Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
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.
†) 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 explicite blend addition will be detailed within the context of complexes.
°) The solids marked by this sign are orbiform, 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 for segmentochora.
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.
Facets being {3} only  Facets being {3} and {4}  Facets being {3}, {4}, and {5} 
J12  †) tridpy  trigonal dipyramid J13  †) pedpy  pentagonal dipyramid J17  †) gyesqidpy  gyroelongated square dipyramid J51  †) tautip  triaugmented trigonal prism J84  snadow  snub disphenoid 
J1  °) squippy  square pyramid J7  †) etripy  elongated trigonal pyramid J8  †) esquipy  elongated square pyramid J10  †) gyesp  gyroelongated square pyramid J14  †) etidpy  elongated trigonal dipyramid J15  †) esquidpy  elongated square dipyramid J16  †) epedpy  elongated pentagonal dipyramid J26  †) gybef  gyrobifastegium J27  †°) tobcu  triangular orthobicupola J28  †) squobcu  square orthobicupola J29  †) squigybcu  square gyrobicupola J35  †) etobcu  elongated triangular orthobicupola J36  †) etigybcu  elongated triangular gyrobicupola J37  †°) esquigybcu  elongated square gyrobicupola J44  †) gyetibcu  gyroelongated triangular bicupola J45  †) gyesquibcu  gyroelongated square bicupola J49  †) autip  augmented triangular prism J50  †) bautip  biaugmented triangular prism J85  snisquap  snub square antiprism J86  waco  sphenocorona J87  †) auwaco  augmented sphenocorona J88  wamco  sphenomegacorona J89  hawmco  hebesphenomegacorona J90  dawci  disphenocingulum 
J9  †) epeppy  elongated pentagonal pyramid J30  †) pobcu  pentagonal orthobicupola J31  †) pegybcu  pentagonal gyrobicupola J32  †) pocuro  pentagonal orthocupolarotunda J33  †) pegycuro  pentagonal gyrocupolarotunda J38  †) epobcu  elongated pentagonal orthobicupola J39  †) epigybcu  elongated pentagonal gyrobicupola J40  †) epocuro  elongated pentagonal orthocupolarotunda J41  †) epgycuro  elongated pentagonal gyrocupolarotunda J42  †) epobro  elongated pentagonal orthobirotunda J43  †) epgybro  elongated pentagonal gyrobirotunda J46  †) gyepibcu  gyroelongated pentagonal bicupola J47  †) gyepcuro  gyroelongated pentagonal cupolarotunda J52  †) aupip  augmented pentagonal prism J53  †) baupip  biaugmented pentagonal prism J72  °) gyrid  gyrated rhombicosidodecahedron J73  °) pabgyrid  parabigyrated rhombicosidodecahedron J74  °) mabgyrid  metabigyrated rhombicosidodecahedron J75  °) tagyrid  trigyrated rhombicosidodecahedron J91  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 cupola J20  †) epcu  elongated pentagonal cupola J21  †) epro  elongated pentagonal rotunda J24  †) gyepcu  gyroelongated pentagonal cupola J68  †) autid  augmented truncated dodecahedron J69  †) pabautid  parabiaugmented truncated dodecahedron J70  †) mabautid  metabiaugmented truncated dodecahedron J71  †) tautid  triaugmented truncated dodecahedron J76  °) dirid  diminished rhombicosidodecahedron J77  °) pagydrid  paragyrate diminished rhombicosidodecahedron J78  °) magydrid  metagyrate diminished rhombicosidodecahedron J79  °) bagydrid  bigyrate diminished rhombicosidodecahedron J80  °) pabidrid  parabidiminished rhombicosidodecahedron J81  °) mabidrid  metabidiminished rhombicosidodecahedron J82  °) gybadrid  gyrated bidiminished rhombicosidodecahedron J83  °) tedrid  tridiminished rhombicosidodecahedron 
J3  °) tricu  triangular cupola J18  †) etcu  elongated triangular cupola J22  †) gyetcu  gyroelongated triangular cupola J54  †) auhip  augmented hexagonal prism J55  †) pabauhip  parabiaugmented hexagonal prism J56  †) mabauhip  metabiaugmented hexagonal prism J57  †) tauhip  triaugmented hexagonal prism J65  †) autut  augmented truncated tetrahedron 
Facets being {3}, {4}, and {8}  Facets being {3} and {5}  Facets being {3}, {5}, and {10} 
J4  °) squacu  square cupola J19  †°) escu  elongated square cupola J23  †) gyescu  gyroelongated square cupola J66  †) autic  augmented truncated cube J67  †) bautic  biaugmented truncated cube 
J2  °) peppy  pentagonal pyramid J11  †°) gyepip  gyroelongated pentagonal pyramid J34  †°) pobro  pentagonal orthobirotunda J48  †) gyepabro  gyroelongated pentagonal birotunda J58  †) aud  augmented dodecahedron J59  †) pabaud  parabiaugmented dodecahedron J60  †) mabaud  metabiaugmented dodecahedron J61  †) taud  triaugmented dodecahedron J62  °) mibdi  metabidiminished icosahedron J63  °) teddi  tridiminished icosahedron J64  †) auteddi  augmented tridiminished icosahedron 
J6  °) pero  pentagonal rotunda J25  †) gyepro  gyroelongated pentagonal rotunda 
Further reading: since, the restriction of strict convexity was released. V. Zalgaller (and his students) enlisted the set of convex regularfaced polyhedra with conditional edges. And the corresponding list then is provided here.
