Stott expansion (resp. contraction)

Within Verhandlingen der koninklijke Akademie van Wetenschappen, eerste Sectie, Deel XI (Amsterdam, 1913) Mrs. A. Boole Stott published the "Geometrical deduction of semiregular from regular polytopes and space fillings". What is outlined there essentially boils down, by using Dynkin symbols, how polytopes are related which differ just in the mark of a single node.

If applied onto convex shapes only one might consider the polytope being contained within the rubber skin of a balloon. Then pulling apart the vertices, all members of an edge class, those of a face class, or of any other bounding elements would just result in new edges (and higher dimensional elements too) which simultanuously increase from zero to non-zero size (expansion). The inverse process, reducing one specific edge class in size down to zero is called contraction.

       o3x3o            o3x3x
       
x3x3o            x3x3x







       o3o3o            o3o3x
       
x3o3o            x3o3x

       o3x4o            o3x4x
       
x3x4o            x3x4x







       o3o4o            o3o4x
       
x3o4o            x3o4x

       o3x5o            o3x5x
       
x3x5o            x3x5x







       o3o5o            o3o5x
       
x3o5o            x3o5x

A corresponding continuous transition can interactively be done in this VRML.

Despite of those few provided examples, Stott expansion applies to all dimensions and all curvatures, i.e. spherical geometries as well as euclideans or hyperbolics.

Partial Stott expansion (resp. contraction)

To guarantee "semiregularity" (as she states it – nowadays we would ask for uniformity), the application was meant to be restricted to actions under the full symmetry of the to be considered polytopes, resp. to those symmetries the unmarked Dynkin symbols represent.

Exactly 100 years thereafter, i.e. in 2013, Klitzing extended her ideas onto partial applications as well, i.e. onto sets of bounding elements, which form a complete equivalence class under some subsymmetry only. – In general this would result in non-uniform polytopes.

The main concern however was to result in CRF (convex regular faced) polytopes at least. According sequences are displayed here below.

• Spherical Geometry
  • 3D
  • 4D
  • 5D
• Euclidean Geometry
  • 3D
  • 4D

Only series which contain true CRFs or which are used within such of the next dimension.
† = subdimensional;   ° = uniform;   * = scaliform;   _ = Johnson solid / CRF

Spherical Geometry

----
 3D
----

axial subsymmetry

o3o4o = point †°  ↔  edge †°   ↔  {4} †°   ↔  o3o4x = cube °[1]

x3o4o = oct °     ↔  esquidpy  ↔  squobcu  ↔  x3o4x = sirco °

tetrahedral subsymmetry

o3o4o = point †°  ↔  tet °  ↔  o3x4o = co °[2]

x3o4o = oct °     ↔  tut °  ↔  x3x4o = toe °[3]

   | o4o3a    patex-o4o3a  o4x3a  :: tetrahedral
---+----------------------------
its 1st type of cells     (in the right case for any "a", in the left only for "a" = "x"):
 4 | . o3a    . o3a     -> . x3a  :: triangular
 4 | . o3a -> . x3a        . x3a
---+----------------------------
its 2nd type of cells     (only for "a" = "x"):
12 | o . a    o . a        o . a  :: none
---+----------------------------
its 3rd type of cells     (in the right case ever, in the left one never):
 6 | o4o . -> pex-o4o   -> o4x .  :: axial

