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---- 3D ----
This page is available sorted by complexity (below) or by point-group symmetry 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.
| o3o3o | o3o4o | o3o5o | o ono | o o o |
x3o3o - tet o3x3o - oct |
x3o4o - oct o3x4o - co o3o4x - cube |
x3o5o - ike o3x5o - id o3o5x - doe |
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 x x - cube |
| o3o3o | o3o4o | o3o5o | o ono |
x3x3o - tut x3o3x - co x3x3x - toe |
x3x4o - toe x3o4x - sirco o3x4x - tic x3x4x - girco |
x3x5o - ti x3o5x - srid o3x5x - tid x3x5x - grid |
x x3x - hip x x4x - op x x5x - dip x x6x - twip x xnx - 2n-p |
a3b3c - (general variant) |
a3b4c - (general variant) |
a3b5c - (general variant) |
Especially the Grünbaumians can be best understood, if the incidence matrices of those degenerate polyhedra 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.)
| o3/2o3o3*a (µ=2) | o3/2o3o (µ=3) | o3/2o3/2o (µ=5) |
x3/2o3o3*a - 2tet o3/2o3x3*a - 2tet x3/2x3o3*a - 2oct x3/2o3x3*a - oho x3/2x3x3*a - 2tut |
x3/2o3o - tet o3/2x3o - oct o3/2o3x - tet x3/2x3o - 3tet x3/2o3x - 2thah o3/2x3x - tut x3/2x3x - cho+4{6/2} |
x3/2o3/2o - tet o3/2x3/2o - oct x3/2x3/2o - 3tet x3/2o3/2x - co x3/2x3/2x - 2oct+6{4} |
| o3/2o3/2o3/2*a (µ=6) | o3/2o4o4*a (µ=2) | o4/3o3o4*a (µ=4) |
x3/2o3/2o3/2*a - 2tet x3/2x3/2o3/2*a - 2oct x3/2x3/2x3/2*a - 6tet |
x3/2o4o4*a - oct+6{4} o3/2o4x4*a - 2cube x3/2x4o4*a - 2co x3/2o4x4*a - socco x3/2x4x4*a - 2tic |
x4/3o3o4*a - 2cube o4/3x3o4*a - oct+6{4} o4/3o3x4*a - oct+6{4} x4/3x3o4*a - gocco x4/3o3x4*a - socco o4/3x3x4*a - 2cho x4/3x3x4*a - cotco |
| o3/2o4o (µ=5) | o4/3o3o (µ=7) | o4/3o3/2o (µ=11) |
x3/2o4o - oct o3/2x4o - co o3/2o4x - cube x3/2x4o - 2oct+6{4} x3/2o4x - querco o3/2x4x - tic x3/2x4x - sroh+8{6/2} |
x4/3o3o - cube o4/3x3o - co o4/3o3x - oct x4/3x3o - quith x4/3o3x - querco o4/3x3x - toe x4/3x3x - quitco |
x4/3o3/2o - cube o4/3x3/2o - co o4/3o3/2x - oct x4/3x3/2o - quith x4/3o3/2x - sirco o4/3x3/2x - 2oct+6{4} x4/3x3/2x - groh+8{6/2} |
| o4/3o4/3o3/2*a (µ=14) | o5/2o3o3*a (µ=2) | o3/2o5o5*a (µ=2) |
x4/3o4/3o3/2*a - oct+6{4} o4/3x4/3o3/2*a - 2cube x4/3x4/3o3/2*a - gocco x4/3o4/3x3/2*a - 2co x4/3x4/3x3/2*a - 2quith |
