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In 1966 Norman Johnson defined and enumerated a new set of polyhedra, nowadays bearing his name: they are bound to be convex, built from regular polygons, but not being uniform.Here they are grouped into sets according to the types of facets they use.
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†) The solids marked by this sign are (external)blends from easier elemental components (i.e. they can be sliced into those). Often this can be read from its full (descriptive) name too. Those elemental components used would be:tet,squippy,peppy;oct,squap,pap,hap,oap,dap;trip,cube,pip,hip,op,dip;doe,tut,tic,tid;tricu,squacu,pecu;pero,teddi,waco.The explicit blend addition will be detailed within the context ofcomplexes.
°) The solids marked by this sign areorbiform, that is, have a unique circumradius.In fact, those generally are diminishings of uniform polyhedra and/or just have one or more caps rotated.Those solids (in addition to the uniforms) would be valid bases forsegmentochora.
There are only 8 Johnson solids, which bow to neither of these descriptions:bilbiro,dawci,hawmco,snadow,snisquap,thawro,waco,wamco. Those kind of are the true findings of this set. Sometimes these therefore get refered to ascrown jewels.
| Facets being {3} only | Facets being {3} and {4} | Facets being {3}, {4}, and {5} |
J12 - †)tridpy - trigonal dipyramidJ13 - †)pedpy - pentagonal dipyramidJ17 - †)gyesqidpy - gyroelongated square dipyramidJ51 - †)tautip - triaugmented trigonal prismJ84 -snadow - snub disphenoid | J1 - °)squippy - square pyramidJ7 - †)etripy - elongated trigonal pyramidJ8 - †)esquipy - elongated square pyramidJ10 - †)gyesp - gyroelongated square pyramidJ14 - †)etidpy - elongated trigonal dipyramidJ15 - †)esquidpy - elongated square dipyramidJ16 - †)epedpy - elongated pentagonal dipyramidJ26 - †)gybef - gyrobifastegiumJ27 - †°)tobcu - triangular orthobicupolaJ28 - †)squobcu - square orthobicupolaJ29 - †)squigybcu - square gyrobicupolaJ35 - †)etobcu - elongated triangular orthobicupolaJ36 - †)etigybcu - elongated triangular gyrobicupolaJ37 - †°)esquigybcu - elongated square gyrobicupolaJ44 - †)gyetibcu - gyroelongated triangular bicupolaJ45 - †)gyesquibcu - gyroelongated square bicupolaJ49 - †)autip - augmented triangular prismJ50 - †)bautip - biaugmented triangular prismJ85 -snisquap - snub square antiprismJ86 -waco - sphenocoronaJ87 - †)auwaco - augmented sphenocoronaJ88 -wamco - sphenomegacoronaJ89 -hawmco - hebesphenomegacoronaJ90 -dawci - disphenocingulum | J9 - †)epeppy - elongated pentagonal pyramidJ30 - †)pobcu - pentagonal orthobicupolaJ31 - †)pegybcu - pentagonal gyrobicupolaJ32 - †)pocuro - pentagonal orthocupolarotundaJ33 - †)pegycuro - pentagonal gyrocupolarotundaJ38 - †)epobcu - elongated pentagonal orthobicupolaJ39 - †)epigybcu - elongated pentagonal gyrobicupolaJ40 - †)epocuro - elongated pentagonal orthocupolarotundaJ41 - †)epgycuro - elongated pentagonal gyrocupolarotundaJ42 - †)epobro - elongated pentagonal orthobirotundaJ43 - †)epgybro - elongated pentagonal gyrobirotundaJ46 - †)gyepibcu - gyroelongated pentagonal bicupolaJ47 - †)gyepcuro - gyroelongated pentagonal cupolarotundaJ52 - †)aupip - augmented pentagonal prismJ53 - †)baupip - biaugmented pentagonal prismJ72 - °)gyrid - gyrated rhombicosidodecahedronJ73 - °)pabgyrid - parabigyrated rhombicosidodecahedronJ74 - °)mabgyrid - metabigyrated rhombicosidodecahedronJ75 - °)tagyrid - trigyrated rhombicosidodecahedronJ91 -bilbiro - bilunabirotunda |
| Facets being {3}, {4}, {5}, and {6} | Facets being {3}, {4}, {5}, and {10} | Facets being {3}, {4}, and {6} |
J92 -thawro - triangular hebesphenorotunda | J5 - °)pecu - pentagonal cupolaJ20 - †)epcu - elongated pentagonal cupolaJ21 - †)epro - elongated pentagonal rotundaJ24 - †)gyepcu - gyroelongated pentagonal cupolaJ68 - †)autid - augmented truncated dodecahedronJ69 - †)pabautid - parabiaugmented truncated dodecahedronJ70 - †)mabautid - metabiaugmented truncated dodecahedronJ71 - †)tautid - triaugmented truncated dodecahedronJ76 - °)dirid - diminished rhombicosidodecahedronJ77 - °)pagydrid - paragyrate diminished rhombicosidodecahedronJ78 - °)magydrid - metagyrate diminished rhombicosidodecahedronJ79 - °)bagydrid - bigyrate diminished rhombicosidodecahedronJ80 - °)pabidrid - parabidiminished rhombicosidodecahedronJ81 - °)mabidrid - metabidiminished rhombicosidodecahedronJ82 - °)gybadrid - gyrated bidiminished rhombicosidodecahedronJ83 - °)tedrid - tridiminished rhombicosidodecahedron | J3 - °)tricu - triangular cupolaJ18 - †)etcu - elongated triangular cupolaJ22 - †)gyetcu - gyroelongated triangular cupolaJ54 - †)auhip - augmented hexagonal prismJ55 - †)pabauhip - parabiaugmented hexagonal prismJ56 - †)mabauhip - metabiaugmented hexagonal prismJ57 - †)tauhip - triaugmented hexagonal prismJ65 - †)autut - augmented truncated tetrahedron |
| Facets being {3}, {4}, and {8} | Facets being {3} and {5} | Facets being {3}, {5}, and {10} |
J4 - °)squacu - square cupolaJ19 - †°)escu - elongated square cupolaJ23 - †)gyescu - gyroelongated square cupolaJ66 - †)autic - augmented truncated cubeJ67 - †)bautic - biaugmented truncated cube | J2 - °)peppy - pentagonal pyramidJ11 - †°)gyepip - gyroelongated pentagonal pyramidJ34 - †°)pobro - pentagonal orthobirotundaJ48 - †)gyepabro - gyroelongated pentagonal birotundaJ58 - †)aud - augmented dodecahedronJ59 - †)pabaud - parabiaugmented dodecahedronJ60 - †)mabaud - metabiaugmented dodecahedronJ61 - †)taud - triaugmented dodecahedronJ62 - °)mibdi - metabidiminished icosahedronJ63 - °)teddi - tridiminished icosahedronJ64 - †)auteddi - augmented tridiminished icosahedron | J6 - °)pero - pentagonal rotundaJ25 - †)gyepro - gyroelongated pentagonal rotunda |
Further reading: since, the restriction of strict convexity was released. V. Zalgaller (and his students) enlisted the set ofconvex regular-faced polyhedra withconditional edges. And the corresponding list then is providedhere.
In 1979 Roswitha Blind applied this idea to higher dimensional polytopes: those are bound to be convex, built from regular facet-polytopes, but not being uniform themselves.Here they are grouped into sets according to the types of facets they use.
*) The ones marked by this sign do exist for any higher dimension as well.
†) These, just as for the 3d case, are obtained as blends of more elemental hypersolids.
°) These, just as for the 3d analog, are orbiform.
| Facets beingtet only (cf.Tetrahedrochora) | Facets beingtet andoct | Facets beingtet andike |
tete = pt||tet|| pt - *†) tetrahedral bipyramidite = pt||ike|| pt - †) icosahedral bipyramid | octpy = pt||oct - *°) octahedral pyramidaurap = pt||oct||tet - †) augmented rectified pentachoron | ikepy = pt||ike - °) icosahedral pyramidand:millions more or less asymmetric diminishings ofex like e.g.sadi andidimex. |
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It might be noted that all these Blind polytopes of 4D and above turn out to have triangles for faces only. Thence all these can clearly beambified and the result still would come out to have a single edge size throughout (which thus can be re-addressed to be unity again).In fact the occuring (faces of) vertex figures either are flat themselves or those of theirblended components are such, thereby dissecting the new vertex figure facet into two facets instead.However even the joining pseudo polytopes here would had been using triangles only, thence the given claim follows.
