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Definitions and Asides

Compounds are defined to be multicovers of a single spot by more than just a single polytope. Sure there are infinitely many such possibilities, even continuously related. So the more interesting case comes up, when there is an overall symmetry (not needed to be the same of the used polytopes), under the action of which the total compound is preserved. Sure, there are slightly shifted copies too, which make up such a symmetrical compound: the center of that shift vector serves for inversonal symmetry. Thus, the symmetrical compounds of real interest would require an overall symmetry, which at least is the same (or probably larger) than that of its components. Examples here are all compounds of dual pairs (which by definition have the same overall symmetry as each component – except for selfduals, where symmetry is increased by the exchange of those 2 components) or even the ones given in the lower part of plate 7 (there, in the first 2 cases, componenets are considered under pyritohedral symmetry only, while the compound symmetry comes out to be full cubical).

Compounds moreover are called uniform, if within the definition of polytopal uniformity we replace vertex transivity by vertex figures transivity. This is because for compounds it might be possible to have still no completely coincident elements of subdimension larger than 0, but having coincident vertices none the less. There, this stated demand asks for some symmetry, which interchanges those coincident vertices of the related components as well.

As an aside one might ask whether this uniformity already requires compounds to be isohedral. This can be answered kind of depending: Consider 2 (or more) vertex coincident isohedral compounds (or polytopes), both of the same edge-length, and take the (mere) compound of those. The above stated demand now would require some symmetry which interchanges the vertex figures of the components. This generally would not work. But if we thereafter would identify coincident vertices, the picture changes. We no longer have coincident vertices with single vertex figures each, but we get single vertices with compound vertex figures. Consequently that additional above requirement brakes down, as there is nothing left to be interchanged. (It only remains the question, whether we still have true compounds, or if we would have changed this status, slightly towards mere polytopes, by the application of identification of coinciding vertices.)

Accordingly, more generally we could consider compounds of (possibly) different components, whenever those components have the same circumradius and moreover are inscribed into a common convex hull, while the subset of components of each type already makes up the full overall symmetry of the total compound, and further all those types of components will have the same edge length. (This will brake down in 2D, as different polygons, with the same edge-length, always have different circumradii.) Just to provide examples:

Note that for such uniform compounds (with identified vertices) the demand on uniformity of individual components would not hold! Just consider the (mere) compound of 12 (scaliform) pentagonal pyramids, vertex-inscribed into an icosahedron (which itself will not belong to that compound). That compound thus even qualifies as isohedral. If it additionally is asked to be uniform, from the damand on vertex figure transitivity it becomes clear that vertices need to be identified. – Thus we then constructed a uniform compound (with identified vertices) from non-uniform components! (In fact a hollow and moreover Grünbaumian one, as triagles are completely coincident by 3 each.)

Thus, as long as vertex identification does not take place, uniform compounds need not only be isogonal (by definition), but come out to be isohedral as well. And likewise their components too will be uniform polytopes only.

A compound can be defined to be regular, just like a polytope, when it is transitive on all sub-dimensional elements. Moreover, there are further qualifiers of that type, introduced by Coxeter, vertex-regular and (dually) face-regular (or rather using a corresponding qualifier for the (n-1)-dimensional element). These are achieved, when the vertices (resp. the facet-planes) belong to a regular polytope. That is, when the encasing convex hull (resp. the common intersection kernel) would be a regular polytope. Or, again stated in an other way, vertex-regular compounds occure as facetings of regular polytopes, while face-regular compounds occure as stellations of regular polytopes.

For (isohedral) vertex-regular compounds and their duals Coxeter introduced the following notation:

a P [b Q] c R

where Q denotes the (regular) component polytopes (therefore, like P and R too, usually written as Schläfli symbol), b is the count of components. If vertex-regular, one has an encasing regular polytope P, the vertices of which will be used by a components each. Conversely, if face(t)-regular, there is a common intersection kernel which is a further regular polytope R, again possibly with its facet-planes being used by c components each. If vertex-regular but not face(t)-regular, the final part behind the closing square bracket will be omitted; conversely, if face(t)-regular but not vertex-regular the part before the opening one is omitted. (Examples will be given below, as far as applicable. The facet-regular case for Q being a simplex and b=2 will be considered through all dimensions at the end.)

