Acronym cid Name small complex icosidodecahedron Circumradius sqrt[(5+sqrt(5))/8] = 0.951057 Vertex figure [(3/2,5)5] (type A) or [(3,5)5]/3 (type B) Snub derivation General of army ike Colonel of regiment ike Confer non-Grünbaumian masters: gad   ike   Grünbaumian relatives: ike+2gad   2ike+gad   ike+3gad   3ike+gad   3ike+3gad   2ike+4gad   4ike+2gad   convex uniform variant: tid   variations: a3b5c Externallinks

As abstract polytope cid is isomorphic to gacid, thereby replacing pentagons by pentagrams.

Looks like a compound of the icosahedron (ike) and the great dodecahedron (gad), and indeed edges coincide by pairs, but vertices are identified. Note that without edge-doubling it would be a tetradic figure (type C).

Through the replacement of normal pentagons by complete pentagons (-x)5f, i.e. additional f-edges, this same set of faces provides an even further type D. Vertices coincide by 5 then, x-edges by pairs.

Incidence matrix according to Dynkin symbol

```x3/2o5o5*a (type A)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
x   . .    |  2 | 60 |  1  1
-----------+----+----+------
x3/2o .    |  3 |  3 | 20  *
x   . o5*a |  5 |  5 |  * 12
```

```o5/3o3x5*a (type B)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x5*a |  5 |  5 |  * 12
```

```x3/2o5/2o5*a (type B)

.   .   .    | 12 | 10 |  5  5
-------------+----+----+------
x   .   .    |  2 | 60 |  1  1
-------------+----+----+------
x3/2o   .    |  3 |  3 | 20  *
x   .   o5*a |  5 |  5 |  * 12
```

```o5/4x3o5*a (type A)

.   . . | 12 | 10 |  5  5
--------+----+----+------
.   x . |  2 | 60 |  1  1
--------+----+----+------
o5/4x . |  5 |  5 | 12  *
.   x3o |  3 |  3 |  * 20
```

```o5/4o3x5*a (type A)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x5*a |  5 |  5 |  * 12
```

```x5/4o5/2o3*a (type B)

.   .   .    | 12 | 10 |  5  5
-------------+----+----+------
x   .   .    |  2 | 60 |  1  1
-------------+----+----+------
x5/4o   .    |  5 |  5 | 12  *
x   .   o3*a |  3 |  3 |  * 20
```

```o5/4x3/2o5/3*a (type B)

.   .   . | 12 | 10 |  5  5
----------+----+----+------
.   x   . |  2 | 60 |  1  1
----------+----+----+------
o5/4x   . |  5 |  5 | 12  *
.   x3/2o |  3 |  3 |  * 20
```

```x5/4o5/4o3/2*a (type A)

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
x   .   .      |  2 | 60 |  1  1
---------------+----+----+------
x5/4o   .      |  5 |  5 | 12  *
x   .   o3/2*a |  3 |  3 |  * 20
```

```β3o5o (type A)

both( . . . ) | 12 | 10 |  5  5
--------------+----+----+------
sefa( β3o . ) |  2 | 60 |  1  1
--------------+----+----+------
β3o .   ♦  3 |  3 | 20  *
sefa( β3o5o ) |  5 |  5 |  * 12

starting figure: x3o5o
```

```(Type C)

12 |  5 |  5  5
---+----+------
2 | 30 |  2  2  :4 incident faces
---+----+------
3 |  3 | 20  *
5 |  5 |  * 12
```

```as uniform compound

12 |  5  5 |  5  5 || 1 1
-----+-------+-------++----
2 | 30  * |  2  0 || 1 0
2 |  * 30 |  0  2 || 0 1
-----+-------+-------++----
3 |  3  0 | 20  * || 1 0
5 |  0  5 |  * 12 || 0 1
-----+-------+-------++----
♦ 12 | 30  0 | 20  0 || 1 *
♦ 12 |  0 30 |  0 20 || * 1
```

```o3(-x)5f (Type D)

.   .  . | 60 |  2  1 |  1  2
---------+----+-------+------
.  -x  . |  2 | 60  * |  1  1
.   .  f |  2 |  * 30 |  0  2
---------+----+-------+------
o3(-x) . |  3 |  3  0 | 20  *
. (-x)5f | 10 |  5  5 |  * 12
```

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