In 1979 Roswitha Blind applied this idea to higher dimensional polytopes: those are bound to be convex, built from regular facetpolytopes, 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.
°) Alike, those are orbiform.
Facets being tet only (cf. Tetrahedrochora)  Facets being tet and oct  Facets being tet and ike 
tedpy = pt  tet  pt  *†) tetrahedral bipyramid ikedpy = pt  ike  pt  †) icosahedral bipyramid 
octpy = pt  oct  *°) octahedral pyramid aurap = pt  oct  tet  †) augmented rectified pentachoron 
ikepy = pt  ike  °) icosahedral pyramid and: millions more or less asymmetric ex diminishings between sadi and ex 
There is also a different possibility to extrapolate from the set of Johnson solids into higher dimensions. Instead of requiring (n1)dimensional facets to be regular, one rather could stick to 2dimensional faces being regular only. That type of research meanwhile is known as CRF (convex & regular faced).
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 vertextransitivity: 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, a subset 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 of convex segmetochora.
Further examples of 4D CRF polytopes will follow below. In higher dimensional spaces even less is known. Cf. eg.
(unsorted collection only – beyond those already contained within other classes, esp. like Wythoffians, axial polytopes (esp. named types of), or segmentochora. 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 socalled expanded kaleidofacetings, are described in more detail on a own page. And thus just some of those are contained below too – then mainly for historical reasons.)
A 4D CRF list (downloadable spreadsheet) is also available. Additionally it complements the below provided listing of 4D cases with all the here excluded ones. Moreover it provides individual cell counts each.
CRFs  Remarks 

• nonorbiform monostratics  (top of CRF) 
npy  inv gyro npy (2<n<6) = external blend of 2 nappies ncu  inv gyro ncu (6/5<n<6) = external blend of 2 npufs line  bilbiro {3}  thawro {5}  pocuro {3}  gybef = external blend of traf and trippy) npy  inv ortho npy (2<n<6) = external blend of 2 npipies ncu  inv ortho ncu (6/5<n<6) = external blend of 2 npufs {4}  squobcu = external blend of 2 squacufs xfox oxfo3ooox&#xt (3mibdiwedge) xfoxo oxfox5ooofx&#xt (5mibdiwedge) monoaugmented 3mibdiwedge biaugmented 3mibdiwedge xofox ooxfo3oxoox&#xt (triaugmented 3mibdiwedge) 
In 2012 Quickfur came up with 2 nonorbiform monostratic families, described in more detail here, 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's expanded kaleidofacetings construction device. In 2015 J. Roth and the author still found several more. – Thereby it also turned out too as an aside that the external blend of the ncuf and the npuf generally would provide corealmic lacings, i.e. the 2 trips would join into a prism on a rhombic base, and the squippies unite with the adjoining tets into sheered digonal cupolae with rhombic lacings. Therefore their blends would not qualify as further CRFs In 2016 Quickfur succeded to complete an idea of the author, using 3 mibdies 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 either mibdi by an according pyramid, without losing the CRF property. The corresponding triaugmentation then is e.g. xofox ooxfo3oxoox&#xt. Note that this triaugmentation then incorporates a very special edge, having 6 incident peppies. 
• axials  (top of CRF) 
ooo3ooo4oxo&#xt  Even so cute clearly is a mere stack of segmentochora, by itself it is an interesting shape. E.g. ico can be decomposed into 8 such building blocks. And, because of having a tiptotip distance of exactly one edge unit, it can be decomposed itself in turn into 6 squascs. 
ofx3xoo3ooo&#xt ofx3xoo4ooo&#xt ofx3xoo5ooo&#xt = icosienadiminished rotunda of ex ofx3xoo xxx&#xt ofx3xoo3xxx&#xt ofx3xoo4xxx&#xt ofx3xoo5xxx&#xt ofxo3xooo3oooo&#xt ofxo3xooo4oooo&#xt ofxo3xooo5oooo&#xt = icosidiminished rotunda of ex ofxo3xooo xxxx&#xt ofxo3xooo3xxxx&#xt ofxo3xooo4xxxx&#xt ofxo3xooo5xxxx&#xt xofo3ooox3oxoo&#xt xoBo3ofox3xooo&#xt xofo3ooox5oxoo&#xt = monodiminished rotunda of ex xxFVF(Vx)fox3ofxxf(oF)xxx3xoooo(xo)xfo&#xt 42diminished sidpixhi 42diminished 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, and z is either o or x). In fact those all are extrapolations from the pentagon (2D: ofx&#xt), via the teddi (3D: ofx3xoo&#xt), into 4D. Because teddi itself was nicknamed teddy sometimes, this small family of teddipolychora (and their higher dimensional analogues) winkingly was attributed the name of ursachora (resp. ursatopes). Similar to teddi itself (yielding auteddi), 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 of auteddi itself exists as well. Like teddi can be augmented at its pentagons, resulting there back in the ike, the tetrahedral, octahedral, and icosahedral ursachoron can be augmented at its teddies as well. Even so, the tetrahedral can be augmented at any teddi at the same time, the octahedral just at nonneighbouring teddies in order to remain convex. For the icosahedral the same would hold true at the first sight, but the cavities at the peppyjoin then could be bridged there by additional 5tetrosettes. In 2016 Quickfur and the author came up with a cellwise expanded version of the tetrahedral ursachoron, which then was the first nontrivial CRF containing tedrid. – An axially octahedral symmetric relative does not exist because there is no 4fold relative of teddi. But then again an axially icosahedral symmetric version can be considered, assuming the 5fold relative of teddi to be doe. So, starting with srid and 20 tedrids one observes that this one comes back to be a 21diminishing of sidpixhi. It then even will be possible to apply that to both hemiglomes at the same time, resulting in an according 42diminishing. – In contrast to sidpixhi its contracted version, ex can be split into 2 rotundae in a CRF way. This is how ofx3xoo5ooo&#xt can be reconsidered a 21diminishing of the ex rotunda. 
