Acronym id
TOCID symbol ID
Name icosidodecahedron,
rectified icosahedron,
rectified dodecahedron

` © ©`
Vertex figure [(3,5)2] = x f
Vertex layers
 Layer Symmetry Subsymmetries o3o5o o3o . o . o . o5o 1 o3x5o o3x .{3} first o . overtex first . x5o{5} first 2 x3f . x . fvertex figure . o5f 3 F3o . F . x . x5x 4 f3f . f . F . f5o 5a o3F . o . V . o5xopposite {5} 5b V . o 6 f3x . f . F 7 x3o .opposite {3} F . x 8 x . f 9 o . oopposite vertex
(F=ff=x+f=2x+v=x(10,3), V=F+v=2f=2x+2v=x(10,5))
Coordinates
1. (τ, 0, 0)   & all permutations, all changes of sign
(vertex inscribed fq-oct)
2. 2/2, τ/2, 1/2)   & even permutations, all changes of sign
where τ = (1+sqrt(5))/2
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral members: sidhid   seihid – other edge facetings)
Dual rhote
Dihedral angles
• between {3} and {5}:   arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632°
Confer
Grünbaumian relatives:
2id   2id+40{3}   id+seihid+sidhid
related Johnson solids:
pero   pobro   epgybro
External

As abstract polytope id is isomorphic to gid, thereby replacing pentagons by pentagrams.

Incidence matrix according to Dynkin symbol

```o3x5o

. . . | 30 |  4 |  2  2
------+----+----+------
. x . |  2 | 60 |  1  1
------+----+----+------
o3x . |  3 |  3 | 20  *
. x5o |  5 |  5 |  * 12

snubbed forms: o3β5o
```

```o3/2x5o

.   . . | 30 |  4 |  2  2
--------+----+----+------
.   x . |  2 | 60 |  1  1
--------+----+----+------
o3/2x . |  3 |  3 | 20  *
.   x5o |  5 |  5 |  * 12
```

```o5/4x3o

.   . . | 30 |  4 |  2  2
--------+----+----+------
.   x . |  2 | 60 |  1  1
--------+----+----+------
o5/4x . |  5 |  5 | 12  *
.   x3o |  3 |  3 |  * 20
```

```o5/4x3/2o

.   .   . | 30 |  4 |  2  2
----------+----+----+------
.   x   . |  2 | 60 |  1  1
----------+----+----+------
o5/4x   . |  5 |  5 | 12  *
.   x3/2o |  3 |  3 |  * 20
```

```xoxfo5ofxox&#xt   → outer heights = sqrt[(5-sqrt(5))/10] = 0.525731
inner heights = sqrt[(5+sqrt(5))/10] = 0.850651
({5} || pseudo dual f-{5} || pseudo {10} || pseudo f-{5} || dual {5})

o....5o....     | 5 *  * * * | 2  2  0 0 0  0  0 0 | 1 2 1 0 0 0 0 0
.o...5.o...     | * 5  * * * | 0  2  2 0 0  0  0 0 | 0 1 2 1 0 0 0 0
..o..5..o..     | * * 10 * * | 0  0  1 1 1  1  0 0 | 0 0 1 1 1 1 0 0
...o.5...o.     | * *  * 5 * | 0  0  0 0 0  2  2 0 | 0 0 0 0 2 1 1 0
....o5....o     | * *  * * 5 | 0  0  0 0 0  0  2 2 | 0 0 0 0 1 0 2 1
----------------+------------+---------------------+----------------
x.... .....     | 2 0  0 0 0 | 5  *  * * *  *  * * | 1 1 0 0 0 0 0 0
oo...5oo...&#x  | 1 1  0 0 0 | * 10  * * *  *  * * | 0 1 1 0 0 0 0 0
.oo..5.oo..&#x  | 0 1  1 0 0 | *  * 10 * *  *  * * | 0 0 1 1 0 0 0 0
..x.. .....     | 0 0  2 0 0 | *  *  * 5 *  *  * * | 0 0 0 1 1 0 0 0
..... ..x..     | 0 0  2 0 0 | *  *  * * 5  *  * * | 0 0 1 0 0 1 0 0
..oo.5..oo.&#x  | 0 0  1 1 0 | *  *  * * * 10  * * | 0 0 0 0 1 1 0 0
...oo5...oo&#x  | 0 0  0 1 1 | *  *  * * *  * 10 * | 0 0 0 0 1 0 1 0
..... ....x     | 0 0  0 0 2 | *  *  * * *  *  * 5 | 0 0 0 0 0 0 1 1