Nevertheless fortete and all its dimensional analogs (includingtridpy – and similarily forpedpy too) it comes out that the result won't be convex anymore: pairs of medial section based pyramids will include there a concavedihedral. While forite that one turns out to be convex as well. Similarily for theambification ofaurap the dihedral at its joining squares would turn out to be concave.Still, just as the non-ambified ones too, those of bipyramids and that augmentation are nothing but externalblends of more elemental polytopes. For the remaining ones and these components we get:
rect( ox3oo3oo&#x (tetpy = pen) ) = xo3ox3oo&#x (tetaoct = rap)rect( ox3oo4oo&#x (octpy) ) = xo3ox4oo&#x (octaco)rect( ox3oo5oo&#x (ikepy) ) = xo3ox5oo&#x (ikaid)rect( xo3ox3oo&#x (rap) ) = oxx3xxo3oox&#xt (srip)rect(sadi ) =risadirect(idimex ) =ridimex...rect( ox3oo3oo3oo&#x (penpy = hix) ) = xo3ox3oo3oo&#x (penarap = rix)rect( ox3oo3oo4oo&#x (hexpy) ) = xo3ox3oo4oo&#x (hexaico)...
Quite similarily a meretruncation would have worked out as well. The same considerations then apply here.
trunc( ox3oo3oo&#x (tetpy = pen) ) = xux3oox3ooo&#xt (tettum = tip)trunc( ox3oo4oo&#x (octpy) ) = xux3oox4ooo&#xt (octum)trunc( ox3oo5oo&#x (ikepy) ) = xux3oox5ooo&#xt (iktum)trunc( xo3ox3oo&#x (rap) ) = xuxx3xxux3ooox&#xt (grip)trunc(sadi ) =tisaditrunc(idimex ) =tidimex...trunc( ox3oo3oo3oo&#x (penpy = hix) ) = xux3oox3ooo3ooo&#x (hixtum = tix)trunc( ox3oo3oo4oo&#x (hexpy) ) = xux3oox3ooo4ooo&#x (hextum)...
There is also a different possibility toextrapolate from the set of Johnson solids into higher dimensions.Instead of requiring (n-1)-dimensionalfacets to be regular, one rather could stick to 2-dimensionalfaces being regular only.That type of research meanwhile is known asCRF (convex ®ularfaced).
Of course, the set of Blind polytopes would be contained within this much broader set.Note that the CRF not only encompasses all the (convex) uniform polytopes, but even all the (convex)scaliforms would belong to that class.(In fact, relaxing within the definition of scaliforms the requirement for vertex-transitivity: this is what those CRFs truely are.)
On the other hand the set of (convex)orbiform polytopes would be enclosed, which thereby describe only those CRF which have their vertices on the hypersphere.(But note, even in 3D there exist Johnson solids which do not bow to this restriction.) –Those remarks should show that it would be rather unwildy to start an exhaustive research of the class of CRF, even for 4D.
None the less, asubset of the above was already listed in 2000 by Klitzing: the set of convex monostratic orbiform polychora,from then on also known under the name ofconvex segmetochora.
Further examples of 4D CRF polytopes and some5D ones will follow below. Inhigher dimensional spaces even less is known. Cf. eg.
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(unsorted collection only –beyond those already contained within other classes, esp. likeWythoffians,axial polytopes (esp. named types of), orsegmentochora.Segmentochoral stacks are mainly suppressed here either.)
(Several CRFs, both axial ones and of higher symmetry, were found to be constructable in a quite specific way as initiated in 2014 by W. Gefaert.Those, now being so-calledexpanded kaleido-facetings, are described in more detail on a own page. And thus just some of those are contained below too – then mainly for historical reasons.)
A4D CRF list (downloadable spreadsheet) is also available. Additionally it complements the below provided listing of 4D caseswith all the here excluded ones. Moreover it provides individual cell counts each.
| CRFs | Remarks |
|---|---|
| • non-orbiform monostratics | (top of CRF) |
n-py || inv gyro n-py (2<n<6) = externalblend of 2n-appiesn-cu || inv gyro n-cu (6/5<n<6) = externalblend of 2n-pufsline || bilbiro{3} || thawro{5} || pocuron-py || inv ortho n-py (2<n<6) = externalblend of 2n-pipiesn-cu || inv ortho n-cu (6/5<n<6) = externalblend of 2n-pufs{4} || squobcu = externalblend of 2squacufsxfox oxfo3ooox&#xt (3-mibdi-wedge)xfoxo oxfox5ooofx&#xt (5-mibdi-wedge)mono-augmented 3-mibdi-wedgebi-augmented 3-mibdi-wedgexofox ooxfo3oxoox&#xt (tri-augmented 3-mibdi-wedge) | In 2012 Quickfur came up with 2 non-orbiformmonostratic families, described in more detailhere, esp. their close relationship to segmentochora. In 2014 three close relatives to pyramids have been found, which are not orbiform just because their base is not. –Those had been found as one of the first applications of Gevaert'sexpanded kaleido-facetings construction device. In 2015 J. Roth and the author still found several more.– Thereby it also turned out too as an aside that the externalblend of then-cuf andthen-pufgenerally would provide corealmic lacings, i.e. the 2tripswould join into a prism on a rhombic base, and thesquippies unite with the adjoiningtets into sheered digonal cupolae with rhombic lacings. Therefore these blends wouldnot qualify as further CRFs In 2016 Quickfur succeded to complete an idea of the author, using 3mibdies around an edge. The author in the sequel found a 5fold counterpart, which then asks for a further layer. (A 4fold version does not exist.) –Shortly thereafter the author found that the 3fold case could moreover be independently augmented at eithermibdiby an according pyramid, without losing the CRF property. The corresponding triaugmentationthen is e.g.xofox ooxfo3oxoox&#xt. Note that this triaugmentation then incorporates a very special edge, having 6 incidentpeppies. |
| • axials | (top of CRF) |
ooo3ooo4oxo&#xt | Althoughcute clearly is a mere stack ofsegmentochora, by itself it is an interesting shape. E.g.ico can bedecomposed into 8 such building blocks. And, because of having a tip-to-tip distance of exactly one edge unit, it can be decomposed itself in turn into6squascs. |
ofx3xoo3ooo&#xt = tetuofx3xoo4ooo&#xt = octuofx3xoo5ooo&#xt = iku = icosiena-diminishedrotunda of exofx3xoo xxx&#xt = teddipeofx3xoo3xxx&#xt = coatutuofx3xoo4xxx&#xt = sirco aticuofx3xoo5xxx&#xt = sridatiduofxo3xooo3oooo&#xtofxo3xooo4oooo&#xtofxo3xooo5oooo&#xt = icosi-diminishedrotunda of exofxo3xooo xxxx&#xtofxo3xooo3xxxx&#xtofxo3xooo4xxxx&#xtofxo3xooo5xxxx&#xtxofo3ooox3oxoo&#xtxoBo3ofox3xooo&#xtxofo3ooox5oxoo&#xt = mono-diminishedrotunda of exxxFVF(Vx)fox-3-ofxxf(oF)xxx-3-xoooo(xo)xfo-&#xt42-diminished sidpixhi42-diminished ex | W. Krieger found a family of genuinely bistratic figures, which do belong there:ofx3xooPzzz&#xt (where P can variously be 2, 3, 4, or 5, andz is eithero orx). In fact those all are extrapolations from the pentagon (2D:ofx&#xt), via theteddi (3D:ofx3xoo&#xt), into 4D. Because teddi itself was nicknamedteddy sometimes, this small family of teddi-polychora (and their higher dimensional analogues) winkingly was attributed the name ofursachora (resp. ursatopes). – The existence of the infinite series of non-expanded members with simplicial symmetry even was predated by A. Weimholt in 2004. He then already pointed out theorbiformity of any of this subset. There is a separate section onursatopes wrt. general dimensions too. Similar toteddi itself (yieldingauteddi), the ursachora can be augmented at their smaller base too with an attached pyramid.Same holds true then for the expanded versions each, then augmenting with cupolae for sure. And the prism ofauteddi itself exists as well. Liketeddi can be augmented at its pentagons, resulting there back in theike,the tetrahedral, octahedral, and icosahedral ursachoron can be augmented at itsteddies as well.Even though, the tetrahedral can be augmented at any teddi at the same time, the octahedral just at non-neighbouring teddies in order to remain convex.For the icosahedral the same would hold true at the first sight, but the cavities at thepeppy-join then could bebridged there by additional5-tet-rosettes. In 2016 Quickfur and the author came up with acellwise expanded version of the tetrahedral ursachoron, which thenwas the first non-trivial CRF containingtedrids. – An axially octahedral symmetric relative does not exist because there is no 4fold relative ofteddi.But then again an axially icosahedral symmetric version can be considered, assuming the 5fold relative ofteddi to bedoe. So, starting withsrid and 20tedridsone observes that this one comes back to be a 21-diminishing ofsidpixhi. It then even will bepossible to apply that to both hemiglomes at the same time, resulting in an according 42-diminishing. – In contrast tosidpixhi its contracted version,excan be split into 2 rotundae in a CRF way. This is howofx3xoo5ooo&#xt can bereconsidered a 21-diminishing of theex rotunda. |
elongated ico | A special case of augmentation occurs when glueing 2oct-first rotundae ofico at both sides of acope. Clearly a tristratic polychoron. The peculiar clue here is that thesquippies becomeco-realmic to thecubes, thereby blending intoesquidpies (J15)! |
xux3oox3ooo&#xt = tip (uniform)xux3oox3xxx&#xt = coatotum = bistraticco-cap ofprip (tut-diminishedprip)xux3oox4ooo&#xt = oox3xux3oox&#xt = octum =oct-firstrotunda ofthexxux3oox4xxx&#xt = sircoa gircotum = bistraticsirco-cap ofpritoxofo3oooox5ooxoo&#xt = vertex-firstrotunda ofexooo3xox5ofx&#xt = bistraticid-cap ofrahioxxx3xxox5oofx&#xt = tristraticid-cap ofsrixxux3oox5ooo&#xt = iktum = bistraticike-cap oftexxux3oox5xxx&#xt = srida gridtum = bistraticsrid-cap ofprahioxxx3xxox4ooqo&#xt = tristraticco-cap ofrico (dirico =co-diminishedrico)x(ou)x3o(xo)x x(uo)x&#xt = bistratictrip-cap ofsrip ({3}-diminishedsrip)...oqo3ooo4xox&#xt = bistratic vertex-first central segment ofico (pabdico = parabidiminishedico)xxx3xox4oqo&#xt = bistraticco-first central segment ofrico (pabdirico = parabidiminishedrico)oqo3xxx4xox&#xt = bistraticoct-first central segment ofspic (dapabdi spic = deep parabidiminishedspic)xxxx3xoox4xwwx&#xt = tristratictic-first central segment ofproh (pabdiproh = parabidiminishedproh)....xxx3.xox5.ofx&#xt = bistraticid-subsegment ofsrix (diminished tristraticid-cap).xofo3.ooox5.oxoo&#xt = tristratic vertex-first subsegment ofex (mono-diminishedrotunda of ex)...xoxFofo3oxoofox5ooxooxx&#xt = hexastraticike-cap ofroxfxoo2ofVx3xxoo5xoof&#zx = equatorial tetrastratic segment ofroxfx.o2of.x3xx.o5xo.f&#zx = dodeca-diminished equatorial tetrastratic segment ofrox...x.o.o .o.x.fo3o.f.x....f.o.oo3o.o.f....x.f.ox&#xt (hoddatedex) = (shallow-)hexa-octa-diminished deep-tetra-diminishedex | Genuinemultistratic segments of Wythoffians. For segments of Wythoffians, with one of its hyperplane beingtangential, the more specifictermcap is used. A cap with its other hyperplane moreover beingequatorial, will be defined arotunda. The class oftutsatopes belongs here generally. In fact, those were defined quite similar to the ursatopes, but then were found to be bistratic caps of uniforms in general. |
oxo..3ooo..5oox..&#xt = bistratic vertex-first cap ofex = * pt ||ike ||doeoxo.o3ooo.x5oox.o&#xt = dodeca-diminishedrotunda of ex = pt ||ike ||doe ||id.xo.o3.oo.x5.ox.o&#xt = trideca-diminishedrotunda of ex =ike ||doe ||idox.fo3oo.ox5oo.oo&#xt = icosi-diminishedrotunda of ex.x.fo3.o.ox5.o.oo&#xt = icosiena-diminishedrotunda of ex..ofo3..oox5..xoo&#xt = deep mono-diminishedrotunda of ex = 12-augmenteddoe || id..o.o3..o.x5..x.o&#xt = deep trideca-diminishedrotunda of ex = *doe ||id...ike || tet-dim-doe || idike || cube-dim-doe || idxxx3ooo3oxo&#xt = gyratedspid (ortho bicupola) = *tet ||co ||tetoxux3xxoo3xxxx&#xt = gyratedpripoxux3xxoo3oooo&#xt = augmentedtip...ike || id || srid = bistratic cap ofrox (=biscrox)gyepip || ... || dirid = arsd biscroxpap || ... || pabidrid =arspabd biscroxmibdi || ... || mabidrid = arsmabd biscroxteddi ||tepdid || tedrid =arsted biscroxpeppy || ... || pecu = chopped off part itself =gyepip || pero (segmentochoron)ike || f-ike || id || doe = dia-disdodeca-icosa-diminishedex (= diddidex)gyepip || f-gyepip || ... || doe = arsd diddidexpap || f-pap || ... || doe = arspabd diddidexmibdi || f-mibdi || ... || doe = arsmabd diddidexteddi ||f-teddi ||tepdid || doe =arsted diddidexdoe || (id \ fq-oct) || f-ike || doepybiscrox (= ike || (id \ fq-oct) || srid)xoxFofof(Vx)fofoFxox-3-oxoofoxx(oo)xxofooxo-5-ooxooxxo(of)oxxooxoo-&#xt =rox......o.(.x)fo......-3-......x.(.o)xx......-5-......x.(.f)ox......-&#xt =arsdod depabdirox | Diminishings thereof, gyrations, ... (Those marked by * either come out to besegmentochora themselves or are mere stacks of those, i.e. externalblends of such.) In 2016 the author investigated around-symmetrical diminishings ("ars..d.."). Here generally multistratic caps are considered, where all layers are diminished individually, but throughout in the same fashion. Then it took until 2019 before further around-symmetrical diminishings ("ars..d..") were found by "ndl", this time based on 4 selected layers ofex. Surely there are millions of subsymmetrical diminishings ofex.A pleasing one with axially pyritohedral symmetry was derived in 2016 by the author.It shows up 2 opposing deep cuts (doe). While onto one of these there attach 12peppies,onto the other there attach 12mibdies. Into the 6 pyritohedrally arranged gaps of pairs ofmibdies then a furthermibdi each is inserted. After quite some while, at the start of 2021, a guy calling himself "puffer fish" applied that samepyritohedral diminishing of theid section, as occurs within the former, ontobiscrox as well.And in mid 2021 he derived an aroundsymmetrical dodecadiminishing of the deep parabidiminishedrox (arsdod depabdirox). |
