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---- 3D ----

This page is available sorted by point-group symmetry (below) or by complexity or by similarity.



This dimension is accessible for pictures. Thus most of the following uniform polyhedra pages provide such. Further all pictures bear links to VRML models.

For most of those, which are derivable as any kind of snubs, pictures (and VRMLs) on their derivation are provided in addition. There the color coding is: red are the elements to be alternated, yellow are the faceting faces underneath (sefa( . )), the starting figure is given as wire frame. Those figures in general do not show uniform representants, it is the starting figure which is chosen to be uniform.

Especially the Grünbaumians can be best understood, if the (abstract) incidence matrices of those degenerate polyhedra with (geometrical) complete coincidence of some elements are not investigated individually, but independently of the special symmetry, i.e. by considering simultanuously the general Schwarz triangle o-p-o-q-o-r-*a and deriving therefrom the individual cases. (For the notation of virtual nodes like *a see here.)

Just as for the Grünbaumians, especially the holosnubs with ...β3... elements are better understood from the consideration of ...βn... with general odd n. Note moreover, that only for the holosnubs (due to their construction advice) in the followings those sections titled "convex" this attribute rather should be reduced to locally convex instead.

*) Some of the partial snubs which contain both s and x nodes, respectively partial holosnubs which contain both β and x nodes, do not have a uniform representation. Those are only possible with different edge lengths, e.g. as mere alternated faceting from their uniform starting figure. This latter representation surely is always possible.



Tetrahedral Symmetries   (up)

©
  o3o3o (convex) o3/2o3o3*a (µ=2) o3/2o3o (µ=3)
quasiregulars
x3o3o - tet
o3x3o - oct
x3/2o3o3*a - 2tet (?)
o3/2o3x3*a - 2tet (?)
x3/2o3o - tet
o3/2x3o - oct
o3/2o3x - tet
other
Wythoffians
x3x3o - tut
x3o3x - co
x3x3x - toe

a3b3c - (general variant)
x3/2x3o3*a - 2oct (?)
x3/2o3x3*a - oho
x3/2x3x3*a - 2tut (?)
x3/2x3o - 3tet (?)
x3/2o3x - 2thah (?)
o3/2x3x - tut
x3/2x3x - cho+4{6/2} (?)
(partial)
snubs and
holosnubs
β3o3o - 2tet (?)
o3β3o - oct+6{4} (?)
β3x3o - 2oct (?)
x3β3o - (?) *)
β3β3o - 2oct+8{3} (?)
β3o3x - oho
β3o3β - (?) *)
β3x3x - 2tut (?)
x3β3x - 2co (?)
β3β3x - 2co (?)
β3x3β - (?) *)
s3s3s - ike
β3/2o3o3*a - 4tet (?)
o3/2o3β3*a - 4tet (?)
β3/2x3o3*a - 2oct (?)
β3/2β3o3*a - 2oct+12{4} (?)
β3/2o3x3*a - oho+8{3} (?)
x3/2o3β3*a - 6tet (?)
β3/2o3β3*a - 2oct+12{4} (?)
β3/2x3x3*a - oho+8{3} (?)
x3/2x3β3*a - 4oct (?)
β3/2β3x3*a - (?) *)
β3/2x3β3*a - 4oct (?)
s3/2s3s3*a - 2oct+8{3} (?)
β3/2o3o - 2tet (?)
o3/2β3o - oct+6{4} (?)
o3/2o3β - 2tet (?)
β3/2x3o - 2tet (?)
x3/2β3o - 2tet (?)
β3/2β3o - 6tet (?)
β3/2o3x - 2oct (?)
x3/2o3β - 2oct (?)
β3/2o3β - (?) *)
o3/2β3x - (?) *)
o3/2x3β - 2oct (?)
o3/2β3β - 2oct+8{3} (?)
β3/2x3x - oho
x3/2β3x - (?) *)
x3/2x3β - 6tet (?)
β3/2β3x - 4thah (?)
β3/2x3β - 6tet (?)
...
  o3/2o3/2o (µ=5) o3/2o3/2o3/2*a (µ=6)  
quasiregulars
x3/2o3/2o - tet
o3/2x3/2o - oct
x3/2o3/2o3/2*a - 2tet (?)
 
other
Wythoffians
x3/2x3/2o - 3tet (?)
x3/2o3/2x - co
x3/2x3/2x - 2oct+6{4} (?)
x3/2x3/2o3/2*a - 2oct (?)
x3/2x3/2x3/2*a - 6tet (?)
 