----
 4D
----

axial subsymmetry

o3o3o4o = point †°  ↔  edge †°  ↔  {4} †°           ↔  cube †°      ↔  o3o3o4x = tes °[4]

x3o3o4o = hex °     ↔  pex hex   ↔  quawros         ↔  pacsid pith  ↔  x3o3o4x = sidpith °

o3x3o4o = ico °     ↔  pexic     ↔  bicyte ausodip  ↔  pacsrit      ↔  o3x3o4x = srit °

x3x3o4o = thex °    ↔  pex thex  ↔  pabex thex      ↔  pacprit      ↔  x3x3o4x = prit °

hexadecachoral subsymmetry

o3o4o3o = point †°      ↔  hex °      ↔  rit °      ↔  o3x4o3o = rico °[5]

x3o4o3o = ico °         ↔  thex °     ↔  tah °      ↔  x3x4o3o = tico °[6]

o3o4o3x = (dual) ico °  ↔  poxic      ↔  pocsric    ↔  o3x4o3x = srico °

x3o4o3x = spic °        ↔  owau prit  ↔  poc prico  ↔  x3x4o3x = prico °

   | a3o4o3b    pox-a3o4o3b    poc-a3o4x3b    a3o4x3b  :: hexadecachoral
---+-------------------------------------------------
its 1st type of cells     (in the right case for any "b", in the left only for "b" = "x"):
 8 | . o4o3b -> . o4o3b        patex-o4o3b    . o4x3b  :: tetrahedral
 8 | . o4o3b    patex-o4o3b -> patex-o4o3b    . o4x3b
 8 | . o4o3b    patex-o4o3b    . o4x3b     -> . o4x3b
---+-------------------------------------------------
its 2nd type of cells     (in the right case for any "b" and "a" = "x", in the left only for "a" = "b" = "x"):
32 | a . o3b    a . o3b        a . o3b     -> a . x3b  :: triangular
32 | a . o3b    a . o3b     -> a . x3b        a . x3b
32 | a . o3b -> a . x3b        a . x3b        a . x3b
---+-------------------------------------------------
its 3rd type of cells     (only for "a" = "b" = "x"):
96 | a3o . b    a3o . b        a3o . b        a3o . b  :: none
---+-------------------------------------------------
its 4th type of cells     (in the right case ever, in the left only for "a" = "x"):
24 | a3o4o . -> pex-a3o4o   -> pac-a3o4x   -> a3o4x .  :: axial

----
 5D
----

axial subsymmetry

o3o3o3o4o = point †°  ↔  edge †°  ↔  {4} †°  ↔  cube †°  ↔  tes †°  ↔  o3o3o3o4x = pent °[7]

x3o3o3o4o = tac °     ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  x3o3o3o4x = scant °

o3x3o3o4o = rat °     ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  o3x3o3o4x = span °

o3o3x3o4o = nit °     ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  o3o3x3o4x = sirn °

x3x3o3o4o = tot °     ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  x3x3o3o4x = cappin °

x3o3x3o4o = sart °    ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  x3o3x3o4x = carnit °

o3x3x3o4o = bittit °  ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  o3x3x3o4x = prin °

x3x3x3o4o = gart °    ↔  ...      ↔  ...     ↔  ...      ↔  ...     ↔  x3x3x3o4x = cogart °

This set of series seems to exist for all dimensions. (The other set of sequences provided in lower dimensions, seems to stick to those only. In fact, one clearly could extend that one step on to the family o3o3o *b3o *b3o (cf. the corresponding footnotes), but this one would then be a flat euclidean tetracomb, not a true 5D polytope.)

Euclidean Geometry

----
 3D  = honeycombs
----

axial subsymmetry

o4o3o4o = point †°  ↔  aze †°   ↔  squat †°  ↔  o4o3o4x = chon °[8]

x4o3o4o = chon °    ↔  chon °   ↔  chon °    ↔  x4o3o4x = chon °[9]

o4x3o4o = rich °    ↔  pexrich  ↔  pacsrich  ↔  o4x3o4x = srich °

x4x3o4o = tich °    ↔  pextich  ↔  pacprich  ↔  x4x3o4x = prich °


o3o3x *b4o = octet °  ↔  pextoh *  ↔  pacratoh *  ↔  o3o3x *b4x = ratoh °

The last sequence could be considered as an alternating faceting of the second sequence: we have s4o3o4y = o3o3x *b4y (where y=o or y=x). Similarily one might ask about the further alternated faceting case. But that one does not provide anything new, within this symmetry it just reproduces the 3rd sequence: s4x3o4y = o4x3o4y.

