Acronym	sishi, spD
Name	small stellated hecatonicosachoron, stellated polydodecahedron
Cross sections	©
Circumradius	1
Inradius	(1+sqrt(5))/4 = 0.809017
Density	4
Vertex figure	doe
Coordinates	(1, 0, 0, 0)                                         & all permutations, all changes of sign (vertex inscribed q-hex) (1/2, 1/2, 1/2, 1/2)                             & all permutations, all changes of sign (vertex inscribed tes) ((1+sqrt(5))/4, (sqrt(5)-1)/4, 1/4, 0)   & all even permutations, all changes of sign (vertex inscribed v-sadi)
General of army	ex
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: co ditdid gad gidtid gike ike oct oho pip sidtid sissid stip tet thah trip sadixhix (compound) 300 0 0 0 0 0 0 0 0 0 0 0 0 600 0 gadixhix (compound) 300 0 0 0 0 0 0 0 0 0 0 0 0 600 0 idhi 0 120 0 0 0 0 0 0 0 0 0 0 0 0 0 gaghi 0 0 120 0 0 0 0 0 0 0 0 0 0 0 0 didhi 0 0 0 120 120 0 0 0 0 0 0 0 0 0 0 gitpodady (fissary) 0 0 0 120 0 120 0 0 0 0 0 0 0 0 0 sridixhi 0 0 0 120 0 0 0 0 0 0 0 0 600 0 0 dittady 0 0 0 0 120 120 0 0 0 0 0 0 0 0 0 sifdahihox (fissary) 0 0 0 0 120 0 0 300 0 0 0 0 0 0 0 sitpodady (fissary) 0 0 0 0 120 0 0 0 0 120 0 0 0 0 0 sidtixhi 0 0 0 0 120 0 0 0 0 0 0 0 600 0 0 sridaphi 0 0 0 0 120 0 0 0 0 0 0 0 0 0 1200 gifdahihox (fissary) 0 0 0 0 0 120 0 300 0 0 0 0 0 0 0 ofiddady 0 0 0 0 0 120 0 0 0 120 0 0 0 0 0 gidtixhi 0 0 0 0 0 120 0 0 0 0 0 0 600 0 0 gridaphi 0 0 0 0 0 120 0 0 0 0 0 0 0 0 1200 dox (compound) 0 0 0 0 0 0 600 0 0 0 0 0 0 0 0 paphacki 0 0 0 0 0 0 0 0 720 0 0 0 0 0 0 gridixhi 0 0 0 0 0 0 0 0 0 120 0 0 600 0 0 sishi 0 0 0 0 0 0 0 0 0 0 120 0 0 0 0 paphicki 0 0 0 0 0 0 0 0 0 0 0 720 0 0 0 getut 0 0 0 0 0 0 0 0 0 0 0 0 480 480 480 setut 0 0 0 0 0 0 0 0 0 0 0 0 480 480 480 & others)
Dual	fix
Dihedral angles	at {5/2} between sissid and sissid:   144°
Face vector	120, 1200, 720, 120
Confer	Grünbaumian relatives: sishi+gaghi+120did   sishi+gaghi+idhi   sishi+gridixhi   sishi+idhi   sishi+ofiddady   sishi+paphicki+gridaphi   2sishi   2sishi+2paphicki   3sishi   6sishi   uniform relatives: gudap   rappisdi   related segmentochora: gikaike   decompositions: pt || sishi   general polytopal classes: Wythoffian polychora   regular   noble polytopes
External links

As abstract polytope sishi is isomorphic to gaghi, thereby replacing pentagrams by pentagons, resp. sissid by gad, resp. replacing doe vertex figures by gissid ones.

Here the sissids form 12 swirling rings of 10 each.

Note that the edge skeleton supports not only sissid or gike cells, but also gad or ike cells. Therefore any single such ring of 10 sissids could well be replaced by a ring of starps (5/3-antiprisms). The then open faces could be closed by a complemental ring of 10 paps (5-antiprisms) – up to some "glue" of 50 tets: This is how gudap could be obtained herefrom as a subsymmetrical faceting.

If considered with according densities, then sishi can be thought of as the external blend of 120 sissidpies. This decomposition is described as the degenerate segmentoteron ox5/2oo5oo3oo&#x.

Incidence matrix according to Dynkin symbol

x5/2o5o3o

.   . . . | 120 ♦   20 |  30 |  12
----------+-----+------+-----+----
x   . . . |   2 | 1200 |   3 |   3
----------+-----+------+-----+----
x5/2o . . |   5 |    5 | 720 |   2
----------+-----+------+-----+----
x5/2o5o . ♦  12 |   30 |  12 | 120

x5/2o5o3/2o

.   . .   . | 120 ♦   20 |  30 |  12
------------+-----+------+-----+----
x   . .   . |   2 | 1200 |   3 |   3
------------+-----+------+-----+----
x5/2o .   . |   5 |    5 | 720 |   2
------------+-----+------+-----+----
x5/2o5o   . ♦  12 |   30 |  12 | 120

x5/2o5/4o3o

.   .   . . | 120 ♦   20 |  30 |  12
------------+-----+------+-----+----
x   .   . . |   2 | 1200 |   3 |   3
------------+-----+------+-----+----
x5/2o   . . |   5 |    5 | 720 |   2
------------+-----+------+-----+----
x5/2o5/4o . ♦  12 |   30 |  12 | 120

x5/2o5/4o3/2o

.   .   .   . | 120 ♦   20 |  30 |  12
--------------+-----+------+-----+----
x   .   .   . |   2 | 1200 |   3 |   3
--------------+-----+------+-----+----
x5/2o   .   . |   5 |    5 | 720 |   2
--------------+-----+------+-----+----
x5/2o5/4o   . ♦  12 |   30 |  12 | 120

x5/3o5o3o

.   . . . | 120 ♦   20 |  30 |  12
----------+-----+------+-----+----
x   . . . |   2 | 1200 |   3 |   3
----------+-----+------+-----+----
x5/3o . . |   5 |    5 | 720 |   2
----------+-----+------+-----+----
x5/3o5o . ♦  12 |   30 |  12 | 120

x5/3o5o3/2o

.   . .   . | 120 ♦   20 |  30 |  12
------------+-----+------+-----+----
x   . .   . |   2 | 1200 |   3 |   3
------------+-----+------+-----+----
x5/3o .   . |   5 |    5 | 720 |   2
------------+-----+------+-----+----
x5/3o5o   . ♦  12 |   30 |  12 | 120

x5/3o5/4o3o

.   .   . . | 120 ♦   20 |  30 |  12
------------+-----+------+-----+----
x   .   . . |   2 | 1200 |   3 |   3
------------+-----+------+-----+----
x5/3o   . . |   5 |    5 | 720 |   2
------------+-----+------+-----+----
x5/3o5/4o . ♦  12 |   30 |  12 | 120

x5/3o5/4o3/2o

.   .   .   . | 120 ♦   20 |  30 |  12
--------------+-----+------+-----+----
x   .   .   . |   2 | 1200 |   3 |   3
--------------+-----+------+-----+----
x5/3o   .   . |   5 |    5 | 720 |   2
--------------+-----+------+-----+----
x5/3o5/4o   . ♦  12 |   30 |  12 | 120