Site Map Polytopes Dynkin Diagrams Vertex Figures, etc. Incidence Matrices Index

---- 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.



quasiregular polyhedra and (multi)prisms of regulars   (up)

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


other Wythoffian polyhedra   (up)

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)
 


other kaleidoscopical uniform polyhedra (possibly Grünbaumian)   (up)

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


snubs and other non-kaleidoscopical uniform polyhedra   (up)

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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