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Johnson solids

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 regular-faced polyhedra with conditional edges. And the corresponding list then is provided here.

Blind polytopes   (up)

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.
°) 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
     e.g. idimex.

CRFs   (up)

There is also a different possibility to extrapolate from the set of Johnson solids into higher dimensions. Instead of requiring (n-1)-dimensional facets to be regular, one rather could stick to 2-dimensional 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 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, 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.

Known 4D CRFs

(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 so-called expanded 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.)

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.

 • non-orbiform monostratics(top of CRF)
n-py || inv gyro n-py   (2<n<6)
   = external blend of 2 n-appies
n-cu || inv gyro n-cu   (6/5<n<6)
   = external blend of 2 n-pufs

line || bilbiro
{3} || thawro
{5} || pocuro

{3} || gybef
   = external blend of traf and trippy)
n-py || inv ortho n-py   (2<n<6)
   = external blend of 2 n-pipies
n-cu || inv ortho n-cu   (6/5<n<6)
   = external blend of 2 n-pufs
{4} || squobcu
   = external blend of 2 squacufs

xfox oxfo3ooox&#xt (3-mibdi-wedge)
xfoxo oxfox5ooofx&#xt (5-mibdi-wedge)

mono-augmented 3-mibdi-wedge
bi-augmented 3-mibdi-wedge
xofox ooxfo3oxoox&#xt (tri-augmented 3-mibdi-wedge)

In 2012 Quickfur came up with 2 non-orbiform 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 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 external blend of the n-cuf and the n-puf 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)
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 tip-to-tip distance of exactly one edge unit, it can be decomposed itself in turn into 6 squascs.
ofx3xoo3ooo&#xt = tetu
ofx3xoo4ooo&#xt = octu
ofx3xoo5ooo&#xt = iku
   = icosiena-diminished rotunda of ex

ofx3xoo xxx&#xt = teddipe
ofx3xoo3xxx&#xt = coatutu
ofx3xoo4xxx&#xt = sirco aticu
ofx3xoo5xxx&#xt = sridatidu

   = icosi-diminished rotunda of ex

ofxo3xooo xxxx&#xt

   = mono-diminished rotunda of ex

42-diminished sidpixhi
42-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, 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 teddi-polychora (and their higher dimensional analogues) winkingly was attributed the name of ursachora (resp. ursatopes). – The existence of the infinite series of non-expanded members with simplectic symmetry even was predated by A. Weimholt in 2004. He then already pointed out the orbiformity of any of this subset.

There is a separate section on ursatopes wrt. general dimensions too.

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 non-neighbouring teddies in order to remain convex. For the icosahedral the same would hold true at the first sight, but the cavities at the peppy-join then could be bridged there by additional 5-tet-rosettes.

In 2016 Quickfur and the author came up with a cellwise expanded version of the tetrahedral ursachoron, which then was the first non-trivial 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 21-diminishing of sidpixhi. It then even will be possible to apply that to both hemiglomes at the same time, resulting in an according 42-diminishing. – 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 21-diminishing of the ex rotunda.

elongated ico
A special case of augmentation occurs when glueing 2 oct-first rotundae of ico at both sides of a cope. Clearly a tristratic polychoron. The peculiar clue here is that the squippies become co-realmic to the cubes, thereby blending into esquidpies (J15)!
xux3oox3ooo&#xt = tip (uniform)
xux3oox3xxx&#xt = coatotum
   = bistratic co-cap of prip
     (tut-diminished prip)
xux3oox4ooo&#xt = oox3xux3oox&#xt = octum
   = oct-first rotunda of thex
xux3oox4xxx&#xt = sircoa gircotum
   = bistratic sirco-cap of prit
   = vertex-first rotunda of ex
   = bistratic id-cap of rahi
   = tristratic id-cap of srix
xux3oox5ooo&#xt = iktum
   = bistratic ike-cap of tex
xux3oox5xxx&#xt = srida gridtum
   = bistratic srid-cap of prahi
   = tristratic co-cap of rico
     (dirico = co-diminished rico)
x(ou)x3o(xo)x x(uo)x&#xt
   = bistratic trip-cap of srip
     ({3}-diminished srip)

   = bistratic vertex-first central segment of ico
     (pabdico = parabidiminished ico)
   = bistratic co-first central segment of rico
     (pabdirico = parabidiminished rico)
   = bistratic oct-first central segment of spic
     (dapabdi spic = deep parabidiminished spic)
   = tristratic tic-first central segment of proh
     (pabdiproh = parabidiminished proh)

   = bistratic id-subsegment of srix
     (diminished tristratic id-cap)
   = tristratic vertex-first subsegment of ex
     (mono-diminished rotunda of ex)

   = hexastratic ike-cap 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.