(partial)
snubs and
holosnubs
β3/2o3/2o - 2tet (?)
o3/2β3/2o - oct+6{4} (?)
s3/2s3/2s - gike
...
β3/2o3/2o3/2*a - 4tet (?)
β3/2x3/2o3/2*a - 2oct (?)
β3/2β3/2o3/2*a - 2oct+12{4} (?)
...
 


Octahedral Symmetries   (up)

©
  o3o4o (convex) o3/2o4o4*a (µ=2) o4/3o3o4*a (µ=4)
quasiregulars
x3o4o - oct
o3x4o - co
o3o4x - cube
x3/2o4o4*a - oct+6{4} (?)
o3/2o4x4*a - 2cube (?)
x4/3o3o4*a - 2cube (?)
o4/3x3o4*a - oct+6{4} (?)
o4/3o3x4*a - oct+6{4} (?)
other
Wythoffians
x3x4o - toe
x3o4x - sirco
o3x4x - tic
x3x4x - girco

a3b4c - (general variant)
x3/2x4o4*a - 2co (?)
x3/2o4x4*a - socco
x3/2x4x4*a - 2tic (?)
x4/3x3o4*a - gocco
x4/3o3x4*a - socco
o4/3x3x4*a - 2cho (?)
x4/3x3x4*a - cotco
(partial)
snubs and
holosnubs
β3o4o - oct+6{4} (?)
o3β4o - (?) *)
o3o4s - tet
β3x4o - 2co (?)
x3β4o - (?) *)
s3s4o - ike
β3o4x - socco
x3o4s - tut
β3o4β - (?) *)
o3β4x - (?) *)
o3x4s - co
o3β4β - 2co+16{3} (?)
β3x4x - 2tic (?)
x3β4x - 2sirco (?)
x3x4s - toe
s3s4x - sirco
β3x4β - (?) *)
x3β4β - 2sirco (?)
s3s4s - snic
β3β4β - disco
...
...
  o3/2o4o (µ=5) o4/3o3o (µ=7) o4/3o3/2o (µ=11)
quasiregulars
x3/2o4o - oct
o3/2x4o - co
o3/2o4x - cube
x4/3o3o - cube
o4/3x3o - co
o4/3o3x - oct
x4/3o3/2o - cube
o4/3x3/2o - co
o4/3o3/2x - oct
other
Wythoffians
x3/2x4o - 2oct+6{4} (?)
x3/2o4x - querco
o3/2x4x - tic
x3/2x4x - sroh+8{6/2} (?)
x4/3x3o - quith
x4/3o3x - querco
o4/3x3x - toe
x4/3x3x - quitco
x4/3x3/2o - quith
x4/3o3/2x - sirco
o4/3x3/2x - 2oct+6{4} (?)
x4/3x3/2x - groh+8{6/2} (?)
(partial)
snubs and
holosnubs
β3/2o4o - oct+6{4} (?)
s3/2s4o - gike
...
o4/3o3β - oct+6{4} (?)
o4/3s3s - ike
...
o4/3o3/2β - oct+6{4} (?)
o4/3s3/2s - gike
...
  o4/3o4/3o3/2*a (µ=14)    
quasiregulars
x4/3o4/3o3/2*a - oct+6{4} (?)
o4/3x4/3o3/2*a - 2cube (?)
   
other
Wythoffians
x4/3x4/3o3/2*a - gocco
x4/3o4/3x3/2*a - 2co (?)
x4/3x4/3x3/2*a - 2quith (?)
   
(partial)
snubs and
holosnubs
...
   