xFoFx3ooooo5xofox&#xtxFoFx3xxxxx5xofox&#xtoxFx3xfox5xoxx&#xtxofxF(Vo)Fxfox3xFxoo(xo)ooxFx5xoxFf(oV)fFxox&#xtxofxfox3xFxoxFx5xoxFxox&#xtoxFx3xoox5oxoo&#xtthawro pyriteoxoofooxo3oooxoxooo5ooxoooxoo&#xt..oofoo..3..oxoxo..5..xooox..&#xt...ofo...3...xox...5...ooo...&#xt (twau iddip)..x.o.o.(..)........3..o.f.x.(..)........5..x.o.x.(..)........&#xt ..x.ofo.(..)........3..o.fox.(..)........5..x.oxx.(..)........&#xt (using: F=ff=x+f, V=2f) | Further genuinemultistratic axials – which arenot segments of Wythoffians. ThexFoFx3ooooo5xofox&#xt was conjectured in 2014 by Quickfur (and proven toexist by Klitzing by means of the linked file). Note that it cannot be a stack of segmentochora as there are unsegmentable cells which reach throughall layers. In fact here those are 30bilbiros – the starting point of Quickfur's research, which then led to that find!–ThexFoFx3xxxxx5xofox&#xt then is nothing but its immediateStott expansion. According to a further idea of Quickfur also in 2014 Klitzing and Gevaert elaborated this CRF, which on the one hand is a further polychoron incorporating J92 (thawroes) for cells(cf. alsothawrorh of sectionhigher symmetric ones), but on the other hand cannearly be derived as a tristraticid-first tropal section of o3o3x5o (rahi), just that the corresponding bottom layer (there being f3f5o, an f-doe) here is replaced by an x3x5x (grid), which thus assures the CRF-ness again. Quickfur then found that the corresponding decastratic medial part ofrahi (in the sense of a deep parabidiminishing) likewise can have these replacements applied. – And M. Čtrnáct in reply found that the tropal tetrastratic part thereof even can be withdrawn therefrom, reconnecting the outer remainder of that parabidiminishing again (with some minor local rearrangments of cells). This find then is special in that it no longer uses anytet for cells.Instead it uses stacks of 3decagonal prisms each, and beside of thethawroes now alsobilbiroes. Quickfur considered in 2014 the still tristraticike-first subsegment ofrox, oxFf3xooo5oxox&#xt. That one is not CRF, because of f-edges in the bottom layer. But that very offending layer can be replaced accordingly, resulting inoxFx3xoox5oxoo&#xt. – What is even more surprising: the thus changed figure still allows for an axial-pyritohedrally symmetric diminishing, then providing thethawro pyrite, described by Gevaert shortly before. Gevaert later in 2014 constructed an axial elongation ofex, which then reads likea layer permutation, but rather pulls the left half of the node symbols to the left and the right ones to the right, thus doubling thecentral one, but thereby he assigns the layers next to the central one to the opposite half. Extension then will pull those "halfs" one unit apart.Thus these next to central layers finally will coincide:oxoofooxo3oooxoxooo5ooxoooxoo&#xt.–At first sight this looks strange, as the circumradius of the central layer then will be smaller as that of the neighbouring ones. Klitzing then proved that this arrangement is convex none the less. The 3 central layers thereof in this run were recognized to be nothing but the external blend (augmentation) ofid || id with 12pt || pip. In mid of 2018 "puffer fish" came up with a further axially tetrahedrally symmetrical lace tower,which both incorporates J63 (teddies) and J92 (thawroes):ooxf3xfox3oxFx&#xt, which sadly shortly thereafter was disproven to be CRF itself.But it clearly can be continued to become such: in fact allB2+## would qualify.Even though, in reply to its "find" Klitzing came up with a new axially icosahedrally symmetrical lace towerxoo3ofx5xox&#xt, which incorporates besides the 2 bases just J63 (teddies), J6 (peroes) and J1 (squippies).That one happens to be a diminishing of a deep tetrastratic subsegment ofrox.There are further related CRFs to that subsegment too. |
ox|Fxo|fx-3-xo|oxF|xo-3-of|xfo|oo-&#xt = B1 + A1ox|Fxo|o--3-xo|oxF|x--3-of|xfo|x--&#xt = B1 + A2ox|Fxo|fo-3-xo|oxF|xx-3-of|xfo|ox-&#xt = B1 + A3ox|Fxo|fx-3-xf|oxF|xo-3-oo|xfo|oo-&#xt = B2 + A1ox|Fxo|o--3-xf|oxF|x--3-oo|xfo|x--&#xt = B2 + A2ox|Fxo|fo-3-xf|oxF|xx-3-oo|xfo|ox-&#xt = B2 + A3where:A1 = ..|Fxo|fx-3-..|oxF|xo-3-..|xfo|oo-&#xtA2 = ..|Fxo|o--3-..|oxF|x--3-..|xfo|x--&#xtA3 = ..|Fxo|fo-3-..|oxF|xx-3-..|xfo|ox-&#xtB1 = ox|Fxo|..-3-xo|oxF|..-3-of|xfo|..-&#xtB2 = ox|Fxo|..-3-xf|oxF|..-3-oo|xfo|..-&#xt(vertical & horizontal lines introduced for comparision purposes only)xfofxx|f|oFxfxo-3-ooxxoF|x|Foxxoo-3-oxfxFo|f|xxfofx&#xt...|Fo|...-3-...|xx|...-3-...|oF|...&#xt =pretasto(these both incorporate most of B1 non-axially!) | Quickfur in 2014 found also an axiallytetrahedral symmetric polychoron, which incorporates bothbilbiro andthawro:oxFxofx3xooxFxo3ofxfooo&#xt ("B1+A1"). "Student5" shortly thereafter found a different continuation of the first "half", also using thosethawroes in the second "half", but in opposite orientation:oxFxoo3xooxFx3ofxfox&#xt ("B1+A2"). Gevaert some weeks later found a different continuation for the second "half" as well. That one no longer usesbilbiroes. Instead it uses further 4thawroes there too.In fact, just as the 2 second halves reverts the orientation of the thawroes, in these 2 first halves the orientation of theteddies becomes inversed. Accordingly this produces 2 further figures here:oxFxofx3xfoxFxo3ooxfooo&#xt ("B2+A1")andoxFxoo3xfoxFx3ooxfox&#xt ("B2+A2"). Only about 4 years later in early 2018 a third alternative for the second half was found by Klitzing. Accordingly this produces 2 further figures here:oxFxofo3xooxFxx3ofxfoox&#xt ("B1+A3")andoxFxofo3xfoxFxx3ooxfoox&#xt ("B2+A3"). The quite interesting part in these 6 figures is, that thebreaking surface here doesnot consist out of a single flat vertex layer. Instead it rather is a scrambled surface from 3 vertex layers: ..|Fxo|..3..|oxF|..3..|xfo|..&#xt, which truely need a 4D embedding in order to remain unit edged (i.e. a flattened 3D representation of that medial part would exist, but being the rectification of aknown "near miss" Fullerene only). Still in 2014 Quickfur found a much taller axially tetrahedral figure. That one so has overall inversional symmetry, and incorporates12bilbiroes and 8thawroes:xfofxxfoFxfxo3ooxxoFxFoxxoo3oxfxFofxxfofx&#xt.–But shortly thereafter he derived therefrom a further one, just by replacing the equatorial layer f3x3f by Fo3xx3oF&#y (some y, no direct lacings there) which destroys the thawroes, but introduces 12 more equatorial bilbiroes. The result,pretasto, moreover then gets fulldemitessic symmetry!(Cf. also sectionhigher symmetrics.) |
oxwQ wxoo3xxxx4oxxo&#zx = ooxwxoo-3-xxxxxxx-4-oxxoxxo&#xtox.. wx..3xx..4ox..&#zx = ..xwx..-3-..xxx..-4-..xox..&#xt | Still in 2014 Klitzing found by "correction" of a negativeEKF resultoxwQ wxoo3xxxx4oxxo&#zx, after all being aStott expansionofpoxic (wrt. the 3rd node position). |
oxux3xxoo4oooo&#xt (rico-cap +thex-rot)oxux3xxoo4xxxx&#xt (proh-cap +bistratic prit-cap) | A case of somehow more interesting stackings ofsegmentochora,rotundae, multistratic sections etc. might be considered, when sections ofdifferent uniforms are joined as anexternal blend. At the left, in every example the corresponding derivation is mentioned additionally. |
oox3ooo4oxx&#xt – with lacingesquipy (J8)oox3xxx4oxx&#xt – with lacingescu (J19) =biscsrico (bistratic segment ofsrico)xox3oxo5oox&#xt – with lacinggyepip (J11) =biscrox (bistratic segment ofrox)oxx.3oox.3xxo.&#xt – with lacinggybef (J26)oxxo3ooxx3xxoo&#xt – with lacinggybef (J26)oxxux3xxuxx3xxooo&#xt – with lacinggybef (J26)oxx xxx xxo&#xt (gybeffip) = stack of 2 gyratedtisdips | A further point of interest here might pop up in stacks of segmentochora whenever lacing cells happen to become corealmic and thus can beblended into combinedJohnson solids. These for sure are esp. interesting if they won't occure from sections of well-known uniforms already. The first of the left stacks, which result in lacinggybefs, was found in 2017 by Klitzing while searching forbistratic lace tegums. Surely that one can be extended as given there, simply by considering where the lower segment derives from.In fact it then happens to be an augmentation ofsrip.The third stack, resulting in lacinggybefs, was found in 2019 by Quickfur, when trying toStott expand the first one, which surely won't be CRF, but then continuing the construction beyond the offending 3rd layer.That one then comes out to be likewise an augmentation ofgrip. |
