```
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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.

*) 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 ``` otherWythoffians ```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 andholosnubs ```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 (?) ``` otherWythoffians ```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 andholosnubs ```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} (?) ``` otherWythoffians ```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 andholosnubs ```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 ``` ```... ``` ```... ``` 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 ``` otherWythoffians ```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 andholosnubs ```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 (?) ``` otherWythoffians ```x4/3x4/3o3/2*a - gocco x4/3o4/3x3/2*a - 2co (?) x4/3x4/3x3/2*a - 2quith (?) ``` (partial)snubs andholosnubs ```... ```

### 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 (?) ``` otherWythoffians ```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 andholosnubs ```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 (?) s3s5s - snid ``` ```... s5/2s3s3*a - seside ``` ```... ``` 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 (?) ``` otherWythoffians ```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 andholosnubs ```... s5/2s5s - siddid ``` ```... s5/3s3s5*a - sided ``` ```... ``` 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 ``` otherWythoffians ```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 andholosnubs ```... s3/2s3s5*a - 5ike+gad (?) ``` ```... ``` ```... 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 ``` otherWythoffians ```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 andholosnubs ```... ``` ```... 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 ``` otherWythoffians ```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 andholosnubs ```... 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 ``` otherWythoffians ```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 andholosnubs ```... ``` ```... ``` ```... ``` 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 (?) ``` otherWythoffians ```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 andholosnubs ```... ``` ```... ``` ```... ``` 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 (?) ``` otherWythoffians ```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 andholosnubs ```... ``` ```... ``` ```... 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 ``` otherWythoffians ```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 andholosnubs ```... 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 (?) ``` otherWythoffians ```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 andholosnubs ```... ``` ```... ``` ```... ``` 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 (?) ``` otherWythoffians ```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 andholosnubs ```... s5/4s5/4s3/2*a - 4ike+2gad (?) ``` ```... ```

### Prismatic Symmetries   (up)

 o ono (convex) o on/do (µ=d) o o o (convex) products ofquasiregulars ```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 ``` otherWythoffians ```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 andholosnubs ```s2s3s - oct s2s4s - squap 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( ? ) - oho 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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