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Scaliformity as such was introduced as a concept in 2000, when comparing the dimensionally recursive definition of uniform polytopes
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):
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.
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) ----
circumradius | scaliform polychora | facet 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) ----
circumradius | scaliform polytera | facet 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) ----
circumradius | scaliform polypeta | facet 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) ----
circumradius | scaliform polyexa | facet 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) ----
circumradius | scaliform polyzetta | facet 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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