oxx3xoo3oxo&#xt = (ico-rot +spid-rot; r = 1)pt || cube || ike = (cubpy +cubaike; r = 1) | Still an other especial case of mere stackedsegmentochora occurs, when these are different, cannot be consideredas consecutive sections of some singleWythoffian polytope, but still happen to provide a commoncircumradius for all vertices of all layers. That is, whenever such towers still areorbiform. |
| • wedges / lunae / rosettes | (top of CRF) |
0.209785-luna ofspid = {6} ||trip(segmentochoron)0.290215-luna ofspid =tet ||tricu(segmentochoron)tet-first rotunda ofspid =tet ||co(segmentochoron)0.419569-luna ofgyspid = {3} ||tricu || {3}1/4-luna ofhex = pt ||squippy(segmentochoron)vertex-first rotunda ofhex = 2/4-luna ofhex = pt ||oct(segmentochoron)1/6-luna ofico = {3} || gyrotricu(segmentochoron)2/6-luna ofico =oct ||tricu(segmentochoron)oct-first rotunda ofico = 3/6-luna ofico =oct ||co(segmentochoron)1/10-luna ofex2/10-luna ofex3/10-luna ofex4/10-luna ofexvertex-first rotunda ofex = 5/10-luna ofexdim. 1/10-luna ofexouter 5-dim. 2/10-luna ofexinner 5-dim. 2/10-luna ofex7-dim. 2/10-luna ofex1/4-luna ofquawros = {8} ||cube(segmentochoron){4}-first rotunda ofquawros = 2/4-luna ofquawros2/5-luna ofstawros = {10} ||pip(segmentochoron)n-cufbil (2≤n≤4) = externalblend of 2n-cufs | Besides of considering 2 parallel hyperplanes cutting out monostratic layers of larger polychora, in 2012 Quickfur and Klitzing considered 2 intersecting equatorial hyperplanes. Those clearly cut out wedges. The specific choice of those hyperplanes made up the terminuslunae. Note, that 2 lunae with complemental fractions generally add to the fullhemispherical polychoron, the so calledrotunda.– In order to do socompletely, in case ofico co-realmic facets then would have to be re-joined. And, in case ofex, any thus formed cavette from pairs ofpeppies furthermore would have to be re-filled (i.e. augmented) by5 tet rosettes. In other words: lunae are nothing but wedge-like dissected rotundae. The sectioning applies at some vertex layer. If thereby some edgesget fractioned too, those will be omitted completely in either luna. Generally, chopped cells are replaced by accordingJohnson solids which assure the remainder to be convex. –Therefore those would have to be replaced conversely when glueing back again. stawros does allow for a2/5-luna (but no lunae with odd numerators, esp. no rotunda). That 2/5-luna would then be convex again. Although all of the first known examples used rational fractions, thedihedral angle between those wedge facetsneedn't be commensurable with 360° in general. Thus we rather have to expect some real numbersr ∈ ]0, ½] (for CRFs) here. Also then-cufbils kind of belong here, being the adjoins of two alike wedges (as for oranges). Those only exist within the realm of 2≤n≤4 as CRFs, beyond those become non-convex. The limitting cases already arespecial,as there either thetets become corealmic (tridpies) resp.then-cues become corealmic (squobcues.The individual cases then are the 1/2-luna (or rotunda) ofquawros (n=4), the 0.419569-luna ofgyspid (n=3) andtetep (n=2). – The corresponding mono wedges, then-cufs, all qualify assegmentochora. |
thawro-wedged 1/6-luna ofex(non-convex)thawro-wedged 2/6-luna ofexdim. thawro-wedged 1/6-luna ofex(non-convex)octa-dim. thawro-wedged 2/6-luna ofex | Not all wedge shaped sections by vertex layers in the sense of lunes are themselves CRFs. Esp. when the wedge facets intersect in something which would require non-unit edges.Sometimes however right that offending part canlocally be re-arranged, so that it becomes CRF again. This e.g. would be the case for the 1/6- and 2/6-lunes ofex, replacing the (non-CRF) 3-fold half-id wedge facets bythawroes, i.e. scaling down the offending centralf3f into a smallerx3x. Still offending then would be the f-edges, showing up in the Fo3oF&#f parts of their lace cities – not occuring withinthese lunes directly, but rather as cavities, which then would show up in their hulls. These however can be recoveredby incident pentagons, that is by accordingly introducedpeppy pairs. The 2/6-lune allows this directly and sobecomes a CRF. Not so for the 1/6-lune. (That one would require for a larger rebuild, cf.thawrorh,which still keeps locally its wedge shape, but then would have to abandon the lune property completely.) Both even can be further diminished. |
bidsrip (bidiminishedsrip)mibdirit (metabidiminishedrit){4} || tet = bidiminishedrap(segmentochoron)damibdrox (deep metabidiminishedrox)didamibdrox (diminished deep metabidiminishedrox) | Inbidsrip the 2 diminishings areneither parallel (kind of ametabidiminishing),nor equatorial. Moreover it qualifies as awedge, as one of theocts ofsrip gets reduced to its equatorial square, i.e. becomes sub-dimensional. Also a metabidiminishedrit has to be mentioned here: the topco thenwill be diminished in turn from both sides down to its equatorial hexagon (thereby becoming degenerate). In the same way{4} || tet could be considered as bidiminishedrap. In 2014 Quickfur found a metabidiminishing 36°-wedge ofrox, which cuts as deep as to itstid sections.Later he noticed, that the singleike, which is opposite to the wedge decagon, could be chopped off monostratically as well.Then there occurs apabidrid (and the whole figure qualifies then as multi-wedge). |
| • augmentations | (top of CRF) |
autepe = trippy + tepe... | At first there surely are plenty of various augmentations of the smaller polychora with quite small elementary components, mostly facet pyramids. esp. the smaller ones get listed to the left. |
omni-peppy-augmented 5,20-dipomni-pecu-augmented 10,20-dipomni-pecu-gyroaugmented 10,20-dipomni-peppy-augmented 5,10-dipomni-pecu-augmented 10,10-dipomni-pecu-gyroaugmented 10,10-dip... | Speaking of augmentations, esp. by those with 4D pyramids, the set ofduoprisms provides lots of possibilities, esp. sub-symmetrical or even assymmetrical ones. –A special nice finding here is theomni-augmented 5,20-dip,because then some dihedral angles would becomeflat, thereby blending thepeppies and the adjoining (un-augmented)pips intoepedpies (J16)!–The same holds true for the 2 kinds of omni-augmentations of the 10,20-dip, thereby blending thepecues and the adjoining (un-augmented)dips either intoepobcues (J38) or intoepigybcues (J39). The same surely applies to theStott contracted cases, based on the 5,10- resp. 10,10-dips each. Here the neighbouringpeppies would blend directly intopedpies (J13), resp. thepecueswould blend into eitherpobcues (J30) or intopegybcues (J31). In 2019 Quickfur even made up for an naming scheme when not all possible cells are being augmented.E.g.1,3,8-gyro-5,10-pentaaugmented 10,20-duoprism, meaning there are augments at positions 1, 3, 5, 8, 10, and augments 5 and 10 are gyrated relative to the other 3 augments.If both rings are augmentable we might have a (1,2,3),(1,3)-pentaaugmented 4,5-duoprism, meaning 3 augments on the 4-membered ring in positions 1, 2, and 3; and 2 augments on the 5-membered ring in positions 1 and 3.Here, as throughout, the start and direction of counting each is to be chosen that the index set uses the lowest possible number touple. Thus e.g.(1,3,5),(gyro-1)-tetraaugmented 10,10-duoprism is equivalent to a (2,4,6),((ortho)-1)-tetraaugmented 10,10-duoprism, but the first then is to be prefered. |
