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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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