Icosahedral Symmetries   (up)

©
  o3o5o (convex) o5/2o3o3*a (µ=2) o3/2o5o5*a (µ=2)
quasiregulars
x3o5o - ike
o3x5o - id
o3o5x - doe
x5/2o3o3*a - sidtid
o5/2o3x3*a - 2ike (?)
x3/2o5o5*a - cid
o3/2o5x5*a - 2doe (?)
other
Wythoffians
x3x5o - ti
x3o5x - srid
o3x5x - tid
x3x5x - grid

a3b5c - (general variant)
x5/2x3o3*a - 2id (?)
x5/2o3x3*a - siid
x5/2x3x3*a - 2ti (?)
x3/2x5o5*a - 2id (?)
x3/2o5x5*a - saddid
x3/2x5x5*a - 2tid (?)
(partial)
snubs and
holosnubs
β3o5o - cid
o3β5o - (?) *)
o3o5β - sidtid
x3β5o - (?) *)
β3x5o - 2id (?)
β3β5o - seside
x3o5β - siid
β3o5x - saddid
β3o5β - (?) *)
o3x5β - 2id (?)
o3β5x - (?) *)
o3β5β - 2id+40{3} (?)
x3x5β - 2ti (?)
x3β5x - 2srid (?)
x3β5β - 2srid (?)
β3x5x - 2tid (?)
β3x5β - (?) *)
β3β5x - 2srid (?)
s3s5s - snid
β3β5β - dissid
...
s5/2s3s3*a - seside
...
s3/2s5s5*a - 2id+40{3} (?)
  o5/2o5o (µ=3) o5/3o3o5*a (µ=4) o5/2o5/2o5/2*a (µ=6)
quasiregulars
x5/2o5o - sissid
o5/2x5o - did
o5/2o5x - gad
x5/3o3o5*a - ditdid
o5/3x3o5*a - gacid
o5/3o3x5*a - cid
x5/2o5/2o5/2*a - 2sissid (?)
other
Wythoffians
x5/2x5o - 3doe (?)
x5/2o5x - raded
o5/2x5x - tigid
x5/2x5x - sird+12{10/2} (?)
x5/3x3o5*a - gidditdid
x5/3o3x5*a - sidditdid
o5/3x3x5*a - ided
x5/3x3x5*a - idtid
x5/2x5/2o5/2*a - 2did (?)
x5/2x5/2x5/2*a - 6doe (?)
(partial)
snubs and
holosnubs
β5/2o5o - 2gad (?)
o5/2β5o - (?) *)
o5/2o5β - 2sissid (?)
x5/2β5o - (?) *)
β5/2x5o - 2did (?)
β5/2β5o - 3sidtid (?)
x5/2o5β - 2did (?)
β5/2o5x - 2sidhid (?)
β5/2o5β - (?) *)
o5/2x5β - 2did (?)
o5/2β5x - (?) *)
o5/2β5β - 3gidtid (?)
x5/2x5β - 6doe (?)
x5/2β5x -
x5/2β5β -
β5/2x5x - 2tigid (?)
β5/2x5β - (?) *)
β5/2β5x -
s5/2s5s - siddid
...
s5/3s3s5*a - sided
...
s5/2s5/2s5/2*a - 3sidtid (?)
  o3/2o3o5*a (µ=6) o5/4o5o5*a (µ=6) o5/2o3o (µ=7)
quasiregulars
x3/2o3o5*a - gidtid
o3/2x3o5*a - 2gike (?)
o3/2o3x5*a - gidtid
x5/4o5o5*a - 2gad (?)
o5/4o5x5*a - 2gad (?)
x5/2o3o - gissid
o5/2x3o - gid
o5/2o3x - gike
other
Wythoffians
x3/2x3o5*a - 3ike+gad (?)
x3/2o3x5*a - 2seihid (?)
o3/2x3x5*a - giid
x3/2x3x5*a - siddy+20{6/2} (?)
x5/4x5o5*a - 2did (?)
x5/4o5x5*a - 2sidhid (?)
x5/4x5x5*a - 2tigid (?)
x5/2x3o - 2gad+ike (?)
x5/2o3x - sicdatrid
o5/2x3x - tiggy
x5/2x3x - ri+12{10/2} (?)
(partial)
snubs and
holosnubs
...
s3/2s3s5*a - 5ike+gad (?)
