Acronym doe
TOCID symbol D
Name dodecahedron,
cosmohedron,
Goldberg polyhedron GP(1,0)

` © ©`
Vertex figure [53] = f3o
Vertex layers
 Layer Symmetry Subsymmetries o3o5o o3o . o . o . o5o 1 o3o5x o3o .vertex first o . xedge first . o5x{5} first 2 o3f .vertex figure f . f . o5fvertex figure 3 f3x . F . o . f5o 4 x3f . x . F . x5oopposite {5} 5 f3o . F . o 6 o3o .opposite vertex f . f 7 o . xopposite edge
(F=ff)
Lace city
in approx. ASCII-art
```         x

f           f
o                 o

F     F

o                 o
f           f

x
```
Coordinates
1. (τ/2, τ/2, τ/2)   & all permutations, all changes of sign
(vertex inscribed f-cube)
2. 2/2, 1/2, 0)   & 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 – no other uniform polyhedral members)
Dual ike
Dihedral angles
• between {5} and {5}:   arccos(-1/sqrt(5)) = 116.565051°
Confer
Grünbaumian relatives:
2doe   3doe   6doe
related Johnson solids:
aud   pabaud
facetings:
tet-dim doe   cube-dim doe   ditti
stellations:
p2p5p   titdi   mibkid   cell of gap dual
isogonal relatives:
odsnic
general polytopal classes:
regular   noble polytopes
External

As abstract polytope doe is isomorphic to gissid, thereby replacing pentagons by pentagrams.

The number of ways to color the dodecahedron with different colors per face is 12!/60 = 7 983 360. – This is because the color group is the permutation group of 12 elements and has size 12!, while the order of the pure rotational icosahedral group is 60. (The reflectional icosahedral group would have twice as many, i.e. 120 elements.)

Incidence matrix according to Dynkin symbol

```o3o5x

. . . | 20 |  3 |  3
------+----+----+---
. . x |  2 | 30 |  2
------+----+----+---
. o5x |  5 |  5 | 12

snubbed forms: o3o5β
```

```o3/2o5x

.   . . | 20 |  3 |  3
--------+----+----+---
.   . x |  2 | 30 |  2
--------+----+----+---
.   o5x |  5 |  5 | 12
```

```x5/4o3o

.   . . | 20 |  3 |  3
--------+----+----+---
x   . . |  2 | 30 |  2
--------+----+----+---
x5/4o . |  5 |  5 | 12
```

```x5/4o3/2o

.   .   . | 20 |  3 |  3
----------+----+----+---
x   .   . |  2 | 30 |  2
----------+----+----+---
x5/4o   . |  5 |  5 | 12
```

```xfoo5oofx&#xt   → outer heights = sqrt[(5+sqrt(5))/10] = 0.850651
inner height = sqrt[(5-sqrt(5))/10] = 0.525731
({5} || pseudo f-{5} || dual pseudo f-{5} || dual {5})

o...5o...     | 5 * * * | 2 1  0 0 0 | 1 2 0 0
.o..5.o..     | * 5 * * | 0 1  2 0 0 | 0 2 1 0
..o.5..o.     | * * 5 * | 0 0  2 1 0 | 0 1 2 0
...o5...o     | * * * 5 | 0 0  0 1 2 | 0 0 2 1
--------------+---------+------------+--------
x... ....     | 2 0 0 0 | 5 *  * * * | 1 1 0 0
oo..5oo..&#x  | 1 1 0 0 | * 5  * * * | 0 2 0 0
.oo.5.oo.&#x  | 0 1 1 0 | * * 10 * * | 0 1 1 0
..oo5..oo&#x  | 0 0 1 1 | * *  * 5 * | 0 0 2 0
.... ...x     | 0 0 0 2 | * *  * * 5 | 0 0 1 1
--------------+---------+------------+--------
x...5o...     | 5 0 0 0 | 5 0  0 0 0 | 1 * * *
xfo. ....&#xt | 2 2 1 0 | 1 2  2 0 0 | * 5 * *
.... .ofx&#xt | 0 1 2 2 | 0 0  2 2 1 | * * 5 *
...o5...x     | 0 0 0 5 | 0 0  0 0 5 | * * * 1
```
```or
o...5o...      & | 10  * |  2  1  0 | 1  2
.o..5.o..      & |  * 10 |  0  1  2 | 0  3