elongated ico  A special case of augmentation occurs when glueing 2 octfirst rotundae of ico at both sides of a cope. Clearly a tristratic polychoron. The peculiar clue here is that the squippies become corealmic to the cubes, thereby blending into esquidpies (J15)! 
xux3oox3xxx&#xt = bistratic cocap of prip (tutdiminished prip) xux3oox4ooo&#xt = octfirst rotunda of thex xux3oox4xxx&#xt = bistratic sircocap of prit oxofo3oooox5ooxoo&#xt = vertexfirst rotunda of ex ooo3xox5ofx&#xt = bistratic idcap of rahi oxxx3xxox5oofx&#xt = tristratic idcap of srix xux3oox5ooo&#xt = bistratic ikecap of tex xux3oox5xxx&#xt = bistratic sridcap of prahi oxxx3xxox4ooqo&#xt = tristratic cocap of rico (dirico = codiminished rico) x(ou)x3o(xo)x x(uo)x&#xt = bistratic tripcap of srip ({3}diminished srip) ... oqo3xox4ooo&#xt = bistratic vertexfirst central segment of ico (pabdico = parabidiminished ico) xxx3xox4oqo&#xt = bistratic cofirst central segment of rico (pabdirico = parabidiminished rico) oqo3xxx4xox&#xt = bistratic octfirst central segment of spic (dapabdi spic = deep parabidiminished spic) xxxx3xoox4xwwx&#xt = tristratic ticfirst central segment of proh (pabdiproh = parabidiminished proh) ... .xxx3.xox5.ofx&#xt = bistratic idsubsegment of srix (diminished tristratic idcap) .xofo3.ooox5.oxoo&#xt = tristratic vertexfirst subsegment of ex (monodiminished rotunda of ex) ... xoxFofo3oxoofox5ooxooxx&#xt = hexastratic ikecap of rox ... 
Genuine multistratic segments of Wythoffians. For segments of Wythoffians, with one of its hyperplane being tangential, the more specific term cap is used. A cap with its other hyperplane moreover being equatorial, will be defined a rotunda. 
oxo..3ooo..5oox..&#xt = bistratic vertexfirst cap of ex = * pt  ike  doe oxo.o3ooo.x5oox.o&#xt = dodecadiminished rotunda of ex = pt  ike  doe  id .xo.o3.oo.x5.ox.o&#xt = tridecadiminished rotunda of ex = ike  doe  id ox.fo3oo.ox5oo.oo&#xt = icosidiminished rotunda of ex .x.fo3.o.ox5.o.oo&#xt = icosienadiminished rotunda of ex ..ofo3..oox5..xoo&#xt = deep monodiminished rotunda of ex = 12augmented doe  id ..o.o3..o.x5..x.o&#xt = deep tridecadiminished rotunda of ex = * doe  id ... ike  tetdimdoe  id ike  cubedimdoe  id xxx3ooo3oxo&#xt = gyrated spid (ortho bicupola) = * tet  co  tet oxux3xxoo3xxxx&#xt = gyrated prip oxux3xxoo3oooo&#xt = augmented tip ... ike  id  srid = bistratic cap of rox (biscrox) gyepip  ...  dirid = arsd biscrox pap  ...  pabidrid = arspabd biscrox mibdi  ...  mabidrid = arsmabd biscrox teddi  ...  tedrid = arsted biscrox peppy  ...  pecu = chopped off part itself = (segmentochoron: gyepip  pero) doe  (id \ fqoct)  fike  doe 
Diminishings thereof, gyrations, ... (Those marked by * either come out to be segmentochora themselves or are mere stacks of those, i.e. external blends of such.) In 2016 the author investigated aroundsymmetrical diminishings ("ars..d.."). Here generally multistratic caps are considered, where all layers are diminished individually, but throughout in the same fashion. Surely there are millions of subsymmetrical diminishings of ex. 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 12 peppies, onto the other there attach 12 mibdies. Into the 6 pyritohedrally arranged gaps of pairs of mibdies then a further mibdi each is inserted. 
