### Scaliformity

Scaliformity as such was introduced as a concept in 2000, when comparing the dimensionally recursive definition of uniform polytopes

uniform – 1:
Its symmetry group is transitive on the vertices
uniform – 2:
For D>1: its edges are congruent
uniform – 3:
For D>2: its facets are uniform

with one of the then just defined and enlisted segmentochora, with tuta. So, according to that definition, e.g. a uniform polychoron not only should provide a single class of symmetry equivalent vertices and a single edge size (which thus could be taken to be unity), but all its cells ought be uniform polyhedra in turn. And for uniform polyhedra that third requirement then just asks that its polygonal faces have to be uniform in turn. But for those 2D elements the there only remaining first 2 requirements already imply the regularity of the elements (provided a local and thus global non-zero curvature of the surface manifold).

That specific tuta (xo3xx3ox&#x), because being a segmentochoron, clearly has a unique circumsphere. Hence its vertices all are the same distance apart from the center. That is, all vertices would follow the first requirement, provided all vertices additionally can be shown to have the identical global symmetry. Well, as this obviously is the case here for the vertices in either base layer (because those bases are uniform polyhedra), it remains just to be checked whether we also have a further symmetry which interchanges these bases. In that example those bases are identical polyhedra, and therefore requirement 1 truely is fulfilled. Next, the second requirement, already is fulfilled by the definition of segmentochora, i.e. all edges already have unit size. Thus we are left with the third requirement here, i.e. whether all cells would be uniform polyhedra. But this one then is broken here, as the lacing cells of that segmentochoron use Johnson solids as well, in fact there are tricues (xo3xx ..&#x).

The very find of tuta gave rise for a new class of such polytopes, which still follow the first 2 requirements of uniformity, but will not bow to the third one. As a working title such polytopes in those days where called weakly uniform. Five years later, in 2005, this rather negatively attributed term was recoined positively by an own term, scaliform. Further, in this run, it was taken into account also its application to according flat, i.e. euclidean polytopes (like honeycombs etc.). This is why an own third requirement was added instead (which in global non-zero curvatures clearly would be deducible from the former ones):

scaliform – 1:
all vertex flags are transitive
scaliform – 2:
For D>1: all edges are same length
scaliform – 3:
For D>2: all elements are circumscribable

Right from this definition it follows that 2D scaliforms already are regular polygons. Furthermore, the 3D scaliforms already are the uniform polyhedra. But beyond 3D this definition clearly is more liberate. The research for scaliform polytopes is still ongoing. A general classification of these polytopes (outside from the above definition) still is pending.

Only in November 2020 it was proven that the non-uniform scaliform polychora include infinite series too: by the find of the (n/d,n/(n-d))-hemiantiprisms.

Below a mere listing of (just some) of the purely scaliform polytopes will be given, i.e. of those which are scaliform, but not uniform. Accordingly this has to start with 4D for the lowest case. There at least one cell type has to be a Johnson solid – or would belong to some counterpart of that set, encompassing according non-convex polyhedra. (Cf. the symbol below).

But note, this remark already provides an example that for scaliforms the recursivity axiom (as being used for uniformity) is broken: the facets of scaliform polytopes need not be scaliform themself. It is just that the local arrangement of such facets at any vertex provides the overall transitiveness of symmetry on the vertex flags.

Proposition
In fact one further derives the observation, that whenever a non-scaliform polytope will be used as a facet for a scaliform one (within the next dimension) and that very facet itself has vk vertices of type k, then with v0 = gcd({vk}) and integers wk = vk/v0 one obtains that at least k wk facets of that type are required at any vertex. In fact at least wk such facets adjoin the (higher polytopal) vertex by their k-th (own) vertex type – or common multiples therefrom.

E.g. for tuta we have 3 tricues per vertex: one adjoining by its top triangle and 2 adjoining by their bottom hexagon.

