Scaliforms

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 v_k vertices of type k, then with v₀ = gcd({v_k}) and integers w_k = v_k/v₀ one obtains that at least ∑_k w_k facets of that type are required at any vertex. In fact at least w_k 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.

purely scaliform polychora (4D) & honeycombs
purely scaliform polytera (5D) & tetracombs
purely scaliform polypeta (6D) & pentacombs
purely scaliform polyexa (7D) & hexacombs
purely scaliform polyzetta (8D) & heptacombs
purely scaliform polyyotta (9D) & octacombs

° - 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.)

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 4D  purely scaliforms (up)
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circumradius	scaliform polychora	facet total
0.615370	otbaquitit	64 tet + 16 tuquith^†2
0.618034	birgax	48 targi^†3
0.618034	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( xo5/2ox5/2oo5/2a&#x by x5/2o5/2o5/2a each )	12 stap + 24 stappy^†2
0.726543	sporraggix	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
	gastakix	3600 tupip^†3
	mesdi	24 gike + 24 ike + 96 dritit^†3
	sistakix	3600 tustip^†3
	tocap	50 squippy^†2 + 10 stip + 50 tet
	gocap	10 pip + 50 squippy^†2 + 50 tet
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
	dexdap	20 doe + 100 ifbah^†3 + 100 tet
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
2.739576	shi - reduced( β3β5o5/2o by 120 β5o5/2o )	120 seside + 1440 stappy^†2
3.077684	spidrox°	120 pap + 120 pip + 600 squippy^†2
3.625437	prishi - reduced( β3β5o5/2x by 120 β5o5/2x )	120 seside + 1440 stiscu^†2 + 2400 trip
scaliform euclidean honeycombs
∞	3Q3-S3-2P6-2P3-gyro°	N hip + N oct + 2N tricu^†2 + 2N trip
	3Q3-S3-2P6-2P3-ortho°	N hip + N oct + 2N tricu^†2 + 2N trip
	3Q4-T-2P8-P4°	N cube + N op + 2N squacu^†2 + 2N tet
	5Y4-3T-3Q3-T3°	12N squippy^†2 + 9N tet + 4N tricu^†2 + N tut
	5Y4-4T-4P4°	N cube + 2N squippy^†2 + 2N tet
	5Y4-4T-6P3-sq-para°	N squippy^†2 + N tet + N trip
	5Y4-4T-6P3-sq-skew°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-0°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-1-alt°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-1-hel°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-2-alt°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-2-hel°	2N squippy^†2 + 2N tet + 2N trip
	5Y4-4T-6P3-tri-3°	2N squippy^†2 + 2N tet + 2N trip
	6Q3-2S3-gyro°	N oct + 2N tricu^†2
	6Q3-2S3-ortho°	N oct + 2N tricu^†2
	6Q4-2T°	2N squacu^†2 + 2N tet
	10Y4-8T-0°	2N squippy^†2 + 2N tet
	10Y4-8T-1-alt°	2N squippy^†2 + 2N tet
	10Y4-8T-1-hel°	2N squippy^†2 + 2N tet
	10Y4-8T-2-alt°	2N squippy^†2 + 2N tet
	10Y4-8T-2-hel°	2N squippy^†2 + 2N tet
	10Y4-8T-3°	2N squippy^†2 + 2N tet
	gyrich° - s∞o2s6o3x	N oct + N tobcu^†2
	squatap° - s2s4o4o	2 squat + 2N squippy^†2 + 2N tet
scaliform hyperbolic honeycombs
...	pristrah°	NM that + NK trat + NMK tricu^†2 + NMK trip
...	s3s4o4o°	NM ike + 3N squat + 3NM squippy^†2
...	s3s4o4x°	NM ike + 3NM squacu^†2 + 3N tosquat + 4NM trip
0.353553 i	s4o3o4s4*b°	2NM hihexat + 6NK squat + 4NMK tet + 4NMK tricu^†2