The class of tutsatopes 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.

   = bistratic vertex-first cap of ex
   = * pt || ike || doe
   = dodeca-diminished rotunda of ex
   = pt || ike || doe || id
   = trideca-diminished rotunda of ex
   = ike || doe || id
   = icosi-diminished rotunda of ex
   = icosiena-diminished rotunda of ex
   = deep mono-diminished rotunda of ex
   = 12-augmented doe || id
   = deep trideca-diminished rotunda of ex
   = *  doe || id
ike || tet-dim-doe || id
ike || cube-dim-doe || id

   = gyrated spid (ortho bicupola)
   = *  tet || co || tet
   = gyrated prip
   = 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 \ fq-oct) || f-ike || 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 around-symmetrical 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.



thawro pyrite

..oofoo..3..oxoxo..5..xooox..&#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 id-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 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 ike-first 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 axial-pyritohedrally 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.

   = A1 + B1
   = A2 + B1
   = A1 + B2
   = A2 + B2
A1 = xf|oxF|...-3-ox|Fxo|...-3-oo|ofx|...&#xt
B1 = ox|Fxo|...-3-xo|oxF|...-3-of|xfo|...&#xt
A2 =  o|oxF|...-3--x|Fxo|...-3--x|ofx|...&#xt
B2 = ox|Fxo|...-3-xf|oxF|...-3-oo|xfo|...&#xt
(vertical & horizontal lines introduced for comparision purposes only)

   = pretasto
(these both incorporate most of B1 non-axially!)

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
   = ooxwxoo-3-xxxxxxx-4-oxxoxxo&#xt
ox.. wx..3xx..4ox..&#zx
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).
   (rico-cap + thex-rot)
   (proh-cap + bistratic prit-cap)
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.
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)
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 well-known uniforms already.
   = (ico-rot + spid-rot; 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)
0.209785-luna of spid 
   = {6} || trip (segmentochoron)
0.290215-luna of spid 
   = tet || tricu (segmentochoron)
tet-first rotunda of spid 
   = tet || co (segmentochoron)

0.419569-luna of gyspid
   = {3} || tricu || {3}

1/4-luna of hex 
   = pt || squippy (segmentochoron)
vertex-first rotunda of hex 
   = 2/4-luna of hex
   = pt || oct (segmentochoron)

1/6-luna of ico
   = {3} || gyro tricu (segmentochoron)
2/6-luna of ico
   = oct || tricu (segmentochoron)
oct-first rotunda of ico
   = 3/6-luna of ico
   = oct || co (segmentochoron)

1/10-luna of ex
2/10-luna of ex
3/10-luna of ex
4/10-luna of ex
vertex-first rotunda of ex
   = 5/10-luna of ex

1/4-luna of quawros
   = {8} || cube (segmentochoron)
{4}-first rotunda of quawros
   = 2/4-luna of quawros

2/5-luna of stawros
   = {10} ||  pip (segmentochoron)

n-cufbil (2≤n≤4)
   = external blend of 2 n-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 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 co-realmic facets then would have to be re-joined. And, in case of ex, any thus formed cavette from pairs of peppies furthermore would have to be re-filled (i.e. augmented) by 5 tet rosettes.

In other words: lunae are nothing but wedge-like 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 2/5-luna (but no lunae with odd numerators, esp. no rotunda). That 2/5-luna 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 n-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 the tets become corealmic (tridpies) resp. the n-cues become corealmic (squobcues. The individual cases then are the 1/2-luna (or rotunda) of quawros (n=4), the 0.419569-luna of gyspid (n=3) and tetep (n=2). – The corresponding mono wedges, the n-cufs, all qualify as segmentochora.

thawro-wedged 1/6-luna of ex (non-convex)
thawro-wedged 2/6-luna of ex

dim. thawro-wedged 1/6-luna of ex (non-convex)
octa-dim. thawro-wedged 2/6-luna 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 non-unit edges. Sometimes however right that offending part can locally be re-arranged, so that it becomes CRF again.