...
s5/4s5s5*a - 3ike+3gad (?)
β5/2o3o - gidtid
o5/2β3o - 
o5/2o3β - gacid
...
s5/2s3s - gosid
  o3/2o5/2o5*a (µ=8) o5/3o5o (µ=9) o5/4o3o5*a (µ=10)
quasiregulars
x3/2o5/2o5*a - cid
o3/2x5/2o5*a - gacid
o3/2o5/2x5*a - ditdid
x5/3o5o - sissid
o5/3x5o - did
o5/3o5x - gad
x5/4o3o5*a - 2doe (?)
o5/4x3o5*a - cid
o5/4o3x5*a - cid
other
Wythoffians
x3/2x5/2o5*a - sidtid+gidtid (?)
x3/2o5/2x5*a - sidditdid
o3/2x5/2x5*a - ike+3gad (?)
x3/2x5/2x5*a - id+seihid+sidhid (?)
x5/3x5o - quit sissid
x5/3o5x - cadditradid
o5/3x5x - tigid
x5/3x5x - quitdid
x5/4x3o5*a - sidtid+ditdid (?)
x5/4o3x5*a - saddid
o5/4x3x5*a - 2gidhei (?)
x5/4x3x5*a - siddy+12{10/4} (?)
(partial)
snubs and
holosnubs
...
...
s5/3s5s - isdid
...
  o5/3o5/2o3*a (µ=10) o3/2o5o (µ=11) o5/3o3o (µ=13)
quasiregulars
x5/3o5/2o3*a - gacid
o5/3x5/2o3*a - 2gissid (?)
o5/3o5/2x3*a - gacid
x3/2o5o - ike
o3/2x5o - id
o3/2o5x - doe
x5/3o3o - gissid
o5/3x3o - gid
o5/3o3x - gike
other
Wythoffians
x5/3x5/2o3*a - gaddid
x5/3o5/2x3*a - 2sidhei (?)
o5/3x5/2x3*a - ditdid+gidtid (?)
x5/3x5/2x3*a - giddy+12{10/2} (?)
x3/2x5o - 2ike+gad (?)
x3/2o5x - gicdatrid
o3/2x5x - tid
x3/2x5x - sird+20{6/2} (?)
x5/3x3o - quit gissid
x5/3o3x - qrid
o5/3x3x - tiggy
x5/3x3x - gaquatid
(partial)
snubs and
holosnubs
...
s5/3s5/2s3*a - gisdid
...
β3/2β5o - sirsid
s3/2s5s - 4ike+gad (?)
...
s5/3s3s - gisid
  o5/4o3o3*a (µ=14) o3/2o5/2o5/2*a (µ=14) o5/4o5/2o3*a (µ=16)
quasiregulars
x5/4o3o3*a - gidtid
o5/4o3x3*a - 2gike (?)
x3/2o5/2o5/2*a - gacid
o3/2o5/2x5/2*a - 2gissid (?)
x5/4o5/2o3*a - cid
o5/4x5/2o3*a - ditdid
o5/4o5/2x3*a - gacid
other
Wythoffians
x5/4x3o3*a - 2gid (?)
x5/4o3x3*a - giid
x5/4x3x3*a - 2tiggy (?)
x3/2x5/2o5/2*a - 2gid (?)
x3/2o5/2x5/2*a - ditdid+gidtid (?)
x3/2x5/2x5/2*a - 2ike+4gad (?)
x5/4x5/2o3*a - 3sissid+gike (?)
x5/4o5/2x3*a - ided
o5/4x5/2x3*a - ike+3gad (?)
x5/4x5/2x3*a - did+sidhei+gidhei (?)
(partial)
snubs and
holosnubs
...
...
...
  o3/2o5/2o (µ=17) o3/2o5/3o3*a (µ=18) o5/3o5/3o5/2*a (µ=18)
quasiregulars
x3/2o5/2o - gike
o3/2x5/2o - gid
o3/2o5/2x - gissid
x3/2o5/3o3*a - 2ike (?)
o3/2x5/3o3*a - sidtid
o3/2o5/3x3*a - sidtid
x5/3o5/3o5/2*a - 2sissid (?)
o5/3x5/3o5/2*a - 2sissid (?)
other
Wythoffians
x3/2x5/2o - 2gike+sissid (?)
x3/2o5/2x - qrid