Compounds need some extension to the "normal" incidence matrix description, which is due to their multiple components. The appropriately extended description can be found here, and will be used within the individual compound files linked below.

Some compounds show up an overall symmetry which is the same as that of the individual components too. Then those compounds can be described by a (stacked) Dynkin symbol. Further, compounds of 2 components occasionally might be described as both of the alternated facetings (i.e. compound of snubs). Then they can be described by an holosnub notation.

Some Related Topics:

Closely related with the topics of (generall, i.e. not necessarily uniform) compounds is the topic of blends, a concept intoduced by Olshevsky. Likewise it might deal with multicovers of a single spot by more than one polytope (infact, here the number generally will be 2, asked by the dyadicity argument used below). For blends this furthermore asks for having at least one of its facets each completely coincident. The blend then will be set up by the reduction of the component polytopes by these coincident facets, re-adjoining these reductions dyadically at those coincident (open) ridges. Therefore, blends (of polytopes) are still true polytopes. Also there are 2 kinds of blends, internal and external, depending on having the polytopal centers on the same resp. on different sides of the facetal hyperplane. For instance, many of the Johnson solids are external blends of easier components. (The idea of blends sure can also be applied to compounds instead of mere polytopes.) For an explicite example of an internal blend consider the one built from 2 pentagonal prisms (pip), re-adjoined (internally) at a lacing square, having the componental axes arranged orthogonally (tupip). – The withdrawn original doubled up facets clearly still hold their shapes, just having no body any more, therefore those could be spoken of as pseudo facets.

It further might occur that subsets of uniform polytopes of the same regiment form a group under the action of blending. Such a group then is called a cohort. For instance co, oho, and cho make up such a cohort of 3. (In fact, here either one can be produced as the blend of the other two.)

Fissary Polytopes,   Complexified Polytopes,   and   Exotic Polytopes

The higher the dimension the count of possible polytopes increases exponentially. So some restrictions onto what should be considered a "true" polytope might be in place. Dyadicity, i.e. every edge has exactly 2 ends, and, dually, every ridge connects exactly 2 facets, is rather generally accepted. Compounds on the other side are excluded. But even then the realm of uniform polytopes becomes rather huge. Even for 4D the count of known uniform polychora runs way beyond 8000, Grünbaumian figures with completely coincident elements not even included. This situation was the reason for Johnson and Bowers to introduce the following attributes for in a stricter sense further on to be excluded "polychoroids".

One being the attribute fissary, kind as a midway between "true" polytopes and compounds, attributed to non-compound polytopes which have a compound vertex figure.

Dual to fissary polytopes would be figures using compounds for facets (i.e. d-1-faces), but still are not themselves compounds. Those are called complexified polytopes. – In fact this situation happens to occur quite often beyond 3D. Consider facets showing up both, the same symmetry and the same circumradius. Accordingly those would have to occur as facets on the same sub-symmetry axis of the polytope of consideration, and on that axis moreover they have to be placed at the same distance. I.e. those become co-realmic. In case they are not identic sub-polytopes and so would be completely coincident (then could be blended out, for instance), those will become compounds (not necessarily uniform ones, even so their components are uniform, confer the semicompounds mentioned above). – Because those complexified polytopes need some multiwrap (more than one facet occupies the same direction) those clearly cannot occur for convex figures. In contrary they come rather close to Grünbaumian figures.

Then the exotic ones, attributed to polytopes with completely coinciding ridges (i.e. d-2-faces). Even so those exotic polytopes are well-behaved dyadic abstract polytopes, their Grünbaumian realisation for some authors would give rise to "see" their coincidic elements being identified. This shows their closeness to non-dyadic "polytopes". Examples here are gidisdrid and cid in 3D (with coincident edges), or seedatepthi in 4D (with coincident pentagons).