xFoFx3ooooo5xofox&#xt xFoFx3xxxxx5xofox&#xt oxFx3xfox5xoxx&#xt xofxF(Vo)Fxfox3xFxoo(xo)ooxFx5xoxFf(oV)fFxox&#xt xofxfox3xFxoxFx5xoxFxox&#xt oxFx3xoox5oxoo&#xt thawro pyrite oxoofooxo3oooxoxooo5ooxoooxoo&#xt ..oofoo..3..oxoxo..5..xooox..&#xt ...ofo...3...xox...5...ooo...&#xt (twau iddip) (using: F=ff=x+f, V=2f) 
Further genuine multistratic axials – which are not segments of Wythoffians. The xFoFx3ooooo5xofox&#xt was conjectured in 2014 by Quickfur (and proven to exist by Klitzing by means of the linked file). Note that it cannot be a stack of segmentochora as there are unsegmentable cells which reach through all layers. In fact here those are 30 bilbiros – the starting point of Quickfur's research, which then led to that find! – The xFoFx3xxxxx5xofox&#xt then is nothing but its immediate Stott 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. also thawrorh of section higher symmetric ones), but on the other hand can nearly be derived as a tristratic idfirst tropal section of o3o3x5o (rahi), just that the corresponding bottom layer (there being f3f5o, an fdoe) here is replaced by an x3x5x (grid), which thus assures the CRFness again. Quickfur then found that the corresponding decastratic medial part of rahi (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 any tet for cells. Instead it uses stacks of 3 decagonal prisms each, and beside of the thawroes now also bilbiroes. Quickfur considered in 2014 the diminished ikefirst still tristratic segment of rox (i.e. oxFf3xooo5oxox&#xt). That one would not be CRF, so. But its bottom layer can be replaced accordingly: oxFx3xoox5oxoo&#xt. – What is even more surprising: this figure allows for an axialpyritohedrally symmetric diminishing, then providing the thawro pyrite, described by Gevaert shortly before. Gevaert later in 2014 constructed an axial elongation of ex, which then reads like a layer permutation, but rather pulls the left half of the node symbols to the left and the right ones to the right, thus doubling the central 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) of id  id with 12 pt  pip. 
oxFxofx3xooxFxo3ofxfooo&#xt = A1 + B1 oxFxoo3xooxFx3ofxfox&#xt = A2 + B1 oxFxofx3xfoxFxo3ooxfooo&#xt = A1 + B2 oxFxoo3xfoxFx3ooxfox&#xt = A2 + B2 where: A1 = xfoxF...3oxFxo...3ooofx...&#xt B1 = oxFxo...3xooxF...3ofxfo...&#xt A2 = ooxF...3xFxo...3xofx...&#xt B2 = oxFxo...3xfoxF...3ooxfo...&#xt (vertical & horizontal lines introduced for comparision purposes only) xfofxxfoFxfxo3ooxxoFxFoxxoo3oxfxFofxxfofx&#xt ...Fo...3...xx...3...oF...&#xt = pretasto (these both incorporate most of B1 nonaxially!) 
Quickfur in 2014 found also an axially tetrahedral symmetric polychoron, which incorporates both bilbiro and thawro: oxFxofx3xooxFxo3ofxfooo&#xt. "Student5" shortly thereafter found a different continuation of the first "half", also using those thawroes in the second "half", but in opposite orientation: oxFxoo3xooxFx3ofxfox&#xt Gevaert some weeks later found a different continuation for the first "half" as well. That one no longer uses bilbiroes. Instead it uses further 4 thawroes there too. In fact, just as the 2 second halves revers the orientation of the thawroes, in these 2 first halves the orientation of the teddies becomes inversed. Accordingly this produces 2 further figures here: oxFxofx3xfoxFxo3ooxfooo&#xt and oxFxoo3xfoxFx3ooxfox&#xt. The quite interesting part in these 4 figures is, that the breaking surface here does not 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 a known "near miss" Fullerene only). Still in 2014 Quickfur found a much taller axially tetrahedral figure. That one so has overall inversional symmetry, and incorporates 12 bilbiroes and 8 thawroes: 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 full demitessic symmetry! (Cf. also section higher symmetrics.) 
oxwQ wxoo3xxxx4oxxo&#zx = ooxwxoo3xxxxxxx4oxxoxxo&#xt ox.. wx..3xx..4ox..&#zx = ..xwx..3..xxx..4..xox..&#xt  Still in 2014 Klitzing found by "correction" of a negative EKF result oxwQ wxoo3xxxx4oxxo&#zx, after all being a Stott expansion of poxic (wrt. the 3rd node position). 
oxux3xxoo4oooo&#xt (ricocap + thexrot) oxux3xxoo4xxxx&#xt (prohcap + bistratic pritcap)  A case of somehow more interesting stackings of segmentochora, rotundae, multistratic sections etc. might be considered, when sections of different uniforms are joined as an external blend. At the left, in every example the corresponding derivation is mentioned additionally. 