o3/2x5/2x - 2gad+ike (?)
x3/2x5/2x - 2gidtid+5cube (?)
x3/2x5/3o3*a - sissid+3gike (?)
x3/2o5/3x3*a - siid
o3/2x5/3x3*a - 2geihid (?)
x3/2x5/3x3*a - giddy+20{6/2} (?)
x5/3x5/3o5/2*a - 2gidhid (?)
x5/3o5/3x5/2*a - 2did (?)
x5/3x5/3x5/2*a - 2quitsissid (?)
(partial)
snubs and
holosnubs
...
...
...
  o5/4o3o (µ=19) o5/4o5/2o (µ=21) o3/2o3/2o5/2*a (µ=22)
quasiregulars
x5/4o3o - doe
o5/4x3o - id
o5/4o3x - ike
x5/4o5/2o - gad
o5/4x5/2o - did
o5/4o5/2x - sissid
x3/2o3/2o5/2*a - sidtid
o3/2x3/2o5/2*a - 2ike (?)
other
Wythoffians
x5/4x3o - 2sissid+gike (?)
x5/4o3x - gicdatrid
o5/4x3x - ti
x5/4x3x - ri+12{10/4} (?)
x5/4x5/2o - 3gissid (?)
x5/4o5/2x - cadditradid
o5/4x5/2x - 3doe (?)
x5/4x5/2x - 2ditdid+5cube (?)
x3/2x3/2o5/2*a - sissid+3gike (?)
x3/2o3/2x5/2*a - 2id (?)
x3/2x3/2x5/2*a - 4ike+2gad (?)
(partial)
snubs and
holosnubs
...
...
...
s3/2s3/2s5/2*a - sirsid
  o3/2o5/3o (µ=23) o3/2o5/3o5/3*a (µ=26) o5/4o5/3o (µ=27)
quasiregulars
x3/2o5/3o - gike
o3/2x5/3o - gid
o3/2o5/3x - gissid
x3/2o5/3o5/3*a - gacid
o3/2o5/3x5/3*a - 2gissid (?)
x5/4o5/3o - gad
o5/4x5/3o - did
o5/4o5/3x - sissid
other
Wythoffians
x3/2x5/3o - 2gike+sissid (?)
x3/2o5/3x - sicdatrid
o3/2x5/3x - quit gissid
x3/2x5/3x - gird+20{6/2} (?)
x3/2x5/3o5/3*a - 2gid (?)
x3/2o5/3x5/3*a - gaddid
x3/2x5/3x5/3*a - 2quitgissid
x5/4x5/3o - 3gissid (?)
x5/4o5/3x - raded
o5/4x5/3x - quit sissid
x5/4x5/3x - gird+12{10/4} (?)
(partial)
snubs and
holosnubs
...
s3/2s5/3s - girsid
...
...
  o5/4o3/2o (µ=29) o5/4o3/2o5/3*a (µ=32) o5/4o3/2o3/2*a (µ=34)
quasiregulars
x5/4o3/2o - doe
o5/4x3/2o - id
o5/4o3/2x - ike
x5/4o3/2o5/3*a - ditdid
o5/4x3/2o5/3*a - cid
o5/4o3/2x5/3*a - gacid
x5/4o3/2o3/2*a - gidtid
o5/4o3/2x3/2*a - 2gike (?)
other
Wythoffians
x5/4x3/2o - 2sissid+gike (?)
x5/4o3/2x - srid
o5/4x3/2x - 2ike+gad (?)
x5/4x3/2x - 2sidtid+5cube (?)
x5/4x3/2o5/3*a - 3sissid+gike (?)
x5/4o3/2x5/3*a - gidditdid
o5/4x3/2x5/3*a - sidtid+gidtid (?)
x5/4x3/2x5/3*a - gid+geihid+gidhid (?)
x5/4x3/2o3/2*a - 2gid (?)
x5/4o3/2x3/2*a - 3ike+gad (?)
x5/4x3/2x3/2*a - 2sissid+4gike (?)
(partial)
snubs and
holosnubs
...
...
...
  o5/4o5/4o3/2*a (µ=38) o5/4o5/4o5/4*a (µ=42)  
quasiregulars
x5/4o5/4o3/2*a - cid
o5/4x5/4o3/2*a - 2doe (?)
x5/4o5/4o5/4*a - 2gad (?)
 
