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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.
©
| 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} ... |
©
| 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 |
... |
©
| 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 |
| 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 β2β5o → stap 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 |
s2s5/2s → stap s2sn/ds - n/d-ap s2s2n/do - n/d-ap s2s2n/dx - 2n/d-p |
s2s2s - tet |
| 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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