oq3oo4xo xo&#zx =tes + 6 (equatorial)cubpiesoq3xx4xo xo&#zx =ticcup + 6 (gyro)squipufsxw3oo4xo xo&#zx =sircope + 6cubpiesxw3xx4xo xo&#zx =gircope + 6 (gyro)squipufsoa3oo5xo xo&#zx =dope + 12pippiesoa3xx5xo xo&#zx =tiddip + 12 (gyro)pepufsxb3oo5xo xo&#zx =sriddip + 12pippiesxb3xx5xo xo&#zx =griddip + 12 (gyro)pepufs(where a = 3/sqrt(5) = 1.341641, b = a+x = 2.341641)oa3xo3oo xo&#zx =ope + 4 (alternate)trippiesoa3xo3xx xo&#zx =tuttip + 4tripufsxb3xo3oo xo&#zx =tuttip + 4trippiesxb3xo3xx xo&#zx =tope + 4 (alternate)tripufs(where a = (2+sqrt(10))/3 = 1.720759, b = a+x = 2.720759)oa3xo4oo xo&#zx =cope + 6cubpiesoa3xo4xx xo&#zx =ticcup + 6 (ortho)squipufsxb3xo4oo xo&#zx =tope + 6cubpiesxb3xo4xx xo&#zx =gircope + 6 (ortho)squipufs(where a = w/q = 1.707107, b = a+x = 2.707107)of3xo5oo xo&#zx =iddip + 12peppiesof3xo5xx xo&#zx =tiddip + 12 (ortho)pepufsxF3xo5oo xo&#zx =tipe + 12peppiesxF3xo5xx xo&#zx =griddip + 12 (ortho)pepufs(where F = ff = f+x)sy3so5so xo&#zx =sniddip + 12peppies(where y ≈ 2.253679)wacope+cubpy-blend = wacope + 1cubpyhawmcope+cubpy-blend = hawmcope + 1cubpysquappip+2cubpy-blend =squappip + 2cubpiessquappip+2squappy-blend =squappip + 2squappies | Beyond the application to duoprisms other still convex augmentations can be found for simpleprisms as well. In 2014 Quickfur omni-augmented the well-known segmentochorondoe || doe by 12 segmentochorapt || pip. In 2021 username5243 found the external blend of the prism ofwaco withcubpy.Shortly thereafter Quickfur found the external blend of the prism ofhawmco withcubpy,the external blend of thesquappip with 2cubpies, and several other similars usingprism pyramidaugmentations or expansions thereof, i.e. usingpucofastegia.Klitzing thereafter also found the external blend of thesquappip with 2squappies. Later in 2014 Gevaert independently considert an axial expansion ofex, which in the run of Klitzing's evaluation of its equatorial bistratic layer resulted in the according twelf-augmentedid || id. Sure these ones then can beStott expanded too. Theiddip case together with all its expansions applies to the othero3oPo o symmetries too.(The degenerate sub-case P=2 then would lead back totautip (J51) andtauhip (J57), resp. their prisms.)But for thedope case with all its expansions the sub-cases for P=4 here already get limiting: neighbouringsquippies orsquacues (in case together with thecubeorop in between for the expanded versions) become corealmic. Therefore these wouldblend into a single joined cell each. Probably because of its non-quadratic roots of its coordinates the according augmentation ofsniddiponly was established in 2024 by username5243. Note that the to be augmented prismso3xPx x occur twice in here: they could be augmented by an according amount of"magnabicupolaic rings" (or "pucofastegia", i.e.{P} ||2P-prism segmentochora) in either orientation:In fact, theP-gons therof could either align in orientation to those ofo3oPx x (ortho augmentations) or to those ofo3xPo x (gyro augmentations). |
quawrosstawros(non-convex) | In early 2012 Quickfur came up with a multiple augmentation oftes, which then is non-orbiform, but likewise allows for those operations: rotunda andluna. In 2013 Klitzing found anon-convex relative of it (stawros), which is an augmentation based onstarpedip, and also does allow for aluna, but not for a rotunda. |
bicyte ausodipcyte cubau sodipcyte opau sodipcytau tes | A further one can be obtained when augmenting alternatecubes ofsodip withcubpy and allops by{4} || op.Although this sounds un-spectacular so far, it comes out, that allsquippies either cobinepairwise intoocts, or unite with the remaining cubes intoesquidpies,and furthermore allsquacues combine pairwise intosquobcues.– In fact, this polychoron comes out to be apartial Stott contraction ofsrit. Sure not both, thecubes and theops have to be augmented simultanuosly. Either one could be augmented separately, resulting in 2 further CRFs,thecubau- and theopau-one. Similarily ates can be augmented bycubpies at one cycle of 4 places.Then neighbouringsquippies would combine intoocts. In fact, that one also could be looked at as a cyclotetradiminishedico. |
octatut,tricuf-blend | In 2015 Gevaert found a non-axial externalblend ofoctatut withtricuf, blending out a lacingtricu each. Similar to theaxial stacks ofsegmentochora here some lacingtrips become corealmic, thus blending in turn (subdimensionally) intogybefs. |
tisdip =trip-prism(uniform)autisdip =tisdip + 1cubpybautisdip =tisdip + 2cubpiestautisdip =tisdip + 3cubpiesautipip =autip-prism =tisdip + 1 orthosquippypauautipip =autipip + 1cubpybauautipip =autipip + 2cubpiesbautipip =bautip-prism =tisdip + 2 orthosquippypsaubautipip =bautipip + 1cubpytautipip =tautip-prism =tisdip + 3 orthosquippypsgyautisdip =tisdip + 1 gyrosquippypautodip =todip + 1squacupe | A3,4-dip clearly can be augmented withcubpies. The full triaugmentation thenistautisdip. Further it was observed that it well could be augmented withsquippyps too. Thereby thesquippies become corealmic to thetrips. Therefore, in fact, it happens to be nothing butautipip. In 2016 Klitzing then even proved that both augmentation types can be used simultanuously, while still remaining convex. Shortly thereafter Quickfur found that thesquippyp could also be placed in a gyrated mode onto3,4-dip. And Klitzing in turn provided then the expanded version thereof. |
baudeca =deca + 2tetatutstetacoa cube =tetaco +cubacotutatoa sirco =tutatoe +sircoatoe | Assorted quite simple augmentations. |
| •scaliforms | (top of CRF) |
spidrox | It is a scaliform polychoron with swirl-symmetry. It was found already in 2000 by G. Olshevsky. |
bidex | It is scaliform and cell-transitive and thus is even anoble polychoron.In fact, its cells are 48teddi only. It has swirl-symmetry too. It was found in 2004 by A. Weimholt. |
prissi | This scaliform polychoron first was found (in 2005 by Klitzing) as being analternated faceting ofprico. Thus, having a Dynkin symbol (s3s4o3x),W. Krieger later showed, that it is a Stott expansion ofsadi. |
tuta(segmentochoron) | Just for completeness: there is also a 4th so far known non-uniform convex scaliform polychoron, the stack of 2 antialignedtuts, connected by 4+4=8 lacingtricues and 6tets. This monostratic figure accordingly has been published already in 2000 within the article onconvex segmentochora. – In fact already shortly before the publication of this article that very polychoron initiated in a private mailing listthe weakening of the axioms of uniformity, what finally became known as the notion of scaliformity. |