-----------------+-------+----------+-----
x... ....      & |  2  0 | 10  *  * | 1  1
oo..5oo..&#x   & |  1  1 |  * 10  * | 0  2
.oo.5.oo.&#x     |  0  2 |  *  * 10 | 0  2
-----------------+-------+----------+-----
x...5o...      & |  5  0 |  5  0  0 | 2  *
xfo. ....&#xt  & |  2  3 |  1  2  2 | * 10
```

```ofxfoo3oofxfo&#xt   → outer heights = sqrt[(3-sqrt(5))/6] = 0.356822
tropal heights = 1/sqrt(3) = 0.577350
inner height = sqrt[(3+sqrt(5))/6] = 0.934172
(pt || pseudo f-{3} || pseudo (f,x)-{6} || pseudo (x,f)-{6} || pseudo dual f-{3} || pt)

o.....3o.....     | 1 * * * * * | 3 0 0 0 0 0 0 | 3 0 0 0
.o....3.o....     | * 3 * * * * | 1 2 0 0 0 0 0 | 2 1 0 0
..o...3..o...     | * * 6 * * * | 0 1 1 1 0 0 0 | 1 1 1 0
...o..3...o..     | * * * 6 * * | 0 0 0 1 1 1 0 | 0 1 1 1
....o.3....o.     | * * * * 3 * | 0 0 0 0 0 2 1 | 0 0 1 2
.....o3.....o     | * * * * * 1 | 0 0 0 0 0 0 3 | 0 0 0 3
------------------+-------------+---------------+--------
oo....3oo....&#x  | 1 1 0 0 0 0 | 3 * * * * * * | 2 0 0 0
.oo...3.oo...&#x  | 0 1 1 0 0 0 | * 6 * * * * * | 1 1 0 0
..x... ......     | 0 0 2 0 0 0 | * * 3 * * * * | 1 0 1 0
..oo..3..oo..&#x  | 0 0 1 1 0 0 | * * * 6 * * * | 0 1 1 0
...... ...x..     | 0 0 0 2 0 0 | * * * * 3 * * | 0 1 0 1
...oo.3...oo.&#x  | 0 0 0 1 1 0 | * * * * * 6 * | 0 0 1 1
....oo3....oo&#x  | 0 0 0 0 1 1 | * * * * * * 3 | 0 0 0 2
------------------+-------------+---------------+--------
ofx... ......&#xt | 1 2 2 0 0 0 | 2 2 1 0 0 0 0 | 3 * * *
...... .ofx..&#xt | 0 1 2 2 0 0 | 0 2 0 2 1 0 0 | * 3 * *
..xfo. ......&#xt | 0 0 2 2 1 0 | 0 0 1 2 0 2 0 | * * 3 *
...... ...xfo&#xt | 0 0 0 2 2 1 | 0 0 0 0 1 2 2 | * * * 3
```
```or
o.....3o.....      & | 2 *  * | 3  0 0 0 | 3 0
.o....3.o....      & | * 6  * | 1  2 0 0 | 2 1
..o...3..o...      & | * * 12 | 0  1 1 1 | 1 2
---------------------+--------+----------+----
oo....3oo....&#x   & | 1 1  0 | 6  * * * | 2 0
.oo...3.oo...&#x   & | 0 1  1 | * 12 * * | 1 1
..x... ......      & | 0 0  2 | *  * 6 * | 1 1
..oo..3..oo..&#x     | 0 0  2 | *  * * 6 | 0 2
---------------------+--------+----------+----
ofx... ......&#xt  & | 1 2  2 | 2  2 1 0 | 6 *
...... .ofx..&#xt  & | 0 1  4 | 0  2 1 2 | * 6
```

```xfoFofx ofFxFfo&#xt   (F=ff) → height(1,2) = height(3,4) = height(4,5) = height(6,7) = 1/2
height(2,3) = height(5,6) = (sqrt(5)-1)/4 = 0.309017
(line || pseudo f-{4} || pseudo ortho ff-line || pseudo (ff,x)-{4} || pseudo ortho ff-line || pseudo f-{4} || line)

o...... o......     | 2 * * * * * * | 1 2 0 0 0 0 0 0 0 0 | 2 1 0 0 0
.o..... .o.....     | * 4 * * * * * | 0 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0
..o.... ..o....     | * * 2 * * * * | 0 0 2 0 1 0 0 0 0 0 | 1 0 2 0 0
...o... ...o...     | * * * 4 * * * | 0 0 0 1 0 1 1 0 0 0 | 0 1 1 1 0
....o.. ....o..     | * * * * 2 * * | 0 0 0 0 1 0 0 2 0 0 | 0 0 2 0 1
.....o. .....o.     | * * * * * 4 * | 0 0 0 0 0 0 1 1 1 0 | 0 0 1 1 1
......o ......o     | * * * * * * 2 | 0 0 0 0 0 0 0 0 2 1 | 0 0 0 1 2
--------------------+---------------+---------------------+----------
x...... .......     | 2 0 0 0 0 0 0 | 1 * * * * * * * * * | 2 0 0 0 0
oo..... oo.....&#x  | 1 1 0 0 0 0 0 | * 4 * * * * * * * * | 1 1 0 0 0
