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Acronym
|
id
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TOCID symbol
|
ID
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Name
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icosidodecahedron,
rectified icosahedron,
rectified dodecahedron,
equatorial cross-section of vertex-first ex
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VRML
|
⭳ ©
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Circumradius
|
(1+sqrt(5))/2 = 1.618034
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Inradius
wrt. {3}
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sqrt[(7+3 sqrt(5))/6] = 1.511523
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Inradius
wrt. {5}
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sqrt[(5+2 sqrt(5))/5] = 1.376382
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Vertex figure
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[(3,5)2] = x f
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Vertex layers
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| o3x5o | o3x .
{3} first | o . o
vertex first | . x5o
{5} first |
| x3f . | x . f
vertex figure | . o5f |
| F3o . | F . x | . x5x |
| f3f . | f . F | . f5o |
| o3F . | o . V | . o5x
opposite {5} |
| V . o |
| f3x . | f . F | |
x3o .
opposite {3} | F . x |
| | x . f |
o . o
opposite vertex |
(F=ff=x+f=2x+v=x(10,3), V=F+v=2f=2x+2v=x(10,5))
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Coordinates
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-
(τ, 0, 0) & all permutations, all changes of sign
(vertex inscribed fq-oct)
-
(τ2/2, τ/2, 1/2) & even permutations, all changes of sign
where τ = (1+sqrt(5))/2
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General of army
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(is itself convex)
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Colonel of regiment
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(is itself locally convex
– other uniform polyhedral members:
sidhid
seihid
– other edge facetings)
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Dual
|
rhote
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Dihedral angles
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- between {3} and {5}: arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632°
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Face vector
|
30, 60, 32
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Confer
|
- Grünbaumian relatives:
-
2id
2id+40{3}
id+seihid+sidhid
- diminishings:
-
pysnic
- facetings:
-
"id-faceting"
- related Johnson solids:
-
pero
pobro
epgybro
- ambification:
-
rid
- ambification pre-image:
-
doe
ike
- general polytopal classes:
-
Wythoffian polyhedra
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External
links
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|
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