| • non-vertex-transitive higher-symmetric ones | (top of CRF) |
sadi = idex(uniform)tisadi = idtexrisadi = idroxidimextidimex = idimtexridimex = idimroxhexdex | It is quite remarkable, that the non-regular (although uniform)sadi allows for several of the operations as regulars do. The result then will no longer belong to the set of uniform figures, but still comes out to be amultiform CRF. E.g.tisadi is thetruncated sadi,risadi is therectified sadi. It shall be pointed out, that tisadi thereby results in a bistratic 24-diminishing oftex,while risadi results again in a monostratic 24-diminishing ofrox. (Note that rox has its first nodenot being ringed, thus the find of these 2 figures in 2004 by A. Weimholtdoes not contradict to the below result of 2012.) Just assadi is a (uniform) 24-diminishing ofex, obtained by vertex diminishings at the positions of an inscribed f-ico, Klitzing found in 2017 an according (orbiform only) most symmetrical 20-diminishing as well, obtained by vertex diminishings at the positions of an inscribed f-spid. That one then likewise allowed fortruncation orrectification, what happens to be even more surprising, becauseidimex not even is uniform! What is quite obvious from the inscribed f-ico for the 24-diminishing, there too is an inscribed f-tesfor an according 16-diminishing ofex too,hexdex. |
bidex(noble) =x3o3o5o - 48ikepiesbidsid pixhi =x3o3o5x - 48doasridssadi(uniform) =x3o3o5o - 24ikepiesidsrix =x3o3x5o - 24idatiesidprix =x3o3x5x - 24tidagridsidsid pixhi =x3o3o5x - 24doasridsidrox = risadi =o3x3o5o - 24ikaidsidsrahi =o3x3o5x - 24sridatidsidtex = tisadi =x3x3o5o - 24 (bistratic)iktumsidprahi =x3x3o5x - 24 (bistratic)srida gridtums...?idimex =x3o3o5o - 20ikepiesidimsrix =x3o3x5o - 20idatiesidimprix =x3o3x5x - 20tidagridsidim sidpixhi =x3o3o5x - 20doasridsidimrox = ridimex =o3x3o5o - 20ikaidsidimsrahi =o3x3o5x - 20sridatidsidimtex = tidimex =x3x3o5o - 20 (bistratic)iktumsidimprahi =x3x3o5x - 20 (bistratic)srida gridtums...? | In 2012 a non-uniform figure with exactly the same symmetry asbidex was found by Klitzing, together with 3 related non-uniform relatives ofsadi. Theid-part of their names relates toicositetra-diminished, i.e. along the symmetry directions of the vertices of theicositetrachoron. Thus immitating the same construction, as sadi can be derived fromex.Resp.bid- relating tobi-icositetra-diminished, just as in the name of bidex. To that time it also was proven that this set is complete, provided one consideresmonostratic diminishings only,and that thestarting figure (the Wythoffian polychoron ofhyic symmetry) has at least the first node of its Dynkin diagram being ringed. In the sequel of their rediscovery oftisadi = idtex andrisadi = idrox in 2014 by Gevaert, Klitzing extended his former result, with respect to monostratic 24-diminishings, toany Wythoffian with hyic symmetry.Besides risadi just one further figure thereby emerged,idsrahi. (A similar construction could be considered to lower symmetries too: e.g. consider the8-diminishing, i.e. along the symmetry directions of the vertices of thehexadecachoron. That one applied ontoico clearly results intes. But then again the higher Wythoffians could be considered here too. Alas, no non-Wythoffian polychoron would result in this case: application ontospic results insrit,application ontosrico results inproh, andapplication ontoprico results ingidpith.) Immediately after the finds ofidimex,tidimex, andridimex (cf. above) "Username5243" pointed out that the applicability within this subsection might be possible too. This clearly is the case, resulting in the various20-diminishings, i.e. along the symmetry directions of the vertices of thesmall prismatodecachoron. |
cyted srit (cyclo-tetra-diminishing)cyte gysrit (cyclo-tetra-gyration)bicyte gysrit (bi-cyclo-tetra-gyration)bipgy srit (bi-para-(bi-)gyration)... | The relation ofsrit to the vertex-inscribedodip, being considered as itsbi-cyclo-tetra-diminishing, give rise to various partial diminishings or gyrations too. |
pex hexquawrospacsid pithpexicbicyte ausodippacsritpex thexpabex thexpacpritpoxicpocsricowau pritpoc prico | A quite powerful procedure is that ofStott expansion resp. contraction. That not only applies when pulling apart (resp. pushing in) all elements of an equivalence class of total symmetry, but in 2013 Klitzing applied it for lower symmetries as well. Cf. the section onpartial Stott expansions for corresponding series; here only the found CRFs are listed. Some of those were discovered independently before (and might therefore be listed in the respective cathegory as well) – or at least could be obtained differently: |
sidsrahisgysrahi | In 2013 Quickfur suggested a swirl-symmerticdiminishedsrahi (the existance of which shortly after was proven by Klitzing). There exists a correspondinggyration (re-placing all the diminished{5} || dip caps ina gyrated way) as well. – Moreover this might be done with respect to any of those 12 swirl cycles independently. |
cypdex,bicypdexcypdrahi,bicypdrahicypdisrix, bicypdisrixcytid ricocytid sricotridicocyted spiccypdrox,bicypdroxcythdexcydadex | Forex,rahi, andsrix there are mono-, bi-, resp. tristraticcyclical multi-diminishings, which provide a regular pentagonal projection shape (forlace city). Additionally there is an orthogonal cycle then, which likewise can be diminished in the same way too. These thus show up some strange relation to the5,5-dip. Somehow similar is a cyclical (monostratic) diminishing ofrico, which provides a regular triangular projection shape.Or a cyclical bistratic diminishing ofsrico, which provides a regular triangular projection shape too.(Here the resp. orthogonal cycle does not lend to further diminishings, as it is too narrow. But in the latter case in here lives a cycle of 6 (un-chopped)sircoes.)The most elemental one here would be the (cyclical monostratic) tri-diminishing ofico itself. The border case here is the cyclical (monostratic) diminishing ofspic, which provides a square projection.(The orthogonal cycle could be diminished here too, but then reproduces justsrit.) (In these cyclo-diminishings any adjacent pair of diminished facets are mutually incident at some subdimensional element. Therefore those polychora would qualify asmulti-wedges too.) In the dawn of 2014 the author showed that alsorox has a bistratic diminishing, which can be applied cyclically. But there the section planes (individually producingsrid sections) would intersect. Thus this results in a pentagonal multi-wedge with 5pabidrids. – Shortly thereafter Quickfur then even constructed the bicyclically diminished version. At the beginning of 2016 Quickfur came up with a bicyclical multi-diminishing ofex in its3,3-dip orientation. In the first ring he applied 3 bistratic diminishings. Here theobtaineddoes happen to be just tip-to-tip. In the orthogonal ring he applied 6 monostraticdiminishings. There the obtainedikes then adjoin face-to-face. Clearly here the uniformgap would qualify as bicyclical decadiminishing ofex.A true CRF so would be the according monocyclical decadiminishingcydadex. |