----
 4D  = tetracombs
----

axial subsymmetry

o4o3o3o4o = point †°   ↔  aze †°   ↔  squat †°  ↔  chon †°  ↔  o4o3o3o4x = test °[10]

x4o3o3o4o = test °     ↔  test °   ↔  test °    ↔  test °   ↔  x4o3o3o4x = test °[11]

o4x3o3o4o = rittit °   ↔  ...      ↔  ...       ↔  ...      ↔  o4x3o3o4x = sidpitit °

o4o3x3o4o = icot °     ↔  ...      ↔  ...       ↔  ...      ↔  o4o3x3o4x = srittit °

x4x3o3o4o = tattit °   ↔  ...      ↔  ...       ↔  ...      ↔  x4x3o3o4x = capotat °

x4o3x3o4o = srittit °  ↔  ...      ↔  ...       ↔  ...      ↔  x4o3x3o4x = scartit °

x4x3x3o4o = grittit °  ↔  ...      ↔  ...       ↔  ...      ↔  x4x3x3o4x = gicartit °

hexadecachoric-tetracombal subsymmetry

o3o3o4o3o = point †°   ↔  hext °   ↔  rittit °   ↔  bricot °  ↔  o3o3o4x3o = ricot °[12]

x3o3o4o3o = hext °     ↔  ...      ↔  ...        ↔  ...       ↔  x3o3o4x3o = spaht °

o3x3o4o3o = icot °     ↔  ...      ↔  ...        ↔  ...       ↔  o3x3o4x3o = sibricot °

o3o3o4o3x = icot °     ↔  thext °  ↔  batitit °  ↔  bithit °  ↔  o3o3o4x3x = ticot °[13]

x3x3o4o3o = thext °    ↔  ...      ↔  ...        ↔  ...       ↔  x3x3o4x3o = pataht °

x3o3o4o3x = scicot °   ↔  ...      ↔  ...        ↔  ...       ↔  x3o3o4x3x = capoht °

o3x3o4o3x = spict °    ↔  ...      ↔  ...        ↔  ...       ↔  o3x3o4x3x = paticot °

x3x3o4o3x = capicot °  ↔  ...      ↔  ...        ↔  ...       ↔  x3x3o4x3x = capticot °

    | a3b3o4o3c    phextex-a3b3o4o3c  pabhextex-a3b3o4o3c  phextco-a3b3o4x3c  a3b3o4x3c  :: hexadecachoric-tetracombal
----+----------------------------------------------------------------------------------
its 1st type of cells     (in the right case for any "b" and "c", in the left only for "b" = "x" or "c" = "x"):
  N | . b3o4o3c -> . b3o4o3c          pox-b3o4o3c          poc-b3o4x3c        . b3o4x3c  :: hexadecachoral
  N | . b3o4o3c    pox-b3o4o3c     -> pox-b3o4o3c          poc-b3o4x3c        . b3o4x3c
  N | . b3o4o3c    pox-b3o4o3c        poc-b3o4x3c       -> poc-b3o4x3c        . b3o4x3c
  N | . b3o4o3c    pox-b3o4o3c        poc-b3o4x3c          . b3o4x3c       -> . b3o4x3c
----+----------------------------------------------------------------------------------
its 2nd type of cells     (in the right case only for "a" = "x", in the left only for "a" = "c" = "x"):
 8N | a . o4o3c    a . o4o3c          a . o4o3c         -> patex-o4o3a-p   -> a . o4x3c  :: tetrahedral
 8N | a . o4o3c    a . o4o3c       -> patex-o4o3a-p        patex-o4o3a-p      a . o4x3c
 8N | a . o4o3c    a . o4o3c          patex-o4o3a-p        patex-o4o3a-p      a . o4x3c
 8N | a . o4o3c -> patex-o4o3a-p      patex-o4o3a-p     -> a . o4x3c          a . o4x3c
 8N | a . o4o3c    patex-o4o3a-p      patex-o4o3a-p        a . o4x3c          a . o4x3c
 8N | a . o4o3c    patex-o4o3a-p   -> a . o4x3c            a . o4x3c          a . o4x3c
----+----------------------------------------------------------------------------------
its 3rd type of cells     (in the right case only for "a" = "x" or "b" = "x", in the left only for ("a" = "x" or "b" = "x") and "c" = "x"):
32N | a3b . o3c    a3b . o3c          a3b . o3c            a3b . o3c       -> a3b . x3c  :: triangular
32N | a3b . o3c    a3b . o3c          a3b . o3c         -> a3b . x3c          a3b . x3c
32N | a3b . o3c    a3b . o3c       -> a3b . x3c            a3b . x3c          a3b . x3c
32N | a3b . o3c -> a3b . x3c          a3b . x3c            a3b . x3c          a3b . x3c
----+----------------------------------------------------------------------------------
its 4th type of cells     (only for ("a" = "x" or "b" = "x") and "c" = "x"):
96N | a3b3o . c    a3b3o . c          a3b3o . c            a3b3o . c          a3b3o . c  :: none
----+----------------------------------------------------------------------------------
its 5th type of cells     (in the right case ever, in the left only for "a" = "x" or "b" = "x"):
12N | a3b3o4o . -> pex-a3b3o4o     -> pabex-a3b3o4o     -> pac-a3b3o4x     -> a3b3o4x .  :: axial