° - such marked scaliforms are convex
†n - such marked elements are not themself uniform and thus qualify the (overall) polytope to be just scaliform; here n provides the number of according vertex types. (Clearly, for n=1 that facet itself would be scaliform.)

```----
4D  purely scaliforms (up)
----
```
0.615370
```otbaquitit
```
```64 tet + 16 tuquith†2
```
0.618034
```birgax
```
```48 targi†3
```
```gatodsap
```
```240 tustarp†3
```
```gypasp
```
```240 stappy†2
```
0.707107
```dastop
```
```12 stap + 24 stiscu†2
```
```(n/d,n/(n-d))-hap
```
```2 n/d-ap + 2 n/(n-d)-ap + 2n bobipyr†2
```
```hatho
```
```4 bobipyr†2 + 4 tet
```
```koho
```
```4 bobipyr†2 + 8 tet
```
```setho
```
```4 bobipyr†2 + 12 tet
```
```hossdap  -  reduced( xo5/2ox5/2oo5/2*a&#x by x5/2o5/2o5/2*a each )
```
```12 stap + 24 stappy†2
```
0.726543
```sistakix
```
```3600 tustip†3
```
```sporaggix
```
```600 squippy†2 + 120 starp + 120 stip
```
0.732444
```hog dhidicup
```
```12 stap + 24 rastacu†2
```
0.790569
```hocucup
```
```6 so + 12 tutrip†2
```
1
```disdi
```
```24 gad + 96 scuffi†3 + 96 scufgi†3 + 24 sissid
```
```mesdi
```
```24 gike + 24 ike + 96 dritit†3
```
1.224745
```tuta°  -  xo3xx3ox&#x
```
```6 tet + 8 tricu†2 + 2 tut
```
1.328131
```prarsi  -  s3/2s4o3x
```
```24 gike + 96 tricu†2 + 96 trip + 24 tut
```
1.618034
```bidex°
```
```48 teddi†3
```
```spysp
```
```240 peppy†2
```
```stodsap
```
```240 tupap†3
```
1.765796
```siida  -  xx3xo5/2ox3*a &#x
```
```2 siid + 12 stap + 40 tricu†2
```
2.149726
```otbott
```
```64 tet + 16 tutic†2
```
2.497212
```prissi°  -  s3s4o3x
```
```24 ike + 96 tricu†2 + 96 trip + 24 tut
```
3.077684
```gastakix
```
```3600 tupip†3
```
```spidrox°
```
```120 pap + 120 pip + 600 squippy†2
```

```----
5D  purely scaliforms (up)
----
```
0.623054
```stadow  -  xo5/2oo ox5/2oo&#x
```
```10 stasc†2
```
0.674163
```shadow  -  xo7/2oo ox7/2oo&#x
```
```14 shasc†2
```
0.680827
```gashia  -  xo5/2oo5oo5/2ox&#x
```
```2 gashi + 240 sissidpy†2 + 1440 stasc†2
```
0.790569
```triddaf°  -  xo ox xo3ox&#x
```
```6 squasc†2 + 4 traf†2
```
0.816497
```tedrix°  -  xxo xox oxx&#x
```
```6 bidrap†2 + 3 tepe
```
0.866025
```tridafup°  -  xo3ox xo3ox&#x
```
```12 traf†2 + 2 triddip
```
0.895420
```squiddaf°  -  xo ox xo4ox&#x
```
```4 squaf†2 + 8 squasc†2
```
1.050501
```icoap°  -  xo3oo4oo3ox&#x
```
```2 ico + 48 octpy†2 + 192 pen
```
1.074481
```sitpodadia  -  oo5/2oo3xo5/2ox3*b &#x
```
```240 gikepy†2 + 120 sidtidap + 2 sitpodady
```
```sishiap
```
```120 hossdap†1 + 2 sishi + 240 sissidpy†2
```
1.190238
```pabex hix°  -  xx3xo xx3ox&#x
```
```2 thiddip + 6 tricuf†2 + 6 tricupe†2
```
1.224745
```sripa°  -  ox3xo3ox3xo&#x
```
```10 octaco†2 + 2 srip + 20 traf†2
```
1.248606
```ragashia  -  oo5/2xo5ox5/2oo&#x
```
```240 gaddadid†2 + 2 ragashi
```
1.274755
```pexhin°  -  xo3xx3ox xo ox&#zx
```
```6 hex + 12 tepe + 16 tricuf†2 + 4 tuta†1
```
```rita°  -  xo3oo3ox *b3xx&#x
```
```8 hex + 2 rit + 24 tepe + 16 tetaco†2
```
1.322876
```tutcupip°  -  xx xo3xx3ox&#x
```
