Grand antiprism analogs

3D wireframe projection of the grand antiprism, pentagonal antiprismatic cells highlighted.
Source

The grand antