### Compounds

##### 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.

##### Blends

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
```
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2D
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```

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
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```

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 {5,3}[5{3,3}]{3,5} e (icosiicosahedron) - S06 = self-dual compound of 10 tet within an encasing doe common intersection being an ike vertices coincide by pairs 2{5,3}[10{3,3}]2{3,5} rhom (rhombihedron) - S09 = compound of 5 cube edge-faceting of sidtid within an encasing doe common intersection being a rhote vertices coincide by pairs 2{5,3}[5{4,3}] se (small icosiicosahedron) - S17 = compound of 5 oct within an encasing id common intersection being an ike [5{3,4}]2{3,5} 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 [2{;3;3}]{3,4} taki (truncated chiro-icosahedron) - S55 = compound of 5 tut common intersection being an ike [5{;3;3}]{3,5} te (truncated icosiicosahedron) - S56 = compound of 10 tut common intersection being an ike [10{;3;3}]2{3,5} 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 {;3,5;}[6{;5/2,2;}] giddird (great dirhombidodecahedron) - S37 = compound of 12 stip within an encasing srid vertices coincide by pairs 2{;3,5;}[12{;5/2,2;}] 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. Axials n/d-prisms n/d-antiprismsd odd n/d-antiprismsd even regularcompoundbases S21 (esp. 2 cubes) S23 (esp. 2 oct) S25 rotationalfreedom S20 S22 S24

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4D
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```

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
{5,3,3}[6{;5,2;5}]

sadtap = compound of 36 dadip
in sidtaxhi regiment, in hi army
vertices coincide by 6, edges coincide by pairs
6{5,3,3}[36{;5,2;5}]

teppix = compound of 60 deca
in dattady regiment, in hi army
vertices coincide by 3
3{5,3,3}[60{3;3;3}]

xehix  = compound of 60 duhd
("hexacosahemihexacosachoron")
in sishi regiment

???    = compound of 6 ex
in rox army
{3;3,5}[6{3,3,5}]

???    = compound of 10 ex
in hi army
vertices coincide by pairs
2{5,3,3}[10{3,3,5}]

firmax = compound of 25 firp
("facetorectified medial hexacosachoron")
in romex regiment

???    = compound of 10 fix
in hi army
vertices coincide by pairs
2{5,3,3}[10{3,5,5/2}]2{3,3,5}

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
2{5,3,3}[10{5,5/2,3}]2{3,3,5}

???    = compound of 10 gahi
in hi army
vertices coincide by pairs
2{5,3,3}[10{5,3,5/2}]2{3,3,5}

???    = compound of 10 gashi
in hi army
vertices coincide by pairs
2{5,3,3}[10{5/2,5,5/2}]2{3,3,5}

???    = compound of 10 gax
in hi army
vertices coincide by pairs
2{5,3,3}[10{3,3,5/2}]

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
2{5,3,3}[10{5/2,3,5}]2{3,3,5}

???    = compound of 10 gofix
in hi army
vertices coincide by pairs
2{5,3,3}[10{3,5/2,5}]2{3,3,5}

???    = compound of 10 gohi
in hi army
vertices coincide by pairs
2{5,3,3}[10{5,5/2,5}]2{3,3,5}

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

haddet = compound of 2 hex
("demidistesseract")
in tes army
{4,3,3}[2{3,3,4}]

stico  = compound of 3 hex
in ico army
{3,4,3}[3{3,3,4}]2{3,4,3}

???    = compound of 75 hex
in ex army
vertices coincide by 5
5{3,3,5}[75{3,3,4}]10{5,3,3}
= compound of 25 stico

???    = compound of 675 hex
in hi army
vertices coincide by 9
9{5,3,3}[675{4,3,3}]18{3,3,5}

affip  = compound of 16 hiddip
in afdec regiment, in cont army
vertices coincide by pairs
2{3,4,3}[16{;6,2;6}]

datap  = compound of 100 hiddip
in dattady regiment, in hi army
vertices coincide by 6, edges coincide by pairs
6{5,3,3}[100{;6,2;6}]

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
{3,3,5}[5{3,4,3}]{5,3,3}

dox    = compound of 25 ico
in sishi regiment, in ex army
common intersection being an ex
vertices coincide by 5
5{3,3,5}[25{3,4,3}]{3,3,5}

???    = compound of 225 ico
in hi army
vertices coincide by 9
9{5,3,3}[225{3,4,3}]

???    = dual compound of 225 ico
common intersection being an ex
9 octs each are co-realmic (building a compound on their own)
[225{3,4,3}]9{3,3,5}

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
("hexacosahexacosachoron")
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
{5,3,3}[20{3,3,3}]

mix    = compound of 120 pen
in mix regiment, in hi army
common intersection being an ex
{5,3,3}[120{3,3,3}]{3,3,5}

???    = compound of 720 pen
in hi army
6{5,3,3}[720{3,3,3}]

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
2{3,3;5}[25{3;4,3}]

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

???    = compound of 10 sishi
in hi army
vertices coincide by pairs
2{5,3,3}[10{5/2,5,3}]2{3,3,5}

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
10{3,3,5}[60{;3,3,3;}]

???    = compound of 3 srit
in spic regiment
vertices coincide by 2
2{;3,4,3;}[3{3;3,4;}]

gadtap = compound of 36 stadidip
in gadtaxady regiment, in hi army
vertices coincide by 6, edges coincide by pairs
6{5,3,3}[36{;10/3,2;10/3}]

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
5{3,3,5}[24{;5,2;5/2}]

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
2{3,4,3}[3{4,3,3}]{3,4,3}

cac    = compound of 15 tes
("chirocubichoron")
in a chiral sub-sishi regiment, in ex army
(not all edges being used)
2{3,3,5}[15{4,3,3}]

dac    = compound of 75 tes
("dodecahedronary cubichoron")
in sishi regiment, in ex army
2{3,3,5}[75{4,3,3}]

???    = compound of 675 tes
in hi army
vertices coincide by 18
18{5,3,3}[675{4,3,3}]9{3,3,5}

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

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

...
```

```
----
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
6{3;3,3,4}[10{;4,2;3,4}]

...
```

```
----
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

...
```

##### 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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