| 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 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 |
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 |
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.
(unsorted collection only – beyond those already contained within other classes, esp. like Wythoffians or segmentochora. Segmentochoral stacks are mainly suppressed here either.)
| CRFs | Remarks |
|---|---|
| • 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) | In 2012 Quickfur came up with 2 non-orbiform monostratic families, described in more detail here, esp. their close relationship to segmentochora. |
| • axials | (top of CRF) |
ofx3xoo3ooo&#xt ofx3xoo4ooo&#xt ofx3xoo5ooo&#xt ofx3xoo xxx&#xt ofx3xoo3xxx&#xt ofx3xoo4xxx&#xt ofx3xoo5xxx&#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. |
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 (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 (parabidiminished rico) xxxx3xoox4xwwx&#xt = tristratic tic-first central segment of proh (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, one of its hyperplane is tangential, the more specific term cap is used. A cap with its other hyperplane 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.) |
| • wedges / lunae / rosettes | (top of CRF) |
1/4-luna of hex = pt || squippy (segmentochoron) vertex-first rotunda of hex = pt || oct (segmentochoron) 2/6-luna of ico = oct || tricu (segmentochoron) 1/6-luna of ico = {3} || gyro tricu (segmentochoron) oct-first rotunda of ico = oct || co (segmentochoron) 4/10-luna of ex 3/10-luna of ex 2/10-luna of ex 1/10-luna of ex vertex-first rotunda of ex |
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. |
quawros 1/4-luna of quawros = {8} || cube (segmentochoron) {4}-first rotunda of quawros |
In early 2012 Quickfur came up with an 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 2/5-luna (but no rotunda). |
bidsrip (bidiminished srip) | 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. |
| • augmented (duo)prisms | (top of CRF) |
omni-augmented 5,20-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)! |
... | Even other prisms could be augmented by various caps too. E.g. any prism x xNx3o could be augmented by an according amount of "magnabicupolaic rings" (i.e. {N} || 2N-prism segmentochora). (N = 4 is somehow special here. In that special case the circumradius of the cap and that of the augmented body are the same, in fact that one comes out to be a special diminishing 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. |
| • 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. (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.) |
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. |
owau prit | In 2012 Quickfur came up with an augmentation of prit by oct || sirco. It comes out to be identic to the Stott expansion of just 8 out of one class of octs, being applied to spic. Note esp. that thereby all 24 octs of the other class become elongated into esquidpies (J15)! |
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 |
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 incident to a subdimensional element. Therefore those polychora would qualify as multi-wedges too.) |
© 2004-2013 | top of page |