oxx3ooo3oox&#xt – with lacing etripy (J7) oxx3xxx3oox&#xt – with lacing etcu (J18) oox3ooo4oxx&#xt – with lacing esquipy (J8) oox3xxx4oxx&#xt – with lacing escu (J19) = biscsrico (bistratic segment of srico) xox3oxo5oox&#xt – with lacing gyepip (J11) = biscrox (bistratic segment of rox) oxx3xox5oxo&#xt – with lacing autip (J49) xoxx3oxox5ooxo&#xt – with lacing autip (J49) and gyepip (J11)  A further point of interest here might pop up in stacks of segmentochora whenever lacing cells happen to become corealmic and thus can be blended into combined Johnson solids. These for sure are esp. interesting if they won't occure from sections of wellknown uniforms already. 
oxx3xoo3oxo&#xt = (icorot + spidrot; r = 1) pt  cube  ike = (cubpy + cubaike; r = 1)  Still an other especial case of mere stacked segmentochora occurs, when these are different, cannot be considered as consecutive sections of some single Wythoffian polytope, but still happen to provide a common circumradius for all vertices of all layers. That is, whenever such towers still are orbiform. 
• wedges / lunae / rosettes  (top of CRF) 
0.209785luna of spid = {6}  trip (segmentochoron) 0.290215luna of spid = tet  tricu (segmentochoron) tetfirst rotunda of spid = tet  co (segmentochoron) 0.419569luna of gyspid = {3}  tricu  {3} 1/4luna of hex = pt  squippy (segmentochoron) vertexfirst rotunda of hex = 2/4luna of hex = pt  oct (segmentochoron) 1/6luna of ico = {3}  gyro tricu (segmentochoron) 2/6luna of ico = oct  tricu (segmentochoron) octfirst rotunda of ico = 3/6luna of ico = oct  co (segmentochoron) 1/10luna of ex 2/10luna of ex 3/10luna of ex 4/10luna of ex vertexfirst rotunda of ex = 5/10luna of ex 1/4luna of quawros = {8}  cube (segmentochoron) {4}first rotunda of quawros = 2/4luna of quawros 2/5luna of stawros = {10}  pip (segmentochoron) ncufbil (2≤n≤4) = external blend of 2 ncufs 
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 terminus lunae. Note, that 2 lunae with complemental fractions generally add to the full hemispherical polychoron, the so called rotunda. – In order to do so completely, in case of ico corealmic facets then would have to be rejoined. And, in case of ex, any thus formed cavette from pairs of peppies furthermore would have to be refilled (i.e. augmented) by 5 tet rosettes. In other words: lunae are nothing but wedgelike dissected rotundae. The sectioning applies at some vertex layer. If thereby some edges get fractioned too, those will be omitted completely in either luna. Generally, chopped cells are replaced by according Johnson solids which assure the remainder to be convex. – Therefore those would have to be replaced conversely when glueing back again. stawros does allow for a 2/5luna (but no lunae with odd numerators, esp. no rotunda). That 2/5luna would then be convex again. Even so all of the first known examples used rational fractions, the dihedral angle between those wedge facets needn't be commensurable with 360° in general. Thus we rather have to expect some real numbers r ∈ ]0, ½] (for CRFs) here. Also the ncufbils 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 nonconvex. The limitting cases already arespecial, as there either the tets become corealmic (tridpies) resp. the ncues become corealmic (squobcues. The individual cases then are the 1/2luna (or rotunda) of quawros (n=4), the 0.419569luna of gyspid (n=3) and tetep (n=2). – The corresponding mono wedges, the ncufs, all qualify as segmentochora. 
thawrowedged 1/6luna of ex (nonconvex) thawrowedged 2/6luna of ex dim. thawrowedged 1/6luna of ex (nonconvex) octadim. thawrowedged 2/6luna of ex 
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 nonunit edges. Sometimes however right that offending part can locally be rearranged, so that it becomes CRF again. This e.g. would be the case for the 1/6 and 2/6lunes of ex, replacing the (nonCRF) 3fold halfid wedge facets by thawroes, i.e. scaling down the offending central f3f into a smaller x3x. Still offending then would be the fedges, showing up in the Fo3oF&#f parts of their lace cities – not occuring within these lunes directly, but rather as cavities, which then would show up in their hulls. These however can be recovered by incident pentagons, that is by accordingly introduced peppy pairs. The 2/6lune allows this directly and so becomes a CRF. Not so for the 1/6lune. (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 (bidiminished srip) mibdirit (metabidiminished rit) {4}  tet = bidiminished rap (segmentochoron) damibdrox (deep metabidiminished rox) didamibdrox (diminished deep metabidiminished rox) 
In bidsrip the 2 diminishings are neither parallel (kind of a metabidiminishing), nor equatorial. Moreover it qualifies as a wedge, as one of the octs of srip gets reduced to its equatorial square, i.e. becomes subdimensional. Also a metabidiminished rit has to be mentioned here: the top co then will 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 bidiminished rap. In 2014 Quickfur found a metabidiminishing 36°wedge of rox, which cuts as deep as to its tid sections. Later he noticed, that the single ike, which is opposite to the wedge decagon, could be chopped off monostratically as well. Then there occurs a pabidrid (and the whole figure qualifies then as multiwedge). 
• augmentations  (top of CRF) 
omnipeppyaugmented 5,20dip omnipecuaugmented 10,20dip omnipecugyroaugmented 10,20dip ... 
