Acronym doe
TOCID symbol D
Name dodecahedron,
cosmohedron
 
 © ©
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

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


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.&#xt | 0 1 1 1 1 1 0 | 0 0 1 1 1 0 1 1 0 0 | * * 4 * *
....... ...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.&#xt    | 0 2 2 1 | 0 0 2 2 1 0 | * * 4

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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