Acronym doe
TOCID symbol D
Name dodecahedron,
cosmohedron,
Goldberg polyhedron GP(1,0)
 
 © ©
Circumradius sqrt[(9+3 sqrt(5))/8] = 1.401259
Edge radius (3+sqrt(5))/4 = 1.309017
Inradius sqrt[(25+11 sqrt(5))/40] = 1.113516
Vertex figure [53] = f3o
Vertex layers
LayerSymmetrySubsymmetries
 o3o5oo3o .o . o. o5o
1o3o5xo3o .
vertex first
o . x
edge first
. o5x
{5} first
2o3f .
vertex figure
f . f. o5f
vertex figure
3f3x .F . o. f5o
4x3f .x . F. x5o
opposite {5}
5f3o .F . o 
6o3o .
opposite vertex
f . f
7 o . x
opposite 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  
stellations:
p2p5p   titdi   mibkid   cell of gap dual  
general polytopal classes:
regular   noble polytopes  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   quickfur   nan ma

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

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