Speaking of augmentations, esp. by those with 4D pyramids, the set of duoprisms provides lots of possibilities, esp. subsymmetrical or even assymmetrical ones. – A special nice finding here is the omniaugmented 5,20dip, because then some dihedral angles would become flat, thereby blending the peppies and the adjoining (unaugmented) pips into epedpies (J16)! – The same holds true for the 2 kinds of omniaugmentations of the 10,20dip, thereby blending the pecues and the adjoining (unaugmented) dips either into epobcues (J38) or into epigybcues (J39). 
oq3oo4xo xo&#zx = tes + 6 (equatorial) cubpies oq3xx4xo xo&#zx = ticcup + 6 (gyro) squipufs xw3oo4xo xo&#zx = sircope + 6 cubpies xw3xx4xo xo&#zx = gircope + 6 (gyro) squipufs oa3oo5xo xo&#zx = dope + 12 pippies oa3xx5xo xo&#zx = tiddip + 12 (gyro) pepufs xb3oo5xo xo&#zx = sriddip + 12 pippies xb3xx5xo xo&#zx = griddip + 12 (gyro) pepufs (where a = 3/sqrt(5) = 1.341641, b = a+x = 2.341641) oa3xo3oo xo&#zx = ope + 4 (alternate) trippies oa3xo3xx xo&#zx = tuttip + 4 tripufs xb3xo3oo xo&#zx = tuttip + 4 trippies xb3xo3xx xo&#zx = tope + 4 (alternate) tripufs (where a = (2+sqrt(10))/3 = 1.720759, b = a+x = 2.720759) oa3xo4oo xo&#zx = cope + 6 cubpies oa3xo4xx xo&#zx = ticcup + 6 (ortho) squipufs xb3xo4oo xo&#zx = tope + 6 cubpies xb3xo4xx xo&#zx = gircope + 6 (ortho) squipufs (where a = w/q = 1.707107, b = a+x = 2.707107) of3xo5oo xo&#zx = iddip + 12 peppies of3xo5xx xo&#zx = tiddip + 12 (ortho) pepufs xF3xo5oo xo&#zx = tipe + 12 peppies xF3xo5xx xo&#zx = griddip + 12 (ortho) pepufs (where F = ff = f+x) 
Beyond the application to duoprisms other still convex augmentations can be found for simple prisms as well. In 2014 Quickfur omniaugmented the wellknown segmentochoron doe  doe by 12 segmentochora pt  pip. Later in 2014 Gevaert independently considert an axial expansion of ex, which in the run of Klitzing's evaluation of its equatorial bistratic layer resulted in the according twelfaugmented id  id. Sure these ones then can be Stott expanded too. The iddip case together with all its expansions applies to the other o3oPo o symmetries too. (The degenerate subcase P=2 then would lead back to tautip (J51) and tauhip (J57), resp. their prisms.) But for the dope case with all its expansions the subcases for P=4 here already get limiting: neighbouring squippies or squacues (in case together with the cube or op in between for the expanded versions) become corealmic. Therefore these would blend into a single joined cell each. Note that the to be augmented prisms o3xPx x occur twice in here: they could be augmented by an according amount of "magnabicupolaic rings" (or "pucofastegia", i.e. {P}  2Pprism segmentochora) in either orientation: In fact, the Pgons therof could either align in orientation to those of o3oPx x (ortho augmentations) or to those of o3xPo x (gyro augmentations). 
quawros stawros (nonconvex) 
In early 2012 Quickfur came up with a multiple augmentation of tes, which then is nonorbiform, but likewise allows for those operations: rotunda and luna. In 2013 Klitzing found a nonconvex relative of it (stawros), which is an augmentation based on starpedip, and also does allow for a luna, but not for a rotunda. 
bicyte ausodip cyte cubau sodip cyte opau sodip cytau tes 
A further one can be obtained when augmenting alternate cubes of sodip with cubpy and all ops by {4}  op. Even so this sounds unspectacular so far, it comes out, that all squippies either cobine pairwise into octs, or unite with the remaining cubes into esquidpies, and furthermore all squacues combine pairwise into squobcues. – In fact, this polychoron comes out to be a partial Stott contraction of srit. Sure not both, the cubes and the ops have to be augmented simultanuosly. Either one could be augmented separately, resulting in 2 further CRFs, the cubau and the opauone. Similarily a tes can be augmented by cubpies at one cycle of 4 places. Then neighbouring squippies would combine into octs. In fact, that one also could be looked at as a cyclotetradiminished ico. 
octatut,tricufblend  In 2015 Gevaert found an external blend of octatut with tricuf, blending out a lacing tricu each. Similar to the axial stacks of segmentochora here some lacing trips become corealmic, thus blending in turn (subdimensionally) into gybefs. 
tisdip = tripprism (uniform) autisdip = tisdip + 1 cubpy bautisdip = tisdip + 2 cubpies tautisdip = tisdip + 3 cubpies autipip = autipprism = tisdip + 1 ortho squippyp auautipip = autipip + 1 cubpy bauautipip = autipip + 2 cubpies bautipip = bautipprism = tisdip + 2 ortho squippyps aubautipip = bautipip + 1 cubpy tautipip = tautipprism = tisdip + 3 ortho squippyps gyautisdip = tisdip + 1 gyro squippyp autodip = todip + 1 squacupe 
A 3,4dip clearly can be augmented with cubpies. The full triaugmentation then is tautisdip. Further it was observed that it well could be augmented with squippyps too. Thereby the squippies become corealmic to the trips. Therefore, in fact, it happens to be nothing but autipip. In 2016 Klitzing then even proved that both augmentation types can be used simultanuously, while still remaining convex. Shortly thereafter Quickfur found that the squippyp could also be placed in a gyrated mode onto 3,4dip. And Klitzing in turn provided then the expanded version thereof. 
baudeca = deca + 2 tetatuts  Assorted quite simple augmentations. 