----------------+------------+---------------------+----------------
x....5o....     | 5 0  0 0 0 | 5  0  0 0 0  0  0 0 | 1 * * * * * * *
xo... .....&#x  | 2 1  0 0 0 | 1  2  0 0 0  0  0 0 | * 5 * * * * * *
..... ofx..&#xt | 1 2  2 0 0 | 0  2  2 0 1  0  0 0 | * * 5 * * * * *
.ox.. .....&#x  | 0 1  2 0 0 | 0  0  2 1 0  0  0 0 | * * * 5 * * * *
..xfo .....&#xt | 0 0  2 2 1 | 0  0  0 1 0  2  2 0 | * * * * 5 * * *
..... ..xo.&#x  | 0 0  2 1 0 | 0  0  0 0 1  2  0 0 | * * * * * 5 * *
..... ...ox&#x  | 0 0  0 1 2 | 0  0  0 0 0  0  2 1 | * * * * * * 5 *
....o5....x     | 0 0  0 0 5 | 0  0  0 0 0  0  0 5 | * * * * * * * 1
```
```or
o....5o....      & | 10  *  * |  2  2  0  0 | 1  2  1  0
.o...5.o...      & |  * 10  * |  0  2  2  0 | 0  1  2  1
..o..5..o..        |  *  * 10 |  0  0  2  2 | 0  0  2  2
-------------------+----------+-------------+-----------
x.... .....      & |  2  0  0 | 10  *  *  * | 1  1  0  0
oo...5oo...&#x   & |  1  1  0 |  * 20  *  * | 0  1  1  0
.oo..5.oo..&#x   & |  0  1  1 |  *  * 20  * | 0  0  1  1
..x.. .....      & |  0  0  2 |  *  *  * 10 | 0  0  1  1
-------------------+----------+-------------+-----------
x....5o....      & |  5  0  0 |  5  0  0  0 | 2  *  *  *
xo... .....&#x   & |  2  1  0 |  1  2  0  0 | * 10  *  *
..... ofx..&#xt  & |  1  2  2 |  0  2  2  1 | *  * 10  *
.ox.. .....&#x   & |  0  1  2 |  0  0  2  1 | *  *  * 10
```

```oxFfofx3xfofFxo&#xt   → height(1,2) = height(3,4) = height(4,5) = height(6,7) = 1/sqrt(3) = 0.577350
(F=ff=x+f)              height(2,3) = height(5,6) = sqrt[(3-sqrt(5))/6] = 0.356822
({3} || pseudo (x,f)-{6} || pseudo dual F-{3} || pseudo f-{6} || pseudo F-{3} || pseudo gyro {x,f)-{6} || dual {3})

o......3o......     & | 6  * * * | 2  2 0  0  0  0 | 1 1 2 0  0
.o.....3.o.....     & | * 12 * * | 0  1 1  1  1  0 | 0 1 1 1  1
..o....3..o....     & | *  * 6 * | 0  0 0  2  0  2 | 0 0 1 1  2
...o...3...o...       | *  * * 6 | 0  0 0  0  2  2 | 0 0 0 2  2
----------------------+----------+-----------------+-----------
....... x......     & | 2  0 0 0 | 6  * *  *  *  * | 1 0 1 0  0
oo.....3oo.....&#x  & | 1  1 0 0 | * 12 *  *  *  * | 0 1 1 0  0
.x..... .......     & | 0  2 0 0 | *  * 6  *  *  * | 0 1 0 1  0
.oo....3.oo....&#x  & | 0  1 1 0 | *  * * 12  *  * | 0 0 1 0  1
.o.o...3.o.o...&#x  & | 0  1 0 1 | *  * *  * 12  * | 0 0 0 1  1
..oo...3..oo...&#x  & | 0  0 1 1 | *  * *  *  * 12 | 0 0 0 1  1
----------------------+----------+-----------------+-----------
o......3x......     & | 3  0 0 0 | 3  0 0  0  0  0 | 2 * * *  *
ox..... .......&#x  & | 1  2 0 0 | 0  2 1  0  0  0 | * 6 * *  *
....... xfo....&#xt & | 2  2 1 0 | 1  2 0  2  0  0 | * * 6 *  *
.x.fo.. .......&#xt & | 0  2 1 2 | 0  0 1  0  2  2 | * * * 6  *
.ooo...3.ooo...&#x  & | 0  1 1 1 | 0  0 0  1  1  1 | * * * * 12
```

```VooFxf oVofFx ooVxfF&#zx (F=ff=x+f, V=2f)   → all heights = 0 (but not all pw. vertex combis exist as lacings)
(tegum sum of 3 orthogonal V-edges and 3 each parallely oriented (x,f,F)-cubes.)

o..... o..... o.....     | 2 * * * * * | 4 0 0 0 0 0 0 0 0 | 2 2 0 0 0 0 0
.o.... .o.... .o....     | * 2 * * * * | 0 4 0 0 0 0 0 0 0 | 0 0 2 2 0 0 0
..o... ..o... ..o...     | * * 2 * * * | 0 0 4 0 0 0 0 0 0 | 0 0 0 0 2 2 0
...o.. ...o.. ...o..     | * * * 8 * * | 1 0 0 1 1 1 0 0 0 | 1 1 0 1 0 0 1
....o. ....o. ....o.     | * * * * 8 * | 0 1 0 0 1 0 1 1 0 | 0 0 1 1 1 0 1
.....o .....o .....o     | * * * * * 8 | 0 0 1 0 0 1 0 1 1 | 1 0 0 0 1 1 1
-------------------------+-------------+-------------------+--------------
o..o.. o..o.. o..o..&#x  | 1 0 0 1 0 0 | 8 * * * * * * * * | 1 1 0 0 0 0 0
.o..o. .o..o. .o..o.&#x  | 0 1 0 0 1 0 | * 8 * * * * * * * | 0 0 1 1 0 0 0
..o..o ..o..o ..o..o&#x  | 0 0 1 0 0 1 | * * 8 * * * * * * | 0 0 0 0 1 1 0
...... ...... ...x..     | 0 0 0 2 0 0 | * * * 4 * * * * * | 0 1 0 1 0 0 0
...oo. ...oo. ...oo.&#x  | 0 0 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 0 0 1
...o.o ...o.o ...o.o&#x  | 0 0 0 1 0 1 | * * * * * 8 * * * | 1 0 0 0 0 0 1
....x. ...... ......     | 0 0 0 0 2 0 | * * * * * * 4 * * | 0 0 1 0 1 0 0
....oo ....oo ....oo&#x  | 0 0 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 1 0 1
...... .....x ......     | 0 0 0 0 0 2 | * * * * * * * * 4 | 1 0 0 0 0 1 0
-------------------------+-------------+-------------------+--------------
...... o..f.x ......&#xt | 1 0 0 2 0 2 | 2 0 0 0 0 2 0 0 1 | 4 * * * * * *
...... ...... o..x..&#x  | 1 0 0 2 0 0 | 2 0 0 1 0 0 0 0 0 | * 4 * * * * *
.o..x. ...... ......&#x  | 0 1 0 0 2 0 | 0 2 0 0 0 0 1 0 0 | * * 4 * * * *
...... ...... .o.xf.&#xt | 0 1 0 2 2 0 | 0 2 0 1 2 0 0 0 0 | * * * 4 * * * tower: B-E-D
..o.xf ...... ......&#xt | 0 0 1 0 2 2 | 0 0 2 0 0 0 1 2 0 | * * * * 4 * * tower: C-F-E
...... ..o..x ......&#x  | 0 0 1 0 0 2 | 0 0 2 0 0 0 0 0 1 | * * * * * 4 *
...ooo ...ooo ...ooo&#x  | 0 0 0 1 1 1 | 0 0 0 0 1 1 0 1 0 | * * * * * * 8
```
```or
o..... o..... o.....     & | 6  * |  4  0  0 |  2  2 0
...o.. ...o.. ...o..     & | * 24 |  1  1  2 |  2  1 1
---------------------------+------+----------+--------
o..o.. o..o.. o..o..&#x  & | 1  1 | 24  *  * |  1  1 0
...... ...... ...x..     & | 0  2 |  * 12  * |  1  1 0
...oo. ...oo. ...oo.&#x  & | 0  2 |  *  * 24 |  1  0 1
---------------------------+------+----------+--------
...... o..f.x ......&#xt & | 1  4 |  2  1  2 | 12  * *
...... ...... o..x..&#x  & | 1  2 |  2  1  0 |  * 12 *
...ooo ...ooo ...ooo&#x    | 0  3 |  0  0  3 |  *  * 8
```

```with tet subsym. (chiral choice)

12 *  * |  2  1  1  0  0 | 1  2  1 0 tetrahedral {3}-vertices
* 6  * |  0  2  0  2  0 | 0  2  2 0 inscribed octahedral vertices
* * 12 |  0  0  1  1  2 | 0  2  1 1 other tetrahedral {3}-vertices
--------+----------------+----------
2 0  0 | 12  *  *  *  * | 1  1  0 0
1 1  0 |  * 12  *  *  * | 0  1  1 0
1 0  1 |  *  * 12  *  * | 0  1  1 0
0 1  1 |  *  *  * 12  * | 0  1  1 0
0 0  2 |  *  *  *  * 12 | 0  1  0 1
--------+----------------+----------
3 0  0 |  3  0  0  0  0 | 4  *  * *
2 1  2 |  1  1  1  1  1 | * 12  * * {5}
1 1  1 |  0  1  1  1  0 | *  * 12 *
0 0  3 |  0  0  0  0  3 | *  *  * 4
```