Aside: Some of the above partial Stott sequences (with uniforms only) can be re-written as classical ones too, using a lower symmetry right from the beginning:

• ↑ [1]: o o o  ↔  x o o  ↔  x x o  ↔  x x x
• ↑ [2]: o3o3o  ↔  x3o3o  ↔  x3o3x
• ↑ [3]: o3x3o  ↔  x3x3o  ↔  x3x3x
• ↑ [4]: o o o o  ↔  x o o o  ↔  x x o o  ↔  x x x o  ↔  x x x x
• ↑ [5]: o3o3o *b3o  ↔  x3o3o *b3o  ↔  x3o3x *b3o  ↔  x3o3x *b3x
• ↑ [6]: o3x3o *b3o  ↔  x3x3o *b3o  ↔  x3x3x *b3o  ↔  x3x3x *b3x
• ↑ [7]: o o o o o  ↔  x o o o o  ↔  x x o o o  ↔  x x x o o  ↔  x x x x o  ↔  x x x x x
• ↑ [8]: o∞o o∞o o∞o  ↔  o∞o o∞o x∞o  ↔  o∞o x∞o x∞o  ↔  x∞o x∞o x∞o
• ↑ [9]: x∞o x∞o x∞o  ↔  x∞o x∞o x∞x  ↔  x∞o x∞x x∞x  ↔  x∞x x∞x x∞x
• ↑ [10]: o∞o o∞o o∞o o∞o  ↔  o∞o o∞o o∞o x∞o  ↔  o∞o o∞o x∞o x∞o  ↔  o∞o x∞o x∞o x∞o  ↔  x∞o x∞o x∞o x∞o
• ↑ [11]: x∞o x∞o x∞o x∞o  ↔  x∞o x∞o x∞o x∞x  ↔  x∞o x∞o x∞x x∞x  ↔  x∞o x∞x x∞x x∞x  ↔  x∞x x∞x x∞x x∞x
• ↑ [12]: o3o3o *b3o *b3o  ↔  x3o3o *b3o *b3o  ↔  x3o3x *b3o *b3o  ↔  x3o3x *b3x *b3o  ↔  x3o3x *b3x *b3x
• ↑ [13]: o3x3o *b3o *b3o  ↔  x3x3o *b3o *b3o  ↔  x3x3x *b3o *b3o  ↔  x3x3x *b3x *b3o  ↔  x3x3x *b3x *b3x