• scaliforms  (top of CRF) 
spidrox  It is a scaliform polychoron with swirlsymmetry. It was found already in 2000 by G. Olshevsky. 
bidex  It is scaliform and celltransitive and thus is even a noble polychoron. In fact, its cells are 48 teddi only. It has swirlsymmetry too. It was found in 2004 by A. Weimholt. 
prissi  This scaliform polychoron first was found (in 2005 by Klitzing) as being an alternated faceting of prico. Thus, having a Dynkin symbol (s3s4o3x), W. Krieger later showed, that it is a Stott expansion of sadi. 
tutcup (segmentochoron)  Just for completeness: there is also a 4th so far known nonuniform convex scaliform polychoron, the stack of 2 antialigned tuts, connected by 4+4=8 lacing tricues and 6 tets. This monostratic figure accordingly has been published already in 2000 within the article on convex segmentochora. – In fact already shortly before the publication of this article that very polychoron initiated in a private mailing list the weakening of the axioms of uniformity, what finally became known as the notion of scaliformity. 
• nonvertextransitive highersymmetric ones  (top of CRF) 
bidsid pixhi idsrix idprix idsid pixhi 
In 2012 a nonuniform figure with exactly the same symmetry as bidex was found by Klitzing, together with 3 related nonuniform relatives of sadi. The idpart of their names relates to icositetradiminished, i.e. along the symmetry directions of the vertices of the icositetrachoron. Thus immitating the same construction, as sadi can be derived from ex. Resp. bid relating to biicositetradiminished, just as in the name of bidex. To that time it also was proven that this set is complete, provided one consideres monostratic diminishings only, and that the starting figure (the Wythoffian polychoron of hyic symmetry) has at least the first node of its Dynkin diagram being ringed. (A similar construction could be considered to a lower symmetry too: the octadiminishing, i.e. along the symmetry directions of the vertices of the hexadecachoron. That one applied onto ico clearly results in tes. But again the higher Wythoffians could be considered here too. Alas, no nonWythoffian polychoron would result: application onto spic results in srit, application onto srico results in proh, and application onto prico results in gidpith.) 
tisadi risadi idsrahi 
It is quite remarkable, that 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 a multiform CRF. E.g. tisadi is the truncated sadi, risadi is the rectified sadi. It shall be pointed out, that tisadi thereby results in a bistratic icositetradiminishing of tex, while risadi results again in a monostratic icositetradiminishing of rox. (Note that rox has its first node not being ringed, thus the find of these 2 figures in 2004 by A. Weimholt does not contradict to the former result.) In the sequel of their rediscovery of those in 2014 by Gevaert, Klitzing extended his former result, with respect to monostratic icositetradiminishings, to any Wythoffian with hyic symmetry. Besides risadi one further figure thereby emerged, idsrahi. 
cyted srit (cyclotetradiminishing) cyte gysrit (cyclotetragyration) bicyte gysrit (bicyclotetragyration) bipgy srit (bipara(bi)gyration) ... 
The relation of srit to the vertexinscribed odip, being considered as its bicyclotetradiminishing, give rise to various partial diminishings or gyrations too. 
pex hex quawros pacsid pith pexic bicyte ausodip pacsrit pex thex pabex thex pacprit poxic pocsric owau prit poc prico 
A quite powerful procedure is that of Stott 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 on partial 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:

sidsrahi sgysrahi  In 2013 Quickfur suggested a swirlsymmertic diminished srahi (the existance of which shortly after was proven by Klitzing). There exists a corresponding gyration (replacing all the diminished {5}  dip caps in a gyrated way) as well. – Moreover this might be done with respect to any of those 12 swirl cycles independently. 
cypdex, bicypdex cypdrahi, bicypdrahi cypdisrix, bicypdisrix cytid rico cytid srico cyted spic cypdrox, bicypdrox cythdex 
For ex, rahi, and srix there are mono, bi, resp. tristratic cyclical multidiminishings, which provide a regular pentagonal projection shape (for lace 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 the 5,5dip. Somehow similar is a cyclical (monostratic) diminishing of rico, which provides a regular triangular projection shape. Or a cyclical bistratic diminishing of srico, 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 (unchopped) sircoes.) The border case here is the cyclical (monostratic) diminishing of spic, which provides a square projection. (The orthogonal cycle could be diminished here too, but then reproduces just srit.) (In these cyclodiminishings any adjacent pair of diminished facets are mutually incident at some subdimensional element. Therefore those polychora would qualify as multiwedges too.) In the dawn of 2014 the author showed that also rox has a bistratic diminishing, which can be applied cyclically. But there the section planes (individually producing srid sections) would intersect. Thus this results in a pentagonal multiwedge with 5 pabidrids. – Shortly thereafter Quickfur then even constructed the bicyclically diminished version. At the beginning of 2016 Quickfur came up with a bicyclical multidiminishing of ex in its 3,3dip orientation. In the first ring he applied 3 bistratic diminishings. Here the obtained does happen to be just tiptotip. In the orthogonal ring he applied 6 monostratic diminishings. There the obtained ikes then adjoin facetoface. 