```6 tepe + 8 tricupe†2 + 2 tuta†1 + 2 tuttip
```
1.618034
```sirgashia  -  xo5/2ox5xo5/2ox&#x
```
```240 dida raded†2 + 2 sirgashi + 1440 stafe†2
```
1.620185
```thexa°  -  xo3xx3ox *b3oo&#x
```
```16 octatut†2 + 2 thex + 8 tuta†1
```
1.658312
```pripa°  -  xx3ox3xo3xx&#x
```
```10 coatut†2 + 2 prip + 20 tricupe†2
```
1.778824
```ricoa°  -  oo3xo4ox3oo&#x
```
```48 cubaco†2 + 2 rico
```
1.870829
```gripa°  -  ox3xx3xx3xo&#x
```
```2 grip + 20 tricuf†2 + 10 tutatoe†2
```
1.917564
```righia  -  oo5xo5/2ox5oo&#x
```
```2 righi + 240 sissidadid†2
```
2.150581
```taha°  -  xo3xx3ox *b3xx&#x
```
```2 tah + 24 tepe + 16 tutatoe†2 + 8 tuta†1
```
2.527959
```sidpippadiap°  -  reduced( xx5oo5/2xo5/2ox5/2*b&#x by . o5/2x5/2o5/2*b each )
```
```240 gaddaraded†2 + 120 hossdap†1 + 2 sid pippady + 720 stappip
```
2.632865
```sricoa°  -  xo3ox4xo3ox&#x
```
```48 coasirco†2 + 2 srico + 192 traf†2
```
2.829949
```rasishia
```
```120 dastop†1 + 240 didadoe†2 + 2 rasishi
```
3.404434
```sirghia  -  xo5ox5/2xo5ox&#x
```
```240 dida raded†2 + 1440 paf†2 + 2 sirghi
```
3.522336
```pricoa°  -  xx3xo4ox3xx&#x
```
```2 prico + 48 sircoatoe†2 + 192 tricupe†2
```
3.855219
```stut phiddixa  -  xx3oo3xo5/2ox3*b &#x
```
```120 sidtidap + 720 stappip + 2 stut phiddix + 1200 tetaco†2
```
4.311477
```gricoa°  -  xo3xx4xx3ox&#x
```
```2 grico + 48 ticagirco†2 + 192 tricuf†2
```
4.923348
```pirghia  -  xx5xo5/2ox5xx&#x
```
```1440 pecupe†2 + 2 pirghi + 240 radeda tigid†2
```
5.345177
```wavhiddixa  -  oo3xx3xo5/2ox3*b &#x
```
```1200 octatut†2 + 120 siida†1 + 2 wavhiddix
```
6.881910
```sphiddixa  -  xx3xx3xo5/2ox3*b &#x
```
```120 siida†1 + 2 sphiddix + 720 stappip + 1200 tutatoe†2
```
...
```n/d-daf  -  xo ox xo-n/d-ox&#x  (° for d=1)
```
```4 n/d-af†2 + 2n squasc†2
```
...
```n,m-dafup°  -  xo-n-ox xo-m-ox&#x
```
```2m n-af†2 + 2 n,m-dip + 2n m-af†2
```
...
```n/d-dow  -  xo-n/d-oo ox-n/d-oo&#x  (° for d=1)
```
```2n n/d-sc†2
```
...
```n/d,2n/d-dipcup  -  xx-n/d-xo xx-n/d-ox&#x  (° for d=1)
```
```2n n/d-cuf†2 + 2n n/d-cupe†2 + 2 n/d,2n/d-dip
```

```----
6D  purely scaliforms (up)
----
```
0.816497
```tedjak°  -  xoo3ooo3oxo *b3oox&#x
```
```24 hexpy†2 + 3 hin + 24 hix
```
```endjak°  -  xo3ox xo3oo ox3oo&#zx
```
```6 hix + 18 squete†2 + 9 triddaf†1
```
0.866025
```rixa°  -  oo3xo3oo3ox3oo&#x
```
```12 dihin†2 + 20 hix + 2 rix
```
```tratetdafup°  -  xo3ox xo3oo3ox&#x
```
```6 tetaf†2 + 2 tratet + 8 trial triddip†2 + 6 triddaf†1
```
1
```oddimo°  -  xo3ox xo3ox xo3ox&#zx
```
```18 tridafup†1 + 54 triddaf†1
```
```xedrag°  -  xxo4ooo xox4ooo oxx4ooo&#zx
```
```12 hexip + 64 tedrix†1
```
1.224745
```spixa°  -  xo3ox3oo3xo3ox&#x
```
```12 rapaspid†2 + 2 spix + 30 teta ope†2 + 20 tridafup†1
```
1.290994
```medrojak°  -  oxx3xox xox3xxo xxo3oxx&#zx
```
```162 cubasquasc†3 + 27 squoct + 54 tedrix†1 + 54 traffip†2 + 18 tratrip + 27 triddaf†1
```
```ritgyt°  -  xxo3ooo3xox *b3oxx&#x
```
```24 hexaco†2 + 3 rita†1 + 24 tedrix†1
```
1.322876
```pexhax°  -  xo3xx3ox xo3oo3ox&#zx
```
```12 hexip + 6 hin + 6 pexhin†1 + 32 pexhix†2 + 8 tratet
