This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined.
The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m. A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon.
Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols.
The material inside the square brackets, [d{p,q}], denotes the components of the compound: d separate {p,q}'s.
[1] If improper regular polyhedra (dihedra and hosohedra) are allowed, then two more compounds are possible: 2{3,4}[3{4,2}]{4,3} and its dual {3,4}[3{2,4}]2{4,3}.
[2] There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane.
In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.
[3] McMullen adds six in his paper New Regular Compounds of 4-Polytopes, in which he also proves that the list is now complete.
In four dimensions, Garner (1970) asserted the existence of {3,3,3,5}[26{5,3,3,5}]{5,3,3,3};[7] although neither justification nor construction was given, McMullen (2019) proved that this claim is correct.
If any more compact compounds exist, they must involve {4,3,3,5} or {5,3,3,5} being inscribed in {5,3,3,3} (the only case not yet excluded).