x5/2o3o3*a - sidtid o5/2o3x3*a - 2ike x5/2x3o3*a - 2id x5/2o3x3*a - siid x5/2x3x3*a - 2ti |
x3/2o5o5*a - cid o3/2o5x5*a - 2doe x3/2x5o5*a - 2id x3/2o5x5*a - saddid x3/2x5x5*a - 2tid |
| o5/2o5o (µ=3) | o5/3o3o5*a (µ=4) | o5/2o5/2o5/2*a (µ=6) |
x5/2o5o - sissid o5/2x5o - did o5/2o5x - gad x5/2x5o - 3doe x5/2o5x - raded o5/2x5x - tigid x5/2x5x - sird+12{10/2} |
x5/3o3o5*a - ditdid o5/3x3o5*a - gacid o5/3o3x5*a - cid x5/3x3o5*a - gidditdid x5/3o3x5*a - sidditdid o5/3x3x5*a - ided x5/3x3x5*a - idtid |
x5/2o5/2o5/2*a - 2sissid x5/2x5/2o5/2*a - 2did x5/2x5/2x5/2*a - 6doe |
| o3/2o3o5*a (µ=6) | o5/4o5o5*a (µ=6) | o5/2o3o (µ=7) |
x3/2o3o5*a - gidtid o3/2x3o5*a - 2gike o3/2o3x5*a - gidtid x3/2x3o5*a - 3ike+gad x3/2o3x5*a - 2seihid o3/2x3x5*a - giid x3/2x3x5*a - siddy+20{6/2} |
x5/4o5o5*a - 2gad o5/4o5x5*a - 2gad x5/4x5o5*a - 2did x5/4o5x5*a - 2sidhid x5/4x5x5*a - 2tigid |
x5/2o3o - gissid o5/2x3o - gid o5/2o3x - gike x5/2x3o - 2gad+ike x5/2o3x - sicdatrid o5/2x3x - tiggy x5/2x3x - ri+12{10/2} |
| o3/2o5/2o5*a (µ=8) | o5/3o5o (µ=9) | o5/4o3o5*a (µ=10) |
x3/2o5/2o5*a - cid o3/2x5/2o5*a - gacid o3/2o5/2x5*a - ditdid x3/2x5/2o5*a - sidtid+gidtid x3/2o5/2x5*a - sidditdid o3/2x5/2x5*a - ike+3gad x3/2x5/2x5*a - id+seihid+sidhid |
x5/3o5o - sissid o5/3x5o - did o5/3o5x - gad x5/3x5o - quit sissid x5/3o5x - cadditradid o5/3x5x - tigid x5/3x5x - quitdid |
x5/4o3o5*a - 2doe o5/4x3o5*a - cid o5/4o3x5*a - cid x5/4x3o5*a - sidtid+ditdid x5/4o3x5*a - saddid o5/4x3x5*a - 2gidhei x5/4x3x5*a - siddy+12{10/4} |
| o5/3o5/2o3*a (µ=10) | o3/2o5o (µ=11) | o5/3o3o (µ=13) |
x5/3o5/2o3*a - gacid o5/3x5/2o3*a - 2gissid o5/3o5/2x3*a - gacid x5/3x5/2o3*a - gaddid x5/3o5/2x3*a - 2sidhei o5/3x5/2x3*a - ditdid+gidtid x5/3x5/2x3*a - giddy+12{10/2} |
x3/2o5o - ike o3/2x5o - id o3/2o5x - doe x3/2x5o - 2ike+gad x3/2o5x - gicdatrid o3/2x5x - tid x3/2x5x - sird+20{6/2} |
x5/3o3o - gissid o5/3x3o - gid o5/3o3x - gike x5/3x3o - quit gissid x5/3o3x - qrid o5/3x3x - tiggy x5/3x3x - gaquatid |
| o5/4o3o3*a (µ=14) | o3/2o5/2o5/2*a (µ=14) | o5/4o5/2o3*a (µ=16) |
x5/4o3o3*a - gidtid o5/4o3x3*a - 2gike x5/4x3o3*a - 2gid x5/4o3x3*a - giid x5/4x3x3*a - 2tiggy |