```
```bitettut°  -  xx3ox3oo xx3xo3oo&#zx
```
```8 hatet + 16 hix + 16 pabexhix†1 + 32 pexhix†2 + 8 tratet
```
```siphina°  -  xo3oo3ox *b3oo3xx&#x
```
```40 hexip + 10 hin + 32 penaspid†2 + 2 siphin + 80 tratet
```
```ritas°  -  xoxo3oooo3oxox *b3xxxx&#xr
```
```24 hexip + 8 hin + 4 rita†1 + 32 tepaco†2 + 32 tratet
```
1.581139
```bittixa°  -  oo3xo3xx3ox3oo&#x
```
```2 bittix + 20 hix + 12 tipadeca†2
```
1.632993
```thexgyt°  -  xoo3xxx3oxo *b3oox&#x
```
```24 hix + 24 octa tutcup†2 + 3 thexa†1
```
1.658312
```cappixa°  -  xx3xo3oo3ox3xx&#x
```
```2 cappix + 20 pabex hix†1 + 12 spidatip†2 + 30 tepeatuttip†2
```
```sirhina°  -  xo3oo3ox *b3xx3oo&#x
```
```32  rapalsrip†2 + 10  rita†1 + 2  sirhin + 80  tratet
```
```pabex hax°  -  xo3xx3ox xo3xx3ox&#zx
```
```32 pabex hix†1 + 16 tratet + 12 pexhin†1 + 24 tutcupip†1
```
1.870829
```bicotoe°  -  xo3xx4oo ox3xx4oo&#zx
```
```16 haco + 64 pabex hix†1 + 36 pent + 12 squaco + 96 squatricu†2
```
1.978437
```redscox°  -  xxo4xxx xox4xxx oxx4xxx&#zx
```
```24 sidpithip + 192 squasquippy†2 + 64 tedrix†1 + 192 tepacube†2 + 64 tracube
```
2.160247
```tahgyt°  -  xxo3xxx3xox *b3oxx&#x
```
```3 taha†1 + 24 tedrix†1 + 24 tutcupa toe†2
```
2.345208
```cograxa°  -  xx3xo3xx3ox3xx&#x
```
```2 cograx + 30 copeatope†2 + 20 pabex hix†1 + 12 pripagrip†2
```

```----
7D  purely scaliforms (up)
----
```
0.831254
```sissiddow  -  xo5/2oo5oo ox5/2oo5oo&#x
```
```24 stasissiddow†2
```
0.866025
```rila°  -  oo3xo3oo3oo3ox3oo&#x
```
```14 hixalrix†2 + 70 octete†2 + 2 ril
```
```jaka°  -  xo3oo3oo3oo3ox *c3oo&#x
```
```72 gee + 432 hop + 2 jak + 54 tacpy†2
```
```trapendafup°  -  xo3ox xo3oo3oo3ox&#x
```
```6 penaf†2 + 2 trapen + 10 trial tratet†2 + 20 tripal triddip†2
```
```idinaq°  -  xo3ox xo3oo3oo3oo3ox&#zx
```
```20 endjak†1 + 6 gee + 36 hop + 90 squepe†2
```
```odinaq°  -  xoo oxo oox oxo3ooo3oox *e3xoo&#zx
```
```24 gee + 96 hexasc†2 + 192 hop + 8 tedjak†1
```
1.224745
```scala°  -  xo3ox3oo3oo3xo3ox&#x
```
```42 pena rappip†2 + 14 rixascad†2 + 2 scal + 70 tratet altroct†2
```
1.322876
```spila°  -  xo3oo3ox3xo3oo3ox&#x
```
```14 dottaspix†2 + 42 spida rappip†2 + 2 spil + 70 tetal tratet†2
```
```hidlin°  -  xo3oo3ox3oo3xo3oo3ox&#zx
```
```16 bril + 56 pabdimo†2 + 56 rixa†1 + 70 hax
```
```rojaka°  -  oo3xo3oo3ox3oo *c3oo&#x
```
```54 hinro†2 + 72 rixa†1 + 2 rojak
```
1.581139
```sabrila°  -  oo3xo3ox3xo3ox3oo&#x
```
```2 sabril + ...
```
1.658312
```hejaka°  -  xo3oo3oo3oo3ox *c3xx&#x
```
```72 gee + 2 hejak + 54 hinasiphin†2 + 720 hixip + 432 spidapenp†2
```
1.936492
```crala°  -  xo3ox3xo3ox3xo3ox&#x
```
```2 cral + ...
```
```shopjaka°  -  xo3ox3oo3xo3ox *c3oo&#x
```
```2 shopjak + ...
```

```----
8D  purely scaliforms (up)
----
```
1
```broca°  -  oo3oo3xo3oo3ox3oo3oo&#x
```
```2 broc + 70 oca + 16 rilalbril†2
```
```kadify°  -  oo3oo3xo3oo3oo3ox3oo3oo&#zx
```
```630 oca + 1260 octepe†2 + 72 rila†1 + 18 roc
```
```codify°  -  xoo3ooo3oxo *b3oox xoo3ooo3oxo *f3oox&#zx
```
```1536 hexete†2 + 384 oca + 48 odinaq†1
```