This e.g. would be the case for the 1/6- and 2/6-lunes of ex, replacing the (non-CRF) 3-fold half-id wedge facets by thawroes, i.e. scaling down the offending central f3f into a smaller x3x. Still offending then would be the f-edges, 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 2/6-lune allows this directly and so becomes 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 (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 sub-dimensional.

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 multi-wedge).

 • augmentations(top of CRF)
omni-peppy-augmented 5,20-dip
omni-pecu-augmented 10,20-dip
omni-pecu-gyroaugmented 10,20-dip

Speaking of augmentations, esp. by those with 4D pyramids, the set of duoprisms provides lots of possibilities, esp. sub-symmetrical or even assymmetrical ones. – A special nice finding here is the omni-augmented 5,20-dip, because then some dihedral angles would become flat, thereby blending the peppies and the adjoining (un-augmented) pips into epedpies (J16)! – The same holds true for the 2 kinds of omni-augmentations of the 10,20-dip, thereby blending the pecues and the adjoining (un-augmented) 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 omni-augmented the well-known 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 twelf-augmented 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 sub-case 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 sub-cases 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} || 2P-prism segmentochora) in either orientation: In fact, the P-gons therof could either align in orientation to those of o3oPx x (ortho augmentations) or to those of o3xPo x (gyro augmentations).


stawros (non-convex)

In early 2012 Quickfur came up with a multiple augmentation of tes, which then is non-orbiform, but likewise allows for those operations: rotunda and luna.

In 2013 Klitzing found a non-convex 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 un-spectacular 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 opau-one.

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.

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     = trip-prism (uniform)
autisdip   = tisdip + 1 cubpy
bautisdip  = tisdip + 2 cubpies
tautisdip  = tisdip + 3 cubpies

autipip    = autip-prism
           = tisdip + 1 ortho squippyp
auautipip  = autipip + 1 cubpy
bauautipip = autipip + 2 cubpies

bautipip   = bautip-prism
           = tisdip + 2 ortho squippyps
aubautipip = bautipip + 1 cubpy

tautipip   = tautip-prism
           = tisdip + 3 ortho squippyps

gyautisdip = tisdip + 1 gyro squippyp
autodip    = todip + 1 squacupe

A 3,4-dip 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 3,4-dip. And Klitzing in turn provided then the expanded version thereof.

baudeca = deca + 2 tetatuts
Assorted quite simple augmentations.
 • scaliforms(top of CRF)
It is a scaliform polychoron with swirl-symmetry. It was found already in 2000 by G. Olshevsky.
It is scaliform and cell-transitive and thus is even a noble polychoron. In fact, its cells are 48 teddi only. It has swirl-symmetry too. It was found in 2004 by A. Weimholt.
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 non-uniform 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.
 • non-vertex-transitive higher-symmetric ones(top of CRF)
sadi = idex (uniform)
tisadi = idtex
risadi = idrox

tidimex = idimtex
ridimex = idimrox

It is quite remarkable, that the non-regular (even so 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 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 24-diminishing of tex, while risadi results again in a monostratic 24-diminishing 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 below result of 2012.)

Just as sadi is a (uniform) 24-diminishing of ex, 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 for truncation or rectification, what happens to be even more surprising, because idimex not even is uniform!

bidex (noble)
   = x3o3o5o - 48 ikepies
bidsid pixhi
   = x3o3o5x - 48 doasrids

sadi (uniform)
   = x3o3o5o - 24 ikepies
   = x3o3x5o - 24 idaties
   = x3o3x5x - 24 tidagrids
idsid pixhi
   = x3o3o5x - 24 doasrids
idrox = risadi
   = o3x3o5o - 24 ikaids
   = o3x3o5x - 24 sridatids

idtex = tisadi
   = x3x3o5o - 24 (bistratic) iktums
   = x3x3o5x - 24 (bistratic) srida gridtums