Uniform compounds

---- 2D ----

For uniform compounds in 2D we only have 2 possibilities. The first possibility occurs whenever any number k of n/d-gons (k and d free of common divisors) is arranged within an encasing (approprately scaled) kn-gon (vertex-regular); cf. the picture below, where k=n=3 and d=1). By self-duality of polygons one likewise could say, that those are derived as stellations of their common intersection, i.e. a smaller copy of that kn-gon (edge-regular). In general this is what Coxeter denotes as {kn}[k{n/d}]{kn}. Obviously these are the regular compounds of this dimensionality. By definition this type of compound clearly is a static one. Even so, it might occure (for k=2) as limiting case of the following, continuos case. – For the cases with k=2 also a closed Dynkin notation can be provided:

{2n}[2{n/d}]{2n} = β-n/d-β = xo-n/d-ox

The other possibility here contains a continuous parameter, but uses 2 components only. Those compounds are obtained by any specific uniform polygon, which is mirrored at any line which itself is not a mirror symmetry of that first component. Obviously, there is a continuous range of such choices for the angle to be taken between 2 such (neighbouring) mirrors. – Instead of 2 copies of some simple polygon as components, also 2 copies of some regular compound, i.e. one of the first kind, could be used alike. Then, for sure, the total number of components will become accordingly twice the component count of either of those 2 involved component compounds.

regular polygonal compound {9}[3{3}]{9} within an encasing polygon (resp. around a polygon) continuously deformable uniform compound of 2 polygons (or of 2 regular compounds)

Some regular ones bear specific own names too. E.g.
•  {6}[2{3}]{6} is known as the Star of David, and
•  {8}[2{4}]{8} is known as the Star of Lakshmi.

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---- 3D ----

The listing of uniform compounds of this dimension was done by Skilling in 1976. There are again those 2 possibilities. They all are given within the following pictures of Bowers. For the static ones the encasing polyhedron no longer needs to be uniform itself, its edges might have different sizes. And for the deformable ones, the number of components no longer is bound to be 2.

The set of regular 3D compounds ia made up from so, ki, e, rhom, and se. As given below, the first three thus would be fully regular (both vertex- and face-regular), while rhom is only vertex-regular, conversely its dual, se, is just face-regular.

so (stella octangula) - S04
   = self-dual compound of 2 tet 
     within an encasing cube
     common intersection being an oct
     {4,3}[2{3,3}]{3,4} cf. details

ki (chiro-icosahedron) - S05
   = self-dual compound of 5 tet 
     within an encasing doe
     common intersection being an ike

e (icosiicosahedron) - S06
   = self-dual compound of 10 tet
     within an encasing doe
     common intersection being an ike
     vertices coincide by pairs

rhom (rhombihedron) - S09
   = compound of 5 cube
     edge-faceting of sidtid
     within an encasing doe
     common intersection being a rhote
     vertices coincide by pairs

se (small icosiicosahedron) - S17
   = compound of 5 oct 
     within an encasing id
     common intersection being an ike

hirki (hemirhombichiro-icosahedron) - S18
   = compound of 5 thah 
     within an encasing id

tisso (truncated stella octangula) - S54
   = compound of 2 tut 
     within an encasing q3o4x
     common intersection being an oct

taki (truncated chiro-icosahedron) - S55
   = compound of 5 tut
     common intersection being an ike

te (truncated icosiicosahedron) - S56
   = compound of 10 tut
     common intersection being an ike

arie (antirhombicosicosahedron) - S59
   = compound of 5 co

iddei (icosidisicosahedron) - S61
   = compound of 5 oho

gari (great antirhombicosahedron) - S60
   = compound of 5 cho
rasseri (rhombisnub rhombicosicoahedron) - S62
   = compound of 5 sirco

rahrie (rhombihyperhombicosicosahedron) - S64
   = compound of 5 socco

rasher (rhombisnub hyperhombihedron) - S63
   = compound of 5 sroh

raquahri (rhombiquasihyperhombicosicosahedron) - S65
   = compound of 5 gocco

rosaqri (rhombisnub quasirhombicosicosahedron) - S67
   (old: rasquahpri = 
         rhombisnub quasihyperpseudorhombicosicosahedron)
   = compound of 5 querco

rasquahr (rhombisnub quasihyperrhombihedron) - S66
   = compound of 5 groh

tar (truncated rhombihedron) - S57
   (old: harie = hyperhombicosicosahedron)
   = compound of 5 tic

