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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 - gyrobifastigium
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
J4  - °) 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 releaced. 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

In 1979 Roswitha Blind applied this idea to higher dimensional polytopes: they are bound to be convex, built from regular facet-polytopes, but not being uniform. 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 dipyramid
ikedpy = pt || ike || pt -  †) icosahedral dipyramid
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


CRFs

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.


Known 4D CRFs

(unsorted collection only – beyond those already contained within other classes, esp. like Wythoffians or segmentochora. Segmentochoral stacks are mainly suppressed here either.)

CRFsRemarks
 • non-orbiform monostratics(top of CRF)
n-py || inv gyro n-py   (2<n<6)
n-cu || inv gyro n-cu   (6/5<n<6)

line || bilbiro
{3} || thawro

In 2012 Quickfur came up with 2 non-orbiform monostratic families, described in more detail here, esp. their close relationship to segmentochora.

In 2014 two close relatives to pyramids have been found, which are not orbiform just because their base is not.

 • axials(top of CRF)
ofx3xoo3ooo&#xt
ofx3xoo4ooo&#xt
ofx3xoo5ooo&#xt

ofx3xoo xxx&#xt
ofx3xoo3xxx&#xt
ofx3xoo4xxx&#xt
ofx3xoo5xxx&#xt

ofxo3xooo xxxx&#xt
ofxo3xooo3oooo&#xt
ofxo3xooo3xxxx&#xt

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).

Similar to teddi itself (yielding auteddi), the tetrahedral ursachoron can be augmented at its tetrahedral base too by an attached pen. Same holds true then for its expanded version. And for sure, the prism of auteddi itself exists as well.

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)!
xux3oox3xxx&#xt
   = bistratic co-cap of prip
     (tut-diminished prip)
xux3oox4ooo&#xt
   = oct-first rotunda of thex
xux3oox4xxx&#xt
   = bistratic sirco-cap of prit
oxofo3oooox5ooxoo&#xt
   = vertex-first rotunda of ex
ooo3xox5ofx&#xt
   = bistratic id-cap of rahi
oxxx3xxox5oofx&#xt 
   = tristratic id-cap of srix
xux3oox5ooo&#xt
   = bistratic ike-cap of tex
xux3oox5xxx&#xt
   = bistratic srid-cap of prahi
oxxx3xxox4ooqo&#xt
   = 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)
...

xxx3xox4oqo&#xt
   = bistratic co-first central segment of rico
     (pabdirico = parabidiminished rico)
oqo3xxx4xox&#xt
   = bistratic oct-first central segment of spic
     (dapabdi spic = deep parabidiminished spic)
xxxx3xoox4xwwx&#xt
   = tristratic tic-first central segment of proh
     (pabdiproh = parabidiminished proh)
...

.xxx3.xox5.ofx&#xt
   = bistratic id-subsegment of srix
     (diminished tristratic id-cap)
.xofo3.ooox5.oxoo&#xt
   = tristratic vertex-first subsegment of ex
     (mono-diminished rotunda of ex)
...

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.o3ooo.x5oox.o&#xt
   = dodeca-diminished rotunda of ex
   = *  pt || ike || doe || id
.xo.o3.oo.x5.ox.o&#xt
   = trideca-diminished rotunda of ex
   = *  ike || doe || id
.x.fo3.o.ox5.o.oo&#xt
   = icosiena-diminished rotunda of ex
..ofo3..oox5..xoo&#xt
   = deep mono-diminished rotunda of ex
..o.o3..o.x5..x.o&#xt
   = deep trideca-diminished rotunda of ex
   = *  doe || id
...

xxx3ooo3oxo&#xt
   = gyrated spid (ortho bicupola)
   = *  tet || co || tet
oxux3xxoo3xxxx&#xt
   = gyrated prip
...

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.)

xFoFx3ooooo5xofox&#xt
xFoFx3xxxxx5xofox&#xt

oxFx3xfox5xoxx&#xt
xofxF(Vo)Fxfox3xFxoo(xo)ooxFx5xoxFf(oV)fFxox&#xt
xofxfox3xFxoxFx5xoxFxox&#xt

oxFx3xoox5oxoo&#xt
thawro pyrite

(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 "student91" 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. Ctrnáct 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 "student91" shortly before.

ox|Fxo|fx-3-xo|oxF|xo-3-of|xfo|oo&#xt
   = A1 + B1
ox|Fxo|o--3-xo|oxF|x--3-of|xfo|x-&#xt
   = A2 + B1
ox|Fxo|fx-3-xf|oxF|xo-3-oo|xfo|oo&#xt
   = A1 + B2
ox|Fxo|o--3-xf|oxF|x--3-oo|xfo|x-&#xt
   = A2 + B2
where:
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)

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 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

"Student91" 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 hex symmetry! (Cf. also section higher symmetrics.)

 • 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)

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 nominators, 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.

bidsrip (bidiminished srip)

mibdirit (metabidiminished rit)

{4} || tet = bidiminished rap (segmentochoron)

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.

 • augmented (duo)prisms(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).

quawros

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
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.
oqo3xxx4xox&#xt (cf. axials)
   = bistratic oct-first central segment of spic
     (dapabdi spic = deep parabidiminished spic)
...
Even other prisms could be augmented by various caps too. E.g. any prism o3xNx x could be augmented by an according amount of "magnabicupolaic rings" (i.e. {N} || 2N-prism segmentochora). In fact, the N-gons therof could either align in orientation to those of o3oNx x (ortho augmentations) or to those of o3xNo x (gyro augmentations).
N = 4 is thereby somehow special: In that special case the circumradius of the cap and that of the augmented body are the same, in fact its gyro augmentation comes out to be the bistratic oct-first central segment of spic.
 • 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 a noble polychoron. In fact, its cells are 48 teddi 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 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, 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 this very polychoron made up that weakening of uniformity in a private mailing list, what thereafter became known as scaliformity.
 • non-vertex-transitive higher-symmetric ones(top of CRF)
bidsid pixhi

idsrix
idprix
idsid pixhi

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.

(A similar construction could be considered to a lower symmetry too: the octa-diminishing, 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 non-Wythoffian 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 icositetra-diminishing of tex, while risadi results again in a monostratic icositetra-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 former result.)

In the sequel of their rediscover of those in 2014 by a student, Klitzing extended his former result, with respect to monostratic icositetra-diminishings, to any Wythoffian with hyic symmetry. Besides risadi one further figure thereby emerged, idsrahi.

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
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:
- 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.

sidsrahi
sgysrahi
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.
bicyp drahi
bicyp disrix

cytid rico
cytid srico

cyted spic

cypdrox

For rahi and srix there are 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 too, showing up some 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.)

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. (Whether here the orthogonal cycle also could be diminished, resp. whether those then would intersect with the former, eludes me so far.)

thawrorh
tewau thawrorh

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. Ctrnáct then found, that it even can be augmented at any thawro independently. The full or tetra-augmented one then is tewau thawrorh.

pretasto

A quite different polychoron with full hexadecachoral 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).

 • Euclidean 3D Honeycombs(top of CRF)
ditoh
editoh
gyeditoh

pextoh
pacratoh

erich
gyrich *
gyerich
pexrich
pacsrich

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 *

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 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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