x3/2o5/2o5/2*a - gacid o3/2o5/2x5/2*a - 2gissid x3/2x5/2o5/2*a - 2gid x3/2o5/2x5/2*a - ditdid+gidtid x3/2x5/2x5/2*a - 2ike+4gad |
x5/4o5/2o3*a - cid o5/4x5/2o3*a - ditdid o5/4o5/2x3*a - gacid x5/4x5/2o3*a - 3sissid+gike x5/4o5/2x3*a - ided o5/4x5/2x3*a - ike+3gad x5/4x5/2x3*a - did+sidhei+gidhei |
| o3/2o5/2o (µ=17) | o3/2o5/3o3*a (µ=18) | o5/3o5/3o5/2*a (µ=18) |
x3/2o5/2o - gike o3/2x5/2o - gid o3/2o5/2x - gissid x3/2x5/2o - 2gike+sissid x3/2o5/2x - qrid o3/2x5/2x - 2gad+ike x3/2x5/2x - 2gidtid+5cube |
x3/2o5/3o3*a - 2ike o3/2x5/3o3*a - sidtid o3/2o5/3x3*a - sidtid x3/2x5/3o3*a - sissid+3gike x3/2o5/3x3*a - siid o3/2x5/3x3*a - 2geihid x3/2x5/3x3*a - giddy+20{6/2} |
x5/3o5/3o5/2*a - 2sissid o5/3x5/3o5/2*a - 2sissid x5/3x5/3o5/2*a - 2gidhid x5/3o5/3x5/2*a - 2did x5/3x5/3x5/2*a - 2quitsissid |
| o5/4o3o (µ=19) | o5/4o5/2o (µ=21) | o3/2o3/2o5/2*a (µ=22) |
x5/4o3o - doe o5/4x3o - id o5/4o3x - ike x5/4x3o - 2sissid+gike x5/4o3x - gicdatrid o5/4x3x - ti x5/4x3x - ri+12{10/4} |
x5/4o5/2o - gad o5/4x5/2o - did o5/4o5/2x - sissid x5/4x5/2o - 3gissid x5/4o5/2x - cadditradid o5/4x5/2x - 3doe x5/4x5/2x - 2ditdid+5cube |
x3/2o3/2o5/2*a - sidtid o3/2x3/2o5/2*a - 2ike x3/2x3/2o5/2*a - sissid+3gike x3/2o3/2x5/2*a - 2id x3/2x3/2x5/2*a - 4ike+2gad |
| o3/2o5/3o (µ=23) | o3/2o5/3o5/3*a (µ=26) | o5/4o5/3o (µ=27) |
x3/2o5/3o - gike o3/2x5/3o - gid o3/2o5/3x - gissid x3/2x5/3o - 2gike+sissid x3/2o5/3x - sicdatrid o3/2x5/3x - quit gissid x3/2x5/3x - gird+20{6/2} |
x3/2o5/3o5/3*a - gacid o3/2o5/3x5/3*a - 2gissid x3/2x5/3o5/3*a - 2gid x3/2o5/3x5/3*a - gaddid x3/2x5/3x5/3*a - 2quitgissid |
x5/4o5/3o - gad o5/4x5/3o - did o5/4o5/3x - sissid x5/4x5/3o - 3gissid x5/4o5/3x - raded o5/4x5/3x - quit sissid x5/4x5/3x - gird+12{10/4} |
| o5/4o3/2o (µ=29) | o5/4o3/2o5/3*a (µ=32) | o5/4o3/2o3/2*a (µ=34) |
x5/4o3/2o - doe o5/4x3/2o - id o5/4o3/2x - ike x5/4x3/2o - 2sissid+gike x5/4o3/2x - srid o5/4x3/2x - 2ike+gad x5/4x3/2x - 2sidtid+5cube |
x5/4o3/2o5/3*a - ditdid o5/4x3/2o5/3*a - cid o5/4o3/2x5/3*a - gacid x5/4x3/2o5/3*a - 3sissid+gike x5/4o3/2x5/3*a - sidtid+gidtid o5/4x3/2x5/3*a - gidditdid x5/4x3/2x5/3*a - gid+geihid+gidhid |
x5/4o3/2o3/2*a - gidtid o5/4o3/2x3/2*a - 2gike x5/4x3/2o3/2*a - 2gid x5/4o3/2x3/2*a - 3ike+gad x5/4x3/2x3/2*a - 2sissid+4gike |