quitar (quasitruncated rhombihedron) - S58
   (old: quahri = quasihyperhombicosicosahedron)
   = compound of 5 quith

siddo (snub disoctahedron) - S46
   = compound of 2 ike
     within an encasing x3f4o

presipsido (pseudoretrosnub pentagonal 
            snub (pseudo)disoctahedron) - S48
   = compound of 2 gad
     within an encasing x3f4o

passipsido (pseudosnub pentagrammatic 
            snub (pseudo)disoctahedron) - S50
   = compound of 2 sissid
     within an encasing x3f4o

sirsido (small retrosnub disoctahedron) - S52
   = compound of 2 gike
     within an encasing x3f4o
sne (snub icosiicosahedron) - S47
   = compound of 5 ike

presipsi (pseudoretrosnub pentagonal 
          snub (pseudo)icosiicosahedron) - S49
   = compound of 5 gad

passipsi (pseudosnub pentagrammattic 
          snub (pseudo)icosiicosahedron) - S51
   = compound of 5 sissid

sirsei (small retrosnub icosiicosahedron) - S53
   = compound of 5 gike

rah (rhombihexahedron) - S08
   = compound of 3 cube
     within an encasing x3w4o
     common intersection being a sirco-stellation
     (the mere edge-truncation of the cube)

risdoh (rhombisnub dishexahedron) - S07
   = compound of 6 cube
   : rotational freedom

ro ((chiral) rhombioctahedron) - S30
   = compound of 4 trip

dro (dirhombioctahedron) - S31
   = compound of 8 trip

gassic ((chiral) great snub cube) - S42
   = compound of 3 squap

gidsac (great disnub cube) - S43
   = compound of 6 squap

griso (great rhombisnub octahedron) - S38
   = compound of 4 hip

sno (snub octahedron) - S12
   = compound of 4 oct 
     within an encasing o3x4q
doso (disnub octahedron) - S11
   = compound of 8 oct
   : rotational freedom, see idso, hidso, odso

idso (inner disnub octahedron) 
     special range of doso

hidso (hexagrammattic disnub octahedron)
     special case of doso
     (lateral {3} become (rotated) 2{3}-compounds)

odso (outer disnub octahedron)
     special range of doso

dissit (disnub tetrahedron) - S10
   = compound of 4 oct
   : rotational freedom, see idsit, hidsit, odsit

idsit (inner disnub tetrahedron)
     special range of dissit

hidsit (hexagrammattic disnub tetrahedron)
     special case of dissit 
     (lateral {3} become (rotated) 2{3}-compounds)

odsit (outer disnub tetrahedron)
     special range of dissit

sis (small snubihedron) - S01
   = compound of 6 tet
   : rotational freedom, see snu

snu (snubihedron) - S03
     special case of sis 
     (2-fold axes become 4-fold)
     within an encasing x3w4o

dis (disnubihedron) - S02
   = compound of 12 tet
   : rotational freedom

rassid (rhombisnub dodecahedron) - S40
   = compound of 6 dip

grassid (great rhombisnub dodecahedron) - S41
   = compound of 6 stiddip

rosi (rhombisnub icosahedron) - S39
   = compound of 10 hip
kred (old: red, chiro-rhombidodecahedron) - S34
   = compound of 6 pip
     within an encasing f3x5o

dird (dirhimbidodecahedron) - S35
   = compound of 12 pip
     within an encasing f3x5o
     vertices coincide by pairs

gikrid (great chirorhombidodecahedron) - S36
   = compound of 6 stip
     within an encasing srid

giddird (great dirhombidodecahedron) - S37
   = compound of 12 stip
     within an encasing srid
     vertices coincide by pairs

kri (chirorhombicosahedron) - S32
   = compound of 10 trip
     within an encasing x3f5o

dri (dirhombicosahedron) - S33
   = compound of 20 trip
     within an encasing x3f5o
     vertices coincide by pairs

gassid (great snub dodecahedron) - S27
   = compound of 6 pap

gadsid (great disnub dodecahedron) - S26
   = compound of 12 pap
   : rotational freedom

