]
    a/ |b c  b/  a\ c/   B   \c /a  \b  c b| \a          height = afy  resp. height = by  resp. height = cfy
   o   o---o       o-----------o       o---o   o     - B3Z + G3c [comp. only if b=af,  else 2 layers: heigth = |b-af|y]
 b/  a/ c /a     b/      D      \b     a\ c \a  \b       height = bffy resp. height = affy
 o   o---o       o---------------o       o---o   o   - D3Z + H3c [comp. only if a=bf,  else 2 layers: heigth = |bf-a|y]
a| /b A  b\  a/   \c     A     c/   \a  /b  A b\ |a      height = ay   resp. height = bfy resp. height = cffy
 o---------o       o-----------o       o---------o   - A3F + F3A [comp. only if a=cff, else 2 layers: heigth = |a-cff|y]
c|     D    \c  /a   \b  c  b/   a\  c/    D     |c      height = cffy resp. height = ay  resp. height = bfy
 o-----------o         o---o         o-----------o   - c3H + Z3D [comp. only if a=bf,  else 2 layers: heigth = |bf-a|y]
 b\     B    |b       a| c |a       b|    B     /b       height = bffy resp. height = affy
   o---------o         o---o         o---------o     - c3G + Z3B [comp. only if b=af, else 2 layers: heigth = |b-af|y]
    c\   b c/  \a  b/    A    \b  a/  \c b   /c          height = cfy  resp. height = by  resp. height = afy
       o---o     o---------------o     o---o         - A3X + F3b [comp. only if c=a,   else 2 layers: heigth = |c-a|fy]
       a\ C  \a /c       D       c\ a/  C /a             height = afy  resp. height = cfy
         o-----o-------------------o-----o           - D3C
          b\  a \b       B       b/ a  /b                height = bfy
             o---o---------------o---o               - B3a
               c\   \c       c/   /c                     height = cy
                   o---o---o---o                     - b3a
                     a   b   a                           where y = sqrt[(3-sqrt(5))/6] = 0.356822
                                                         and   f = (1+sqrt(5))/2 = 1.618034

             c   b   c   b   c                    
           o---o---o---o---o---o             - b5c
        a/c a/  C a| c |a C  \a c\a              height = az
       o---o-------o---o-------o---o         - C5c
    b/  A  |b a b/   A   \b a b|  A  \b          height = bz
   o-------o---o-----------o---o-------o     - a5A
 c/  D   c/ a /c     D     c\ a \c   D  \c       height = cfz
 o-------o---o---------------o---o-------o   - a5D
b|  B  b/  C  \b     B     b/  C  \b  B  |b      height = bfz
 o-----o-------o-----------o-------o-----o   - C5B
c| b /c \a b a/  \c  b  c/  \a b a/ c\ b |c      height = afz resp. height = cz
 o--o    o---o     o---o     o---o    o--o   - b5X + X5b  [comp. only if c=af, else 2 layers: heigth = |c-af|z]
a| C \a/c  B  c\ a/  C  \a /c  B  c\a/ C |a      height = cz  resp. height = afz
 o----o----------o-------o----------o----o   - B5C
 b\ a |b    D    b\  a  /b    D    b| a /b       height = bfz
   o--o------------o---o------------o--o     - D5a
   c\ a\c    A    c| a |c    A    c/a /c         height = cfz
     o--o----------o---o----------o--o       - A5a
     b\ C \b  c  b/  C  \b  c  b/ C /b           height = bz
       o----o----o-------o----o----o         - c5C
        a\   \a   \a   a/   a/   /a              height = az
           o--o----o---o----o--o             - c5b
             b   c   b   c   b                   where z = sqrt[(5-sqrt(5))/10] = 0.525731
                                                 and   f = (1+sqrt(5))/2 = 1.618034

Pattern
(pyritohedral symmetry)

(upper half:)             (lower half: dito + 60)
                 a c b
              _33-34_              
           _37_     _38_           
        _41_  _29-30_  _42_        
     _45    21       22    46_     
   49/     /           \     \50   
  /_25_  _13           14_  _26_\  
 53    17  \           /  18    54 
/        \_ 5_      _ 6_ /        \
57       9     1---2    10       58
|        |     |   |     |        |
60      12_  _ 4---3_  _11       59
\        /  8         7  \        /
 56_  _20_ /           \ _19_  _55 
  \ 28    16           15    27 /  
   52\     \           /     /51   
      48_  _24_     _23_  _47      
         44_  _32-31_  _43         
            40_     _39            
               36-35               

