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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 tutcup. 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 tutcup (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 tutcup 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.



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 tutcup 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) ----

circumradiusscaliform polychorafacet total
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
hatho
4 bobipyr†2 + 4 tet
koho
4 bobipyr†2 + 8 tet
setho
4 bobipyr†2 + 12 tet
0.726543
sistakix
3600 tustip†3
sporaggix
600 squippy†2 + 120 starp + 120 stip
0.754386
hossdap
12 stap + 24 stappy†2
1
disdi
24 gad + 96 scuffi†3 + 96 scufgi†3 + 24 sissid
mesdi
24 gike + 24 ike + 96 sipstar†3
1.224745
tutcup°  -  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
siidcup
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) ----

circumradiusscaliform polyterafacet total
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
1.190238
pabex hix°  -  xx3xo xx3ox&#x
6 tricuf†2 + 6 tricupe†2 + 2 thiddip
1.224745
pabdinit°  -  ox3xo3ox3xo&#x
10 octaco†2 + 2 srip + 20 traf†2
1.274755
ritag rit°  -  xo3oo3ox *b3xx&#x
8 hex + 2 rit + 24 tepe + 16 tetaco†2
1.620185
thexag thex°  -  xo3xx3ox *b3oo&#x
16 octatut†2 + 2 thex + 8 tutcup†1
1.778824
ricoalrico°  -  oo3xo4ox3oo&#x
2 rico + 48 cubaco†2
2.150581
tahagtah°  -  xo3xx3ox *b3xx&#x
2 tah + 24 tepe + 16 tutatoe†2 + 8 tutcup†1
2.632865
sricoalsrico°  -  xo3ox4xo3ox&#x
48 coasirco†2 + 2 srico + 192 traf†2
3.522336
pricoalprico°  -  xx3xo4ox3xx&#x
2 prico + 48 sircoatoe†2 + 192 tricupe†2
4.311477
gricoalgrico°  -  xo3xx4xx3ox&#x
2 grico + 48 ticagirco†2 + 192 tricuf†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,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) ----

circumradiusscaliform polypetafacet total
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
pabdihax°  -  oo3xo3oo3ox3oo&#x
12 dihin†2 + 20 hix + 2 rix
1
oddimo°  -  xo3ox xo3ox xo3ox&#zx
18 tridafup†1 + 54 triddaf†1
1.290994
ritgyt°  -  xxo3ooo3xox *b3oxx&#x
24 coahex†2 + 3 ritag rit†1 + 24 tedrix†1
1.581139
bittixalbittix°  -  oo3xo3xx3ox3oo&#x
2 bittix + 20 hix + 12 tipadeca†2
1.632993
thexgyt°  -  xoo3xxx3oxo *b3oox&#x
24 hix + 24 octa tutcup†2 + 3 thexag thex†1
1.658312
cappixalcappix°  -  xx3xo3oo3ox3xx&#x
2 cappix + 20 pabex hix†1 + 12 spidatip†2 + 30 tepeatuttip†2
2.160247
tahgyt°  -  xxo3xxx3xox *b3oxx&#x
3 tahagtah†1 + 24 tedrix†1 + 24 toa tutcup†2
2.345208
cograxalcograx°  -  xx3xo3xx3ox3xx&#x
2 cograx + 30 copeatope†2 + 20 pabex hix†1 + 12 pripagrip†2


---- 7D purely scaliforms (up) ----

circumradiusscaliform polyexafacet total
0.866025
rilalril°  -  oo3xo3oo3oo3ox3oo&#x
14 hixalrix†2 + 70 octete†2 + 2 ril
jakaljak°  -  xo3oo3oo3oo3ox *c3oo&#x
72 gee + 432 hop + 2 jak + 54 tacpy†2
1.224745
scalal scal°  -  xo3ox3oo3oo3xo3ox&#x
2 scal + ...
1.322876
spilalspil°  -  xo3oo3ox3xo3oo3ox&#x
2 spil + ...
rojakalrojak°  -  oo3xo3oo3ox3oo *c3oo&#x
54 hinro†2 + 72 pabdihax†1 + 2 rojak
1.581139
sabrilal sabril°  -  oo3xo3ox3xo3ox3oo&#x
2 sabril + ...
1.936492
cralalcral°  -  xo3ox3xo3ox3xo3ox&#x
2 cral + ...
shopjakal shopjak°  -  xo3ox3oo3xo3ox *c3oo&#x
2 shopjak + ...


---- 8D purely scaliforms (up) ----

circumradiusscaliform polyzettafacet total
1
brocalbroc°  -  oo3oo3xo3oo3ox3oo3oo&#x
2 broc + 70 oca + 16 rilalbril†2
kadify°  -  oo3oo3xo3oo3oo3ox3oo3oo&#zx
630 oca + 1260 octepe†2 + 72 rilalril†1 + 18 roc


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