   = x3o3o5o - 20 ikepies
   = x3o3x5o - 20 idaties
   = x3o3x5x - 20 tidagrids
idim sidpixhi
   = x3o3o5x - 20 doasrids
idimrox = ridimex
   = o3x3o5o - 20 ikaids
   = o3x3o5x - 20 sridatids

idimtex = tidimex
   = x3x3o5o - 20 (bistratic) iktums
   = x3x3o5x - 20 (bistratic) srida gridtums

In 2012 a non-uniform figure with exactly the same symmetry as bidex was found by Klitzing, together with 3 related non-uniform relatives of sadi. The id-part of their names relates to icositetra-diminished, 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 bi-icositetra-diminished, 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.

In the sequel of their rediscovery of tisadi = idtex and risadi = idrox in 2014 by Gevaert, Klitzing extended his former result, with respect to monostratic 24-diminishings, to any 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 the 8-diminishing, i.e. along the symmetry directions of the vertices of the hexadecachoron. That one applied onto ico clearly results in tes. But then again the higher Wythoffians could be considered here too. Alas, no non-Wythoffian polychoron would result in this case: application onto spic results in srit, application onto srico results in proh, and application onto prico results in gidpith.)

Immediately after the finds of idimex, tidimex, and ridimex (cf. above) "Username5243" pointed out that the applicability within this subsection might be possible too. This clearly is the case, resulting in the various 20-diminishings, i.e. along the symmetry directions of the vertices of the small 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 of srit to the vertex-inscribed odip, being considered as its bi-cyclo-tetra-diminishing, give rise to various partial diminishings or gyrations too.

pex hex
pacsid pith

bicyte ausodip

pex thex
pabex thex


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:
- In early 2012 Quickfur came up with quawros as a multiple augmentation of tes.
- In 2012 Quickfur came up with owau prit as an augmentation of prit by oct || sirco.
- Though bicyte ausodip was obtained as member of those partial Stott expansion series, it well could be obtained when augmenting every alternate cubes of sodip with a cubpy and any op by a {4} || op.
- Though poxic was obtained as member of those partial Stott expansion series too, it well could be obtained when augmenting the 8 full symmetrical cubes of sidpith by cubpies.

In 2013 Quickfur suggested a swirl-symmertic diminished srahi (the existance of which shortly after was proven by Klitzing). There exists a corresponding gyration (re-placing 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



For ex, rahi, and srix there are mono-, bi-, resp. tristratic cyclical multi-diminishings, 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 5,5-dip.

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 (un-chopped) 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 cyclo-diminishings any adjacent pair of diminished facets are mutually incident at some subdimensional element. Therefore those polychora would qualify as multi-wedges 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 multi-wedge with 5 pabidrids. – Shortly thereafter Quickfur then even constructed the bicyclically diminished version.

At the beginning of 2016 Quickfur came up with a bicyclical multi-diminishing of ex in its 3,3-dip orientation. In the first ring he applied 3 bistratic diminishings. Here the obtained does happen to be just tip-to-tip. In the orthogonal ring he applied 6 monostratic diminishings. There the obtained ikes then adjoin face-to-face.

Clearly here the uniform gap would qualify as bicyclical decadiminishing of ex. A true CRF so would be the according monocyclical decadiminishing cydadex.

tewau thawrorh
dim. 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 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. thawro-wedged 1/6-luna of ex. Btw., that latter one by itself is a multi-wedge too, however a non-convex one.

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




Still in 2014 Klitzing multi-applied 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:
0 double quirks result in icau prissi
1 double quirks result in fooo3xxoF3xfxo *b3oxFo&#zx
2 double quirks then become non-CRF
3 double quirks then become non-CRF
single quirk to 1 arm results in icau pretasto
single quirk to 2 arms results in Fxox3xoxf3oFxx *b3oxfo&#zx
single quirk to 3 arms then becomes non-CRF

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.


In early 2016 Quickfur found a further multi-wedge diminishing of ex, cutting off 3 bistratic vertex-first 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 multi-diminishing (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)


gyrich *

5Y4-4T-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 *


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 un-discovered until 2013.

The cube-doe-bilbiro was found in 2004 by A. Weimholt.

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