.oo.... .oo....&#x  | 0 1 1 0 0 0 0 | * * 4 * * * * * * * | 1 0 1 0 0
.o.o... .o.o...&#x  | 0 1 0 1 0 0 0 | * * * 4 * * * * * * | 0 1 1 0 0
..o.o.. ..o.o..&#x  | 0 0 1 0 1 0 0 | * * * * 2 * * * * * | 0 0 2 0 0
....... ...x...     | 0 0 0 2 0 0 0 | * * * * * 2 * * * * | 0 1 0 1 0
...o.o. ...o.o.&#x  | 0 0 0 1 0 1 0 | * * * * * * 4 * * * | 0 0 1 1 0
....oo. ....oo.&#x  | 0 0 0 0 1 1 0 | * * * * * * * 4 * * | 0 0 1 0 1
.....oo .....oo&#x  | 0 0 0 0 0 1 1 | * * * * * * * * 4 * | 0 0 0 1 1
......x .......     | 0 0 0 0 0 0 2 | * * * * * * * * * 1 | 0 0 0 0 2
--------------------+---------------+---------------------+----------
xfo.... .......&#xt | 2 2 1 0 0 0 0 | 1 2 2 0 0 0 0 0 0 0 | 2 * * * *
....... of.x...&#xt | 1 2 0 2 0 0 0 | 0 2 0 2 0 1 0 0 0 0 | * 2 * * *
.ooooo. .ooooo.&#xr | 0 1 1 1 1 1 0 | 0 0 1 1 1 0 1 1 0 0 | * * 4 * *  cycle (BCEFD)
....... ...x.fo&#xt | 0 0 0 2 0 2 1 | 0 0 0 0 0 1 2 0 2 0 | * * * 2 *
....ofx .......&#xt | 0 0 0 0 1 2 2 | 0 0 0 0 0 0 0 2 2 1 | * * * * 2
```
```or
o...... o......      & | 4 * * * | 1 2 0 0 0 0 | 2 1 0
.o..... .o.....      & | * 8 * * | 0 1 1 1 0 0 | 1 1 1
..o.... ..o....      & | * * 4 * | 0 0 2 0 1 0 | 1 0 2
...o... ...o...        | * * * 4 | 0 0 0 2 0 1 | 0 2 1
-----------------------+---------+-------------+------
x...... .......      & | 2 0 0 0 | 2 * * * * * | 2 0 0
oo..... oo.....&#x   & | 1 1 0 0 | * 8 * * * * | 1 1 0
.oo.... .oo....&#x   & | 0 1 1 0 | * * 8 * * * | 1 0 1
.o.o... .o.o...&#x   & | 0 1 0 1 | * * * 8 * * | 0 1 1
..o.o.. ..o.o..&#x     | 0 0 2 0 | * * * * 2 * | 0 0 2
....... ...x...        | 0 0 0 2 | * * * * * 2 | 0 2 0
-----------------------+---------+-------------+------
xfo.... .......&#xt  & | 2 2 1 0 | 1 2 2 0 0 0 | 4 * *
....... of.x...&#xt  & | 1 2 0 2 | 0 2 0 2 0 1 | * 4 *
.ooooo. .ooooo.&#xr    | 0 2 2 1 | 0 0 2 2 1 0 | * * 4  cycle (BCEFD)
```

```oxfF xFfo Fofx&#zx   (F=ff) → existing heights = 0
(tegum sum of 3 mutually perp. (x,F)-{4} and an f-cube)

o... o... o...     | 4 * * * | 1 2 0 0 0 0 | 1 2 0
.o.. .o.. .o..     | * 4 * * | 0 0 1 2 0 0 | 2 0 1
..o. ..o. ..o.     | * * 8 * | 0 1 0 1 1 0 | 1 1 1
...o ...o ...o     | * * * 4 | 0 0 0 0 2 1 | 0 1 2
-------------------+---------+-------------+------
.... x... ....     | 2 0 0 0 | 2 * * * * * | 0 2 0
o.o. o.o. o.o.&#x  | 1 0 1 0 | * 8 * * * * | 1 1 0
.x.. .... ....     | 0 2 0 0 | * * 2 * * * | 2 0 0
.oo. .oo. .oo.&#x  | 0 1 1 0 | * * * 8 * * | 1 0 1
..oo ..oo ..oo&#x  | 0 0 1 1 | * * * * 8 * | 0 1 1
.... .... ...x     | 0 0 0 2 | * * * * * 2 | 0 0 2
-------------------+---------+-------------+------
oxf. .... ....&#xt | 1 2 2 0 | 0 2 1 2 0 0 | 4 * * tower: B-C-A
.... x.fo ....&#xt | 2 0 2 1 | 1 2 0 0 2 0 | * 4 *
.... .... .ofx&#xt | 0 1 2 2 | 0 0 0 2 2 1 | * * 4
```

```with inscribed tet subsym. (chiral choice)

4  * * |  3 0  0 |  3 inscribed tet vertices
* 12 * |  1 1  1 |  3 neighbouring vertices (of 3 further tets)
*  * 4 |  0 0  3 |  3 remaining 5th inscribed tet vertices
-------+---------+---
1  1 0 | 12 *  * |  2
0  2 0 |  * 6  * |  2
0  1 1 |  * * 12 |  2
-------+---------+---
1  3 1 |  2 1  2 | 12
```