xfooxo5xofxoo oxofox5ooxofx&#zx (ex, regular).foo.o5.ofx.o .xof.x5.oxo.x&#zx.foox.5.ofxo. .xofo.5.oxof.&#zx (gap, uniform) | Some at leastorbiform diminishings wrt. duoprismatic subsymmetries. |
thawrorhtewau thawrorhdim. thawro-wedged 1/6-luna of ex(non-convex) | This polychoron was found by Quickfur in 2014 while searching for CRFs incorporating J92 (thawro).Thus its working title was coined accordingly. (Having that rhombical shape, it surely qualifies as multi-wedge too.) M. Čtrnáct then found, that it even can be augmented at any thawro independently. The full or tetra-augmented one then istewau thawrorh. Also in 2014 Klitzing then found, thatthawrorh is not too surprising as such,it just is the hull of the exterior blend of 2dim. thawro-wedged 1/6-luna ofex. Btw., that latter one by itself is a multi-wedge too, however a non-convex one. |
pretastoicau pretastohidicau pretasto | A quite different polychoron with fulldemitessic symmetry and featuringteddies andbilbiroes was found by Quickfur in 2014. It was found as symmetrical completion of a mere axially symmetrical relative (cf.there). Furthermore all 24 bilbiroes can be augmented bybilbiro wedges. Then the former teddiesbecome fullikes again. It was only in 2020 that Klitzing observed thaticau pretasto features some vertices that areteddi andpip. Although those mutually intersect already within either subset,it was possible to select a further subset of 16 of thepip-type vertices to produce another CRF. |
icau prissi | In 2014 Mrs. Krieger suggested to augmentprissi at all itsikes.This then is equivalent to apartial Stott expansion ofex wrt. the sameicoicsubsymmetry, as prissi itself can be derived fromsadi. |
foxo3xxxF3xfoo *b3oxfo&#zx (icau prissi)fooo3xxoF3xfxo *b3oxFo&#zxooxf3foox3oxfo *b3xFxo&#zx (icau pretasto)Fxox3xoxf3oFxx *b3oxfo&#zxoFFxx3xxoof3fooxo3ooffx&#zxxFfoo3xoxxF3fxoxo3ooffx&#zxoFFxx3xxoof3fooxx3xxFFo&#zxxxfoF3oxxFx3xFxxo3Fofxx&#zxxFfxo3xoxoF3fxooo3oofFx&#zxoFfoo3ooxxF3Fxoxo3ooffx&#zxoFFxx3xxoxf3Fxxox3oofFx&#zx | Still in 2014 Klitzing multi-applied the techniques of Gevaert (cf.EKFs) toex, thus producing intricate facetings with demicubic subsymmetry. Ex itself can be rewritten in that subsymmetry asfoxo3ooof3xfoo *b3oxfo&#zx.The used facetings then arefooo3oo(-x)f3xfxo *b3oxFo&#zx (where that "quirks mode" was applied at level 3 first onto the left arm of the diagram and secondly at the center) resp.fo(-x)o3xoxf3(-x)foo *b3oxfo&#zx (where it was applied independently in 2 layers at one different arm each). These allowed for aStott expansions, which then eliminate all introducedretrograde edges again, thus resulting in CRF figures. These results could be concluded as follows: Shortly later Gevaert managed to write ex in pentic subsymmetry asxffoo3oxoof3fooxo3ooffx&#zx. That display then allows for a similar investigation. Single quirks e.g. result in(-x)ffoo3xxoof3fooxo3ooffx&#zx resp.xFfoo3o(-x)oof3fxoxo3ooffx&#zx, theStott expansions of which (given at the left) Klitzing then proved to be CRF. |
3doe-1pap-dim-ex3doe-1pap-10mibdi-10teddi-dim-ex | In early 2016 Quickfur found a further multi-wedge diminishing ofex, cutting off 3 bistratic vertex-first caps, each withdoe base, which are pairwise pentagon adjoined. The single remaining vertex of that former great circle allows for a monostratic cut offpappy. – In fact, this mere multiwedge would still have 80 vertices. Whereas the find of Quickfur only has 60,being then a further multi-diminishing (each monostratic, at vertices outside that mentioned great circle), providing eithermibdies orteddies as additional facets, thus reducing the count of remainingtets to just 10! |
| • Euclidean 3D Honeycombs | (top of CRF) |
ditoheditohgyeditohpextohpacratoherichgyrich *gyerichpexrichpacsrich5Y4-4T-4P4 *5Y4-4T-6P3-sq-para *5Y4-4T-6P3-sq-skew *10Y4-8T-0 *10Y4-8T-1-alt *10Y4-8T-1-hel (r/l) *10Y4-8T-2-alt *10Y4-8T-2-hel (r/l) *10Y4-8T-3 *5Y4-4T-6P3-tri-0 *5Y4-4T-6P3-tri-1-alt *5Y4-4T-6P3-tri-1-hel (r/l) *5Y4-4T-6P3-tri-2-alt *5Y4-4T-6P3-tri-2-hel (r/l) *5Y4-4T-6P3-tri-3 *3Q4-T-2P8-P4 *6Q4-2T *6Q3-2S3-gyro *6Q3-2S3-ortho *3Q3-S3-2P6-2P3-gyro *3Q3-S3-2P6-2P3-ortho *cube-doe-bilbiro | There are some few euclidean 3D honeycombs known, which count as most as CRF. * Those being marked by an asterisk would classify moreover asscaliform. – Most of them have been found in 2005 by J. McNeill. He then called themelementary honeycombs.So, the different stacking modes (-alt, -hel (r), -hel (l)) still remained un-discovered until 2013. Thecube-doe-bilbiro was found in 2004 by A. Weimholt. |
(unsorted collection only –beyond those already contained within other classes, esp. likeWythoffians,axial polytopes (esp. named types of),partial Stott expansions of Wythoffians,scaliforms,orsegmentotera. In summer 2022 the extrapolation ofgyroelongation became a topic of interest. It soon occured that this could be done by squeezing inalterprismsas the generalization of 3Dantiprisms (where those concepts still coincided).This then led the author to findgyetac.(Within retrospective it furthermore occurs to be related to the 5D Dutour polytope.)In reply to that find then Username5243 came up with itspartial Stott expansions in general,and esp. with the moreover orbiformpabgysiphin. Ambification sometimes applies even beyond theWythoffians and still mightproduce CRF and alsoorbiform outcomes. Within 2025 the idea occured, to consider an odd-dimensional extrapolation ofgybef in its sense ofline|| square|| perp line, i.e. thence as{n/d}|| (n/d,m/b)-dip|| perp {m/b},i.e. the(n/d,m/b)-gybef. This higher-dimensional version then likewise happens to be a bistratic externalblend of a ({n/d}, {m/b}-pyr)-duoprism with an ({n/d}-pyr, {m/b})-duoprism,thereby blending out the common({n/d}, {m/b})-duoprismatic base.(It also allows for even higher dimensional extensions too: the ajoin of an (P-pyr, Q)-duoprism with the (P, Q-pyr)-duoprismacross the common (P,Q)-duoprismatic facet. Eg. for P=Q=tet one could obtain the 7Dtet-gybef.) Within several cases high symmetrical diminishings already provide interesting CRFs too.CRFs Remarks • axials (top of CRF) oxo.3ooo.3oox. *b3ooo.&#xt -5D Dutour polyteron type Aoxoo3oooo3ooxo *b3oooo&#xt -gyetacoxoo3oooo3ooxo *b3xxxx&#xt -pabgysiphinoxoo3xxxx3ooxo *b3oooo&#xt -pexgyetacoxoo3xxxx3ooxo *b3xxxx&#xt -pabex gyetacoxo3xoo4oox3oxo&#xtxxo-n/d-ooo oxx-m/b-ooo&#xt -(n/d,m/b)-gybefesp.:xxo3ooo oxx3ooo&#xt -3-gybefxxo4ooo oxx4ooo&#xt -4-gybef
• diminishings (top of CRF) hin:dihin(chopping off 1 vertex)bidhin(chopping off 2 opposite vertices of one layer)tedhin(chopping off the 4 vertices of an inscribed square of one layer)nit:odnit(chopping off 8 vertices)bodnit(chopping off 16 vertices)squadinit(chopping off 4 extremalocts)mibdinit(chopping off 2 (nearer)raps)sripa(chopping off 2 (opposite)raps)rat:tedrat(chopping off 4 vertices)octpy || octaco(1/4 lune)rix:tedrix(chopping off 3 vertices)
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