ti

Acronym ti

TOCID symbol tI

Name truncated icosahedron,
buckyball,
Goldberg polyhedron GP(1,1)

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Circumradius sqrt[(29+9 sqrt(5))/8] = 2.478019

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 . o` edge 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		`V3f .`	`o . Y`	`. o5u`
8		`o3B .`	`Y . x`	`. o5x` opposite {5}
9a		`V3x .`	`V . A`
9b		`V3x .`	`o . Y`
10		`o3A .`	`B . u`
11		`f3u .`	`f . B`
12		`x3x .` opposite {6}	`A . x`
13			`x . V`
14			`u . f`
15			`x . o` opposite 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

Face vector 60, 90, 32

Confer

Grünbaumian relatives:: 2ti
variations:: a3b5c f3x5o x3f5o (-x)3x5o
ambification:: reti
general polytopal classes:: Wythoffian polyhedra

External
links

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

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