gissed (great invertisnub dodecahedron) - S29
   = compound of 6 starp

gidasid (great invertidisnub dodecahedron) - S28
   = compound of 12 starp
   : rotational freedom

sassid ((chiral) small snub dodecahedron) - S44
   = compound of 6 stap

sadsid (small disnub dodecahedron) - S45
   = compound of 12 stap
si (snub icosahedron) - S16
   = compound of 10 oct

addasi (altered disnub icosahedron) - S13
   : rotational freedom, see oddasi, dasi, iddasi, giddasi

oddasi (outer disnub icosahedron)
   = compound of 20 oct
     special range of addasi

dasi (disnub icosahedron) - S14
   = compound of 20 oct
     special case of addasi
     edge-faceting of gidrid
     vertices coincide by pairs

iddasi (inner disnub icosahedron)
   = compound of 20 oct
     special range of addasi

giddasi (great disnub icosahedron)
   = compound of 20 oct
     special range of addasi

gissi (great snub icosahedron) - S15
   = compound of 10 oct 

sapisseri ((chiral) snub pseudosnub rhombicosahedron) - S19
   = compound of 20 thah
     edge-faceting of gidrid
     vertices coincide by pairs

disco (disnub cuboctahedron) - S68
   = compound of 2 snic

dissid (disnub icosidodecahedron) - S69
   = compound of 2 snid

disdid (disnub dodecadodecahedron) - S73
   = compound of 2 siddid

giddasid (great disnub icosidodecahedron) - S70
   = compound of 2 gosid

idisdid (invertidisnub dodecadodecahedron) - S74
   = compound of 2 isdid
gidsid (great invertidisnub icosidodecahedron) - S71
   = compound of 2 gisid

desided (disnub icosidodecadodecahedron) - S75
   = compound of 2 sided

gidrissid (great diretrosnub icosidodecahedron) - S72
   = compound of 2 girsid

The similar compound of 2 enantiomorphic snubs (as given on Bowers last 2 figure plates above) for the case of gisdid there (and usually) is excluded from the listings for the same reason as Grünbaumian figures were excluded from normal polytopes, i.e. because of the complete coincidence of facial elements (pentagrams in that case). That one would be a further edge faceting of gidrid. – Likewise the compound of 3 (orthogonal) op is here not contained. The corresponding blend is nothing but sroh. Conjungately the compound of 3 (orthogonal) stop, blending out the coincident squares, would result in groh.

None the same, what really is missing in these above pictures, are all the axial cases, based on the 2D compounds cf. wikipedia

Axials n/d-prisms n/d-antiprisms
d odd
d even

S21 (esp. 2 cubes)

S23 (esp. 2 oct)





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---- 4D ----

Because the set of uniform polyhedra in 4-dimensional space is not proven to be enlisted completely so far, even less is known about uniform compounds of this space. And even of those known so far only some are listed here ...

sorted alphabetically by components
???    = compound of 6 dadip
         in hi army

sadtap = compound of 36 dadip
         in sidtaxhi regiment, in hi army
         vertices coincide by 6, edges coincide by pairs

teppix = compound of 60 deca
         in dattady regiment, in hi army
         vertices coincide by 3
xehix  = compound of 60 duhd
         in sishi regiment

???    = compound of 6 ex
         in rox army
???    = compound of 10 ex
         in hi army
         vertices coincide by pairs

firmax = compound of 25 firp
         ("facetorectified medial hexacosachoron")
         in romex regiment
???    = compound of 10 fix
         in hi army
         vertices coincide by pairs

firdox (old: fapchix) = compound of 25 frico
         ("facetorectified dodecahedronary-hexacosachoron", 
         old: "facetopentacubic hecatonicosahexacosachoron")
         in rissidtixhi regiment, in rahi army

???    = compound of 10 gaghi
         in hi army
         vertices coincide by pairs

???    = compound of 10 gahi
         in hi army
         vertices coincide by pairs

???    = compound of 10 gashi
         in hi army
         vertices coincide by pairs

???    = compound of 10 gax
         in hi army
         vertices coincide by pairs

afpox  = compound of 60 gippid 
         in affixthi regiment
         vertices coincide by pairs

gepdi  = compound of 3 girdo
         ("great prismatodisicositetrachoron")
         in giddic regiment, in spic army