| o5/4o5/4o3/2*a (µ=38) | o5/4o5/4o5/4*a (µ=42) | o on/do (µ=d) |
x5/4o5/4o3/2*a - cid o5/4x5/4o3/2*a - 2doe x5/4x5/4o3/2*a - sidtid+ditdid x5/4o5/4x3/2*a - 2id x5/4x5/4x3/2*a - 4sissid+2gike |
x5/4o5/4o5/4*a - 2gad x5/4x5/4o5/4*a - 2did x5/4x5/4x5/4*a - 6gissid |
x x5/2o - stip x x8/3o - stop x x10/3o - stiddip x x4/3x - stop x x5/3x - stiddip x xn/do - n/d-p x xn/dx - 2n/d-p |
Just as for the Grünbaumians, especially the holosnubs with ...β3... elements are better understood from the consideration of ...βn... with general odd n.
| snub | partial snub | |||||||||||||||||||||
s3s3s - ike s3s4s - snic s3s5s - snid s2s3s - oct s2sns - n-ap s2s2s - tet s5/2s3s3*a - seside s5/2s5s - siddid s5/3s3s5*a - sided s5/2s3s - gosid s5/3s5s - isdid s5/3s5/2s3*a - gisdid s5/3s3s - gisid s3/2s3/2s5/2*a - sirsid s3/2s5/3s - girsid s3/2s3/2s - gike s2sn/ds - n/d-ap |
s3s4o - ike s3s4x - sirco s4o3o - tet s4x3o - co s4o3x - tut s4x3x - toe s3s4/3o - ike s3/2s4o - gike s3/2s4/3o - gike s2s2no - n-ap s2s2n/do - n/d-ap s2s2nx - 2n-p s2s2n/dx - 2n/d-p x2sns - n-p x2s2no - n-p x2s2nx - 2n-p s2xno - {n} s2onx - {n} s2xnx - {2n} | |||||||||||||||||||||
| holosnub | hemi | |||||||||||||||||||||
β3o3o - 2tet o3β3o - oct+6{4} β3x3o - 2oct x3β3o - (?) *) β3β3o - 2oct+8{3} β3o3x - oho β3o3β - (?) *) β3x3x - 2tut x3β3x - 2co β3β3x - 2co β3x3β - (?) *) β3o4o - oct+6{4} o3β4o - (?) *) β3x4o - 2co x3β4o - (?) *) β3o4x - socco β3o4β - (?) *) o3β4x - (?) *) o3β4β - 2co+16{3} β3x4x - 2tic x3β4x - 2sirco β3x4β - (?) *) x3β4β - 2sirco β3o5o - cid o3β5o - (?) *) o3o5β - sidtid β3x5o - 2id x3β5o - (?) *) β3β5o - seside β3o5x - saddid x3o5β - siid β3o5β - (?) *) o3β5x - (?) *) o3x5β - 2id o3β5β - 2id+40{3} β3x5x - 2tid x3β5x - 2srid x3x5β - 2ti β3β5x - 2srid β3x5β - (?) *) x3β5β - 2srid β2βno - n/2-ap x2βnx - 2n/2-p β2βnx - 2n/2-p *) not possible as uniform representation, only as faceting |
hemi( x3/2o3x ) - thah hemi( o4/3x3x4*a ) - cho hemi( x3/2o3x5*a ) - seihid hemi( o5/4x5x5*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 | other | |||||||||||||||||||||
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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