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 5D  purely scaliforms (up)
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circumradius	scaliform polytera	facet total
0.623054	stadow - xo5/2oo ox5/2oo&#x	10 stasc^†2
0.629640	ogdow - xo8/3oo ox8/3oo&#x	16 ogasc^†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
0.742632	hossidgyt - reduced( xoo5/2oxo5/2oox5/2a&#x by x5/2o5/2o5/2a each )	3 hossdap^†1 + 36 stapepy^†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
1.074481	sishia	120 hossdap^†1 + 2 sishi + 240 sissidpy^†2
1.098185	hodidgyt - reduced( xxo5/2xox5/2oxx5/2a&#x by x5/2x5/2o5/2a each )	3 dastop^†1 + 36 stisaw^†2
1.125151	pirgadia	120 hog dhidicup^†1 + 2 pirgady + 720 stappip + 240 tiggia gaquatid^†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
1.274755	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
1.677756	sutia - xx3ox4xo3xx4/3*a&#x	48 quithaquitco^†2 + 48 sircoatoe^†2 + 2 suti
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
2.352869	ditdia - xx3xo4xx3ox4a4/3c&#x	2 ditdi + 48 goccoa cotco^†2 + 48 toatic^†2
2.527959	sid pippadia - reduced( xx5oo5/2xo5/2ox5/2b&#x, by . o5/2x5/2o5/2b 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
scaliform euclidean tetracombs
∞	capirsit° - s3s3s4o3x	24N ipe + N prissi^†1 + 3N sadi + 96N tricuf^†2 + 32N triddip + 12N tuta^†1
	capshot° - s3s4o3o3x	12N ipe + 3N rit + N sadi + 24N tetaco^†2 + 32N triddip
	capsthat° - s3s4o3x3x	12N ipe + N prissi^†1 + 3N tah + 32N thiddip + 24N tutatoe^†2
	cesratit° - s4o3x3o4s	3N hex + N thex + 8N tuta^†1
	paltite° - s3s4o3x3o	24N octatut^†2 + N prissi^†1 + 3N thex + 32N triddip
scaliform hyperbolic tetracombs
...	rect( o4x4o3a4x4o3a )°	8NKL o3x3x4o4b + 8NP rect( x4o3o4x4b )^†2 + 3NMKLP rit + 8NMKLP tuta^†1
...	trunc( o4x4o3a4x4o3a )°	8NKL o3x3x4x4b + 3NMKLP tat + 8NP trunc( x4o3o4x4b )^†2 + 8NMKLP tuta^†1
0.25 i	o4s4o3a4s4o3a°	3NMKLP hex + 8NKL o3o3o4s4b + 8NP s4o3o4s4b^†1 + 4NMKLP tuta^†1

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 6D  purely scaliforms (up)
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circumradius	scaliform polypeta	facet total
0.707107	gisdiddit	6400 hix + 576 stadow^†1 + 3840 state^†2
0.707107	starpglassdit - oxoo5/2ooox xooo5/2ooxo&#xr	10 bostarpglassdit^†2 + 4 stadow^†1
0.816497	tedjak° - xoo3ooo3oxo *b3oox&#x	24 hexpy^†2 + 3 hin + 24 hix
0.816497	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
	odhax° - xxo xox oxx oox oxo xoo&#zx	8 hix + 12 tedhin^†2 + 8 tedrix^†1
0.925820	ladbril°	7 teddot^†2 + 7 tedrix^†1 + 7 tratet
1	tridafupip° - xo3ox xo3ox xx&#x	12 traffip^†2 + 2 tratrip + 2 tridafup^†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
1.224745	hidbrag° - xo3ox4oo ox3xo4oo&#zx	12 tedrat^†2 + 64 tridafup^†1 + 16 troct
1.290994	medrojak° - oxx3xox xox3xxo xxo3oxx&#zx	162 cubasquasc^†3 + 27 squoct + 54 tedrix^†1 + 54 traffip^†2 + 18 tratrip + 27 triddaf^†1
1.290994	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
1.414214	codibrox° - xxo4ooq xox4oqo oxx4qoo&#zx	12 bodnit^†2 + 48 squatet + 64 tedrix^†1
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
2.263033	big dispox° - xo3ox4xx ox3xo4xx&#zx	12 cydispan^†2 + 144 squatet + 192 traffip^†2 + 16 trasirco + 64 tridafup^†1
2.345208	cograxa° - xx3xo3xx3ox3xx&#x	2 cograx + 30 copeatope^†2 + 20 pabex hix^†1 + 12 pripagrip^†2