???    = compound of 10 gishi
         in hi army
         vertices coincide by pairs

???    = compound of 10 gofix
         in hi army
         vertices coincide by pairs

???    = compound of 10 gohi
         in hi army
         vertices coincide by pairs

ditsop = compound of 12 gudap
         ("ditrigonary swirlprism")
         in sishi regiment, in ex army

haddet = compound of 2 hex 
         in tes army

stico  = compound of 3 hex
         in ico army

???    = compound of 75 hex
         in ex army
         vertices coincide by 5
       = compound of 25 stico

???    = compound of 675 hex
         in hi army
         vertices coincide by 9

affip  = compound of 16 hiddip 
         in afdec regiment, in cont army
         vertices coincide by pairs

datap  = compound of 100 hiddip 
         in dattady regiment, in hi army
         vertices coincide by 6, edges coincide by pairs

stoc   = compound of 2 ico
         (in dual position)

chi    = chiral compound of 5 ico
         in a chiral sub-sishi regiment, in ex army
         (not all edges being used)
         common intersection being a hi

dox    = compound of 25 ico 
         in sishi regiment, in ex army
         common intersection being an ex
         vertices coincide by 5

???    = compound of 225 ico
         in hi army
         vertices coincide by 9

???    = dual compound of 225 ico
         common intersection being an ex
         9 octs each are co-realmic (building a compound on their own)

sadixhix = compound of 25 ihi 
         in sishi regiment, in ex army
         vertices coincide by 5
         (different orientation of ihis than for gadixhix)

gadixhix = compound of 25 ihi 
         in sishi regiment, in ex army
         vertices coincide by 5
         (different orientation of ihis than for sadixhix)

xix    = compound of 25 ini
         in rissidtixhi regiment, in rahi army

???    = [Grünbaumian] compound of 3 odip
         in srit regiment
         vertices coincide by 2
         blending out pairs of coincident squares → garpit

???    = [Grünbaumian] compound of 9 odip
         in spic regiment
         vertices coincide by 4
         blending out pairs of coincident squares → sirc

ostople = compound of 2 ostodip
         ({8}x{8/3} + {8/3}x{8})

pedeple = compound of 2 padedip
         ({5}x{10} + {10}x{5})

???    = compound of 10 paphacki
         in hi army
         in 10-sishi-compound regiment

???    = compound of 10 paphicki
         in hi army
         in 10-sishi-compound regiment

sted   = compound of 2 (dual) pen
         in deca-dual army
         common intersection being a deca
         [2{3,3,3}]{3;3;3} cf. details
???    = compound of 20 pen
         in hi army

mix    = compound of 120 pen
         in mix regiment, in hi army
         common intersection being an ex
???    = compound of 720 pen
         in hi army

pinnix = compound of 120 pinnip
         ("prismatointercepted hexacosachoron")
         in romex regiment

pestideple = compound of 2 pistadedip
         ({5}x{10/3} + {10/3}x{5})

badhidy = non-isohedral compound of quit sishi and tigaghi
         in sabbadipady regiment, in sidpixhi army

???    = compound of 10 raggix
         same regiment as gadsadox
risted = compound of 2 (inverted) rap
         ("rectified stellated decachoron")

romex  = compound of 120 rap
         ("rectified medial hexacosachoron")
         in romex regiment

redox (old: pichix) = compound of 25 rico
         ("rectified dodecahedronary-hexacosachoron", 
         old: "pentacubic hecatonicosahexacosachoron")
         in rissidtixhi regiment, in rahi army

???    = compound of 10 rox
         same regiment as sadsadox

???    = compound of 10 sishi
         in hi army
         vertices coincide by pairs

sepdi  = compound of 3 sirdo
         ("small prismatodisicositetrachoron")
         in spic regiment and army

sistople = compound of 2 sistodip 
         in gittith regiment, in tat army
         ({4}x{8/3} + {8/3}x{4})

sople  = compound of 2 sodip 
         in sidpith regiment and army
         ({4}x{8} + {8}x{4})

dopix  = compound of 60 spid 
         in sishi regiment, in ex army
         vertices coincide by 10, edges coincide by 3