other
Wythoffians
x5/4x5/4o3/2*a - sidtid+ditdid (?)
x5/4o5/4x3/2*a - 2id (?)
x5/4x5/4x3/2*a - 4sissid+2gike (?)
x5/4x5/4o5/4*a - 2did (?)
x5/4x5/4x5/4*a - 6gissid (?)
 
(partial)
snubs and
holosnubs
...
s5/4s5/4s3/2*a - 4ike+2gad (?)
...
s5/4s5/4s5/4*a - 3ike+3gad (?)
 


Prismatic Symmetries   (up)

  o ono (convex) o on/do (µ=d) o o o (convex)
products of
quasiregulars
x x3o  - trip
x x4o  - cube
x x5o  - pip
x x6o  - hip
x x8o  - op
x x10o - dip
x x12o - twip

x xno  - n-p
x x5/2o  - stip
x x8/3o  - stop
x x10/3o - stiddip

x xn/do  - n/d-p
x xn/2o  - n/2-p
x x x - cube
other
Wythoffians
x x3x - hip
x x4x - op
x x5x - dip
x x6x - twip

x xnx - 2n-p
x x4/3x - stop
x x5/3x - stiddip

x xn/dx - 2n/d-p
x xn/2x - (2n)/2-p
 
(partial)
snubs and
holosnubs
x2β5o  - stip
x2s6o  - trip
x2s8o  - cube
x2s10o - pip
x2s12o - hip
x2s3s  - trip
x2s4s  - cube
x2s5s  - pip
x2s6s  - hip
s2s3s  - oct
s2s4s  - squap
s2s5s  - pap

s2xno  - {n}
s2onx  - {n}
s2xnx  - {2n}
x2βnx  - 2n/2-p
β2βno  - n/2-ap
β2βnx  - 2n/2-p
x2sns  - n-p
s2sns  - n-ap

x2s2no - n-p
x2s2nx - 2n-p
s2s2no - n-ap
s2s2nx - 2n-p
s2sn/ds  - n/d-ap
s2s2n/do - n/d-ap
s2s2n/dx - 2n/d-p
s2s2s - tet


other non-kaleidoscopical uniform polyhedra   (up)

hemi reduced others
hemi( x3/2o3x )        - thah
hemi( o4/3x3x4*a )     - cho
hemi( x3/2o3x5*a )     - seihid
hemi( x5/4o5x5*a )     - sidhid
hemi( o5/4x3x5*a )     - gidhei
hemi( x5/3o5/2x3*a )   - sidhei
hemi( o3/2x5/3x3*a )   - geihid
hemi( x5/3x5/3o5/2*a ) - gidhid
reduced( x3/2x3x        , by 4{6/2} )   - cho
reduced( x3/2x4x        , by 8{6/2} )   - sroh
reduced( x4/3x3/2x      , by 8{6/2} )   - groh
reduced( x5/2x5x        , by 12{10/2} ) - sird
reduced( x3/2x3x5*a     , by 20{6/2} )  - siddy
reduced( x5/2x3x        , by 12{10/2} ) - ri
reduced( x3/2x5/2x5*a   , by id )     - seihid & sidhid
reduced( x5/4x3x5*a     , by 12{10/4} ) - siddy
reduced( x5/3x5/2x3*a   , by 12{10/2} ) - giddy
reduced( x3/2x5x        , by 20{6/2} )  - sird
reduced( x5/4x5/2x3*a   , by did )    - gidhei & sidhei
reduced( x3/2x5/3x3*a   , by 20{6/2} )  - giddy
reduced( x5/4x3x        , by 12{10/4} ) - ri
reduced( x3/2x5/3x      , by 20{6/2} )  - gird
reduced( x5/4x5/3x      , by 12{10/4} ) - gird
reduced( x5/4x3/2x5/3*a , by gid )    - geihid & gidhid
reduced( xx3/2ox&#x     , by {6/2} )    - thah
gidrid
gidisdrid


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