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 7D  purely scaliforms (up)
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circumradius	scaliform polyexa	facet total
0.646447	quithdow - xo4/3xo3oo ox4/3ox3oo&#x	12 ogquithdow^†2 + 16 tiquithdow^†2
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
	tedroc°	4 hixip + 8 tidril^†2
0.935414	trahex dafup° - xo3ox xo3oo3ox *d3oo&#x	6 hexaf^†2 + 2 trahex + 8 tratetdafup^†1 + 16 trial tratet^†2
0.935414	tridafupap° - xoxo3oxox xxoo3ooxx&#x	4 triddippa tridafup^†2 + 12 tritra ope^†2
0.968246	tetocta° - oo3ox3xo oo3xo3ox&#x	8 octa tratet^†2 + 2 tetoct + 8 tratet altroct^†2
1	ihdlaq° - xxo xox oxx oox3ooo3oxo *e3xoo&#zx	6 hax + 48 hedjak^†2 + 24 odhax^†1 + 8 tedjak^†1
	sadlaq° - xxxxooo xxoooxx xoxoxox xooxxxo oxxoxxo oxoxxox ooxxoxx&#zx	14 hax + 128 hop + 56 tedjak^†1
	thidlaq° - xo3ox oo3xo3oo3ox3oo&#zx	20 endjak^†1 + 36 rapalpenp^†2 + 6 rixa^†1 + 12 trapen + 30 tratetdafup^†1
	trarapdafup° - xo3ox oo3xo3ox3oo&#x	6 rappaf^†2 + 2 trarap + 10 tratetaltroct^†2
1.040833	titridafup° - xo3ox xo3ox xx3oo&#x	12 titraf^†2 + 3 tridafupip^†1 + 2 trittip>†2
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
	ritgyo° - xoo oxo oox xxo3ooo3xox *e3oxx&#zx	48 hinaco^†2 + 24 odhax^†1 + 8 ritgyt^†1
1.414214	ritgyco° - xxo xox oxx xxo3ooo3xox *e3oxx&#zx	96 hexipaco^†2 + 24 hinnip + 192 pedril^†2 + 6 ritas^†1 + 8 ritgyt^†1 + 24 xedrag^†1
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 + ...
1.936492	shopjaka° - xo3ox3oo3xo3ox *c3oo&#x	2 shopjak + ...
...	(N,M)-dafupap° - xoxoNoxox xxooMooxx&#x	4 (N,M)-dip\|\|(N,M)-dafup^†2 + 2m (N-g,dual N-g,N-p,gyro N-p) lace simplex^†2 + 2n (M-g,dual M-g,M-p,gyro M-p) lace simplex^†2

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 8D  purely scaliforms (up)
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circumradius	scaliform polyzetta	facet total
1	broca° - oo3oo3xo3oo3ox3oo3oo&#x	2 broc + 70 oca + 16 rilalbril^†2
	jakap° - xx xo3oo3oo3oo3ox *d3oo&#x	72 geep + 432 hopip + 2 jaka^†1 + 2 jakip + 54 tacpyp^†2
	kadify° - oo3oo3xo3oo3oo3ox3oo3oo&#zx	630 oca + 1260 octepe^†2 + 72 rila^†1 + 18 roc
	tijakdap° - xo3ox xo3oo3oo3oo3ox *e3oo&#zx	72 idinaq^†1 + 6 jaka^†1 + 1296 squix^†2 + 162 taccasc^†2
	codify° - xoo3ooo3oxo b3oox xoo3ooo3oxo f3oox&#zx	1536 hexete^†2 + 384 oca + 48 odinaq^†1
scaliform euclidean heptacombs
∞	jakoha° - xo3oo3oo3oo3ox *c3oo3oo&#x	N jaka^†1 + 2 jakoh + 2N jakpy^†2 + 72N oca

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 9D  purely scaliforms (up)
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circumradius	scaliform polyyotta	facet total
1.118034	rapdafup° - oo3xo3ox3oo oo3xo3ox3oo&#x	2 rapdip + 20 tetrappal octrap^†2

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Scaliformity

---- 4D purely scaliforms (up) ----

---- 5D purely scaliforms (up) ----

---- 6D purely scaliforms (up) ----

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

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

---- 9D purely scaliforms (up) ----