???    = compound of 3 srit
         in spic regiment
         vertices coincide by 2

gadtap = compound of 36 stadidip 
         in gadtaxady regiment, in hi army
         vertices coincide by 6, edges coincide by pairs

stardeple = compound of 2 stardedip
         ({5/2}x{10} + {10}x{5/2})

starpeple = compound of 2 starpedip
         ({5}x{5/2} + {5/2}x{5})

starpepla = compound of 2 starpedip
         ({5}x{5/2} + inv{5/2}x inv{5})

???    = compound of 4 starpedip
         in gap army

spidy  = compound of 24 starpedip 
         in sishi regiment, in ex army
         vertices coincide by 5

starstideple = compound of 2 stastidedip
         ({5/2}x{10/3} + {10/3}x{5/2})

???    = [Grünbaumian] compound of 9 stodip
         in giddic regiment
         vertices coincide by 4
         blending out pairs of coincident squares → girc

gico   = compound of 3 tes 
         in ico regiment and army
         common intersection being an ico
         vertices coincide by pairs
cac    = compound of 15 tes
         in a chiral sub-sishi regiment, in ex army
         (not all edges being used)
dac    = compound of 75 tes
         ("dodecahedronary cubichoron")
         in sishi regiment, in ex army

???    = compound of 675 tes
         in hi army
         vertices coincide by 18

thiple = compound of 2 thiddip
         ({3}x{6} + {6}x{3})

tidox  = compound of 25 tico 
         in tissidtixhi regiment
         vertices coincide by pairs


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---- 5D ----

Only few is known here so far ...

sorted alphabetically by components
todin = compound of 2 (alternate) hin

stade = compound of 2 (dual) hix
        common intersection being a dot
        [2{;3,3,3,3}]{3,3;3,3} cf. details
bic   = compound of 10 squoct
        in rat regiment
        vertices coincide by 6


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---- 6D ----

Only few is known here so far ...

sorted alphabetically by components
ket  = compound of 36 dotip 
       in mo regiment
       vertices coincide by 20
       20  o3o3o3o3o *c3x [36  o3o3x3o3o x]
gedak  = compound of 2 (alternate) hax
pok  = compound of 36 hixip 
       in jak regiment
       vertices coincide by 16
stef = compound of 2 (dual) hop
       common intersection being a fe
       [2{;3,3,3,3,3}]{3,3;3;3,3} cf. details
madek  = compound of 2 (inverted) jak
ye   = compound of 80 trittip 
       in mo regiment
       vertices coincide by 30

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Intersection Kernels of Facet-Regular Bi-Simplex Compounds

Simplices in general do miss the inversion symmetry. Accordingly the compound of a mutually dual simplex pair exists for any dimension. Those then all are facet-regular compounds. The intersection kernel not only can be provided generally in its combinatrical shape, but well in its correct metric size as well. In fact we have:

0D: intersection kernel of         o         and         o         is         o         when scaled down by 1
1D: intersection kernel of         x         and         x         is         u         when scaled down by 2
2D: intersection kernel of        x3o        and        o3x        is        x3x        when scaled down by 3
3D: intersection kernel of       x3o3o       and       o3o3x       is       o3u3o       when scaled down by 4
4D: intersection kernel of      x3o3o3o      and      o3o3o3x      is      o3x3x3o      when scaled down by 5
5D: intersection kernel of     x3o3o3o3o     and     o3o3o3o3x     is     o3o3u3o3o     when scaled down by 6
6D: intersection kernel of    x3o3o3o3o3o    and    o3o3o3o3o3x    is    o3o3x3x3o3o    when scaled down by 7
7D: intersection kernel of   x3o3o3o3o3o3o   and   o3o3o3o3o3o3x   is   o3o3o3u3o3o3o   when scaled down by 8
8D: intersection kernel of  x3o3o3o3o3o3o3o  and  o3o3o3o3o3o3o3x  is  o3o3o3x3x3o3o3o  when scaled down by 9
9D: intersection kernel of x3o3o3o3o3o3o3o3o and o3o3o3o3o3o3o3o3x is o3o3o3o3u3o3o3o3o when scaled down by 10

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