33-34 = 2-3 = 58-118 = a
34-38 = 2-6 = 54-58  = b
33-93 = 1-2 = 58-59  = c

Coordinates

(c/2, a/2, (a+2b+cf)f/2) & even permutations, all changes of sign
(vertices 34, 33, 93, 94; 58, 59, 119, 118; 2, 3, 4, 1; 62, 63, 64, 61; 35, 36, 96, 95; 57, 60, 120, 117)
((bf+c)/2, (a+b)/2, (a+bf+cf)f/2) & even permutations, all changes of sign
(vertices 38, 37, 97, 98; 54, 55, 115, 114; 6, 7, 8, 5; 66, 67, 68, 65; 39, 40, 110, 99; 53, 56, 116, 113)
((af+bf+c)/2, b/2, (a+bf²+cf²)/2) & even permutations, all changes of sign
(vertices 30, 29, 89, 90; 50, 51, 111, 110, 10, 11, 12, 9; 70, 71, 72, 69; 31, 32, 92, 91; 49, 52, 112, 109)
(f(b+c)/2, (a+b+cf)/2, (a+bf+c)f/2) & even permutations, all changes of sign
(vertices 42, 41, 101, 102; 26, 27, 87, 86; 14, 15, 16, 13; 74, 75, 76, 73; 43, 44, 104, 103; 25, 28, 88, 85)
((b+cf)/2, (a+bf²+cf)/2, (a+b+c)f/2) & even permutations, all changes of sign
(vertices 46, 45, 105, 106; 18, 19, 79, 78; 22, 23, 24, 21; 82, 83, 84, 81; 47, 48, 108, 107; 17, 20, 80, 77)

General of army (is itself convex)

Colonel of regiment (is itself locally convex)

Face vector 120, 180, 62

Especially

grid (a=b=c=x)
f3x5x
v3x5f
f3v5v
x3(-x)5f

ti (a=b=x, c=o)
f3x5o
x3f5o
(-x)3x5o

srid (a=c=x, b=o)
q3o5x
f3o5x
x3o5f
v3o5f

tid (a=o, b=c=x)
o3(-x)5f

Confer

general polytopal classes:: Wythoffian polyhedra isogonal

Incidence matrix according to Dynkin symbol

a3b5c   (a ≠ 0, b ≠ 0, c ≠ 0 : general grid-variant)

. . . | 120 |  1  1  1 |  1  1  1
------+-----+----------+---------
a . . |   2 | 60  *  * |  1  1  0
. b . |   2 |  * 60  * |  1  0  1
. . c |   2 |  *  * 60 |  0  1  1
------+-----+----------+---------
a3b . |   6 |  3  3  0 | 20  *  *
a . c |   4 |  2  0  2 |  * 30  *
. b5c |  10 |  0  5  5 |  *  * 12

a3b5o   (a ≠ 0, b ≠ 0, c = 0 : general ti-variant)

. . . | 60 |  1  2 |  2  1
------+----+-------+------
a . . |  2 | 30  * |  2  0
. b . |  2 |  * 60 |  1  1
------+----+-------+------
a3b . |  6 |  3  3 | 20  *
. b5o |  5 |  0  5 |  * 12

a3o5c   (a ≠ 0, b = 0, c ≠ 0 : general srid-variant)

. . . | 60 |  2  2 |  1  2  1
------+----+-------+---------
a . . |  2 | 60  * |  1  1  0
. . c |  2 |  * 60 |  0  1  1
------+----+-------+---------
a3o . |  3 |  3  0 | 20  *  *
a . c |  4 |  2  2 |  * 30  *
. o5c |  5 |  0  5 |  *  * 12

o3b5c   (a = 0, b ≠ 0, c ≠ 0 : general tid-variant)

. . . | 60 |  2  1 |  1  2
------+----+-------+------
. b . |  2 | 60  * |  1  1
. . c |  2 |  * 30 |  0  2
------+----+-------+------
o3b . |  3 |  3  0 | 20  *
. b5c | 10 |  5  5 |  * 12

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