thawrorh tewau thawrorh dim. thawrowedged 1/6luna of ex (nonconvex) 
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 multiwedge too.) M. Čtrnáct then found, that it even can be augmented at any thawro independently. The full or tetraaugmented one then is tewau thawrorh. Also in 2014 Klitzing then found, that thawrorh is not too surprising as such, it just is the hull of the exterior blend of 2 dim. thawrowedged 1/6luna of ex. Btw., that latter one by itself is a multiwedge too, however a nonconvex one. 
pretasto icau pretasto 
A quite different polychoron with full demitessic symmetry and featuring teddies and bilbiroes was found by Quickfur in 2014. It was found symmetrical completion of a mere axially symmetrical relative (cf. there). Furthermore all 24 bilbiroes can be augmented by bilbiro wedges. Then the former teddies become full ikes again. 
icau prissi 
In 2014 Mrs. Krieger suggested to augment prissi at all its ikes. This then is equivalent to a partial Stott expansion of ex wrt. the same icoic subsymmetry, as prissi itself can be derived from sadi. 
foxo3xxxF3xfoo *b3oxfo&#zx (icau prissi) fooo3xxoF3xfxo *b3oxFo&#zx ooxf3foox3oxfo *b3xFxo&#zx (icau pretasto) Fxox3xoxf3oFxx *b3oxfo&#zx oFFxx3xxoof3fooxo3ooffx&#zx xFfoo3xoxxF3fxoxo3ooffx&#zx oFFxx3xxoof3fooxx3xxFFo&#zx xxfoF3oxxFx3xFxxo3Fofxx&#zx xFfxo3xoxoF3fxooo3oofFx&#zx oFfoo3ooxxF3Fxoxo3ooffx&#zx oFFxx3xxoxf3Fxxox3oofFx&#zx 
Still in 2014 Klitzing multiapplied the techniques of Gevaert (cf. EKFs) to ex, thus producing intricate facetings with demicubic subsymmetry. Ex itself can be rewritten in that subsymmetry as foxo3ooof3xfoo *b3oxfo&#zx. The used facetings then are fooo3oo(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 a Stott expansions, which then eliminate all introduced retrograde edges again, thus resulting in CRF figures.
These results could be concluded as follows: Shortly later Gevaert managed to write ex in pentic subsymmetry as xffoo3oxoof3fooxo3ooffx&#zx. That display then allows for a similar investigation. Single quirks e.g. result in (x)ffoo3xxoof3fooxo3ooffx&#zx resp. xFfoo3o(x)oof3fxoxo3ooffx&#zx, the Stott expansions of which (given at the left) Klitzing then proved to be CRF. 
3doe1papdimex 3doe1pap10mibdi10teddidimex 
In early 2016 Quickfur found a further multiwedge diminishing of ex, cutting off 3 bistratic vertexfirst caps, each with doe base, which are pairwise pentagon adjoined. The single remaining vertex of that former great circle allows for a monostratic cut off pappy. – In fact, this mere multiwedge would still have 80 vertices. Whereas the find of Quickfur only has 60, being then a further multidiminishing (each monostratic, at vertices outside that mentioned great circle), providing either mibdies or teddies as additional facets, thus reducing the count of remaining tets to just 10! 
• Euclidean 3D Honeycombs  (top of CRF) 
ditoh editoh gyeditoh pextoh pacratoh erich gyrich * gyerich pexrich pacsrich 5Y44T4P4 * 5Y44T6P3sqpara * 5Y44T6P3sqskew * 10Y48T0 * 10Y48T1alt * 10Y48T1hel (r/l) * 10Y48T2alt * 10Y48T2hel (r/l) * 10Y48T3 * 5Y44T6P3tri0 * 5Y44T6P3tri1alt * 5Y44T6P3tri1hel (r/l) * 5Y44T6P3tri2alt * 5Y44T6P3tri2hel (r/l) * 5Y44T6P3tri3 * 3Q4T2P8P4 * 6Q42T * 6Q32S3gyro * 6Q32S3ortho * 3Q3S32P62P3gyro * 3Q3S32P62P3ortho * cubedoebilbiro 
There are some few euclidean 3D honeycombs known, which count as most as CRF. * Those being marked by an asterisk would classify moreover as scaliform. – Most of them have been found in 2005 by J. McNeill. He then called them elementary honeycombs. So, the different stacking modes (alt, hel (r), hel (l)) still remained undiscovered until 2013. The cubedoebilbiro was found in 2004 by A. Weimholt. 
© 20042017  top of page 