Acronym ti
TOCID symbol tI
Name truncated icosahedron,
buckyball,
Goldberg polyhedron GP(1,1)

` © ©`
Edge radius 3 (1+sqrt(5))/4 = 2.427051
Vertex figure [5,6,6] = of&#h
Vertex layers
 Layer Symmetry Subsymmetries o3o5o o3o . o . o . o5o 1 x3x5o x3x .{6} first x . oedge first . x5o{5} first 2 u3f . u . f . u5o 3 A3o . x . V . x5f 4 x3V . A . x . u5x 5 B3o . f . B . x5u 6 f3V . B . u . f5x 7a V3f . V . A . o5u 7b o . Y 8 o3B . Y . x . o5xopposite {5} 9a V3x . V . A 9b o . Y 10 o3A . B . u 11 f3u . f . B 12 x3x .opposite {6} A . x 13 x . V 14 u . f 15 x . oopposite edge
(V=2f, A=f+u=f+2x, B=x+V=2f+x, Y=3f)
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dihedral angles
• between {5} and {6}:   arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632°
• between {6} and {6}:   arccos(-sqrt(5)/3) = 138.189685°
Dual pakid
Confer
Grünbaumian relatives:
2ti
variations:
a3b5c   f3x5o   x3f5o   (-x)3x5o
External

As abstract polytope ti is isomorphic to tiggy, thereby replacing pentagons by pentagrams.

This polyhedron allows for the faceting o5/2f5x, which is a non-uniform truncation of gad and as such a variant of tigid.

