Acronym id
TOCID symbol ID
Name icosidodecahedron
 
 © ©
Circumradius (1+sqrt(5))/2 = 1.618034
Vertex figure [(3,5)2] = x f
Vertex layers
LayerSymmetrySubsymmetries
 o3o5oo3o .o . o. o5o
1o3x5oo3x .
{3} first
o . o
vertex first
. x5o
{5} first
2x3f .x . f
vertex figure
. o5f
3F3o .F . x. x5x
4f3f .f . F. f5o
5ao3F .o . V. o5x
opposite {5}
5bV . o
6f3x .f . F 
7x3o .
opposite {3}
F . x
8 x . f
9o . o
opposite 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
links
hedrondude   wikipedia   WikiChoron   mathworld   Polyedergarten   quickfur

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

© 2004-2019
top of page