Incidence matrix according to Dynkin symbol

```x3x5o

. . . | 60 |  1  2 |  2  1
------+----+-------+------
x . . |  2 | 30  * |  2  0
. x . |  2 |  * 60 |  1  1
------+----+-------+------
x3x . |  6 |  3  3 | 20  *
. x5o |  5 |  0  5 |  * 12

snubbed forms: β3x5o, x3β5o, β3β5o
```

```o5/4x3x

.   . . | 60 |  2  1 |  1  2
--------+----+-------+------
.   x . |  2 | 60  * |  1  1
.   . x |  2 |  * 30 |  0  2
--------+----+-------+------
o5/4x . |  5 |  5  0 | 12  *
.   x3o |  6 |  3  3 |  * 20
```

```xuxuxfoo5oofxuxux&#xt   → height(1,2) = height(2,3) = height(6,7) = height(7,8) = sqrt[(5-sqrt(5))/10] = 0.525731
height(3,4) = height(4,5) = height(5,6) = sqrt[(5+sqrt(5))/10] = 0.850651
({5} || pseudo u-{5} || pseudo (x,f)-{10} || pseudo (u,x)-{10} || pseudo (x,u)-{10} || pseudo (f,x)-{10} || pseudo dual u-{5} || dual {5})

o.......5o.......     | 5 *  *  *  *  * * * | 2 1  0 0  0 0  0 0  0 0  0 0 0 | 1 2 0 0 0 0 0 0
.o......5.o......     | * 5  *  *  *  * * * | 0 1  2 0  0 0  0 0  0 0  0 0 0 | 0 2 1 0 0 0 0 0
..o.....5..o.....     | * * 10  *  *  * * * | 0 0  1 1  1 0  0 0  0 0  0 0 0 | 0 1 1 1 0 0 0 0
...o....5...o....     | * *  * 10  *  * * * | 0 0  0 0  1 1  1 0  0 0  0 0 0 | 0 0 1 1 1 0 0 0
....o...5....o...     | * *  *  * 10  * * * | 0 0  0 0  0 0  1 1  1 0  0 0 0 | 0 0 0 1 1 1 0 0
.....o..5.....o..     | * *  *  *  * 10 * * | 0 0  0 0  0 0  0 0  1 1  1 0 0 | 0 0 0 0 1 1 1 0
......o.5......o.     | * *  *  *  *  * 5 * | 0 0  0 0  0 0  0 0  0 0  2 1 0 | 0 0 0 0 0 1 2 0
.......o5.......o     | * *  *  *  *  * * 5 | 0 0  0 0  0 0  0 0  0 0  0 1 2 | 0 0 0 0 0 0 2 1
----------------------+---------------------+--------------------------------+----------------
x....... ........     | 2 0  0  0  0  0 0 0 | 5 *  * *  * *  * *  * *  * * * | 1 1 0 0 0 0 0 0
oo......5oo......&#x  | 1 1  0  0  0  0 0 0 | * 5  * *  * *  * *  * *  * * * | 0 2 0 0 0 0 0 0
.oo.....5.oo.....&#x  | 0 1  1  0  0  0 0 0 | * * 10 *  * *  * *  * *  * * * | 0 1 1 0 0 0 0 0
..x..... ........     | 0 0  2  0  0  0 0 0 | * *  * 5  * *  * *  * *  * * * | 0 1 0 1 0 0 0 0
..oo....5..oo....&#x  | 0 0  1  1  0  0 0 0 | * *  * * 10 *  * *  * *  * * * | 0 0 1 1 0 0 0 0
........ ...x....     | 0 0  0  2  0  0 0 0 | * *  * *  * 5  * *  * *  * * * | 0 0 1 0 1 0 0 0
...oo...5...oo...&#x  | 0 0  0  1  1  0 0 0 | * *  * *  * * 10 *  * *  * * * | 0 0 0 1 1 0 0 0
....x... ........     | 0 0  0  0  2  0 0 0 | * *  * *  * *  * 5  * *  * * * | 0 0 0 1 0 1 0 0
....oo..5....oo..&#x  | 0 0  0  0  1  1 0 0 | * *  * *  * *  * * 10 *  * * * | 0 0 0 0 1 1 0 0
........ .....x..     | 0 0  0  0  0  2 0 0 | * *  * *  * *  * *  * 5  * * * | 0 0 0 0 1 0 1 0
.....oo.5.....oo.&#x  | 0 0  0  0  0  1 1 0 | * *  * *  * *  * *  * * 10 * * | 0 0 0 0 0 1 1 0
......oo5......oo&#x  | 0 0  0  0  0  0 1 1 | * *  * *  * *  * *  * *  * 5 * | 0 0 0 0 0 0 2 0
........ .......x     | 0 0  0  0  0  0 0 2 | * *  * *  * *  * *  * *  * * 5 | 0 0 0 0 0 0 1 1
----------------------+---------------------+--------------------------------+----------------
x.......5o.......     | 5 0  0  0  0  0 0 0 | 5 0  0 0  0 0  0 0  0 0  0 0 0 | 1 * * * * * * *
xux..... ........&#xt | 2 2  2  0  0  0 0 0 | 1 2  2 1  0 0  0 0  0 0  0 0 0 | * 5 * * * * * *
........ .ofx....&#xt | 0 1  2  2  0  0 0 0 | 0 0  2 0  2 1  0 0  0 0  0 0 0 | * * 5 * * * * *
..xux... ........&#xt | 0 0  2  2  2  0 0 0 | 0 0  0 1  2 0  2 1  0 0  0 0 0 | * * * 5 * * * *
........ ...xux..&#xt | 0 0  0  2  2  2 0 0 | 0 0  0 0  0 1  2 0  2 1  0 0 0 | * * * * 5 * * *
....xfo. ........&#xt | 0 0  0  0  2  2 1 0 | 0 0  0 0  0 0  0 1  2 0  2 0 0 | * * * * * 5 * *
........ .....xux&#xt | 0 0  0  0  0  2 2 2 | 0 0  0 0  0 0  0 0  0 1  2 2 1 | * * * * * * 5 *
.......o5.......x     | 0 0  0  0  0  0 0 5 | 0 0  0 0  0 0  0 0  0 0  0 0 5 | * * * * * * * 1
```
```or
o.......5o.......      & | 10  *  *  * |  2  1  0  0  0  0  0 | 1  2  0  0
.o......5.o......      & |  * 10  *  * |  0  1  2  0  0  0  0 | 0  2  1  0
..o.....5..o.....      & |  *  * 20  * |  0  0  1  1  1  0  0 | 0  1  1  1
...o....5...o....      & |  *  *  * 20 |  0  0  0  0  1  1  1 | 0  0  1  2
-------------------------+-------------+----------------------+-----------
x....... ........      & |  2  0  0  0 | 10  *  *  *  *  *  * | 1  1  0  0
oo......5oo......&#x   & |  1  1  0  0 |  * 10  *  *  *  *  * | 0  2  0  0
.oo.....5.oo.....&#x   & |  0  1  1  0 |  *  * 20  *  *  *  * | 0  1  1  0
..x..... ........      & |  0  0  2  0 |  *  *  * 10  *  *  * | 0  1  0  1
..oo....5..oo....&#x   & |  0  0  1  1 |  *  *  *  * 20  *  * | 0  0  1  1
........ ...x....      & |  0  0  0  2 |  *  *  *  *  * 10  * | 0  0  1  1
...oo...5...oo...&#x     |  0  0  0  2 |  *  *  *  *  *  * 10 | 0  0  0  2
-------------------------+-------------+----------------------+-----------
x.......5o.......      & |  5  0  0  0 |  5  0  0  0  0  0  0 | 2  *  *  *
xux..... ........&#xt  & |  2  2  2  0 |  1  2  2  1  0  0  0 | * 10  *  *
........ .ofx....&#xt  & |  0  1  2  2 |  0  0  2  0  2  1  0 | *  * 10  *
..xux... ........&#xt  & |  0  0  2  4 |  0  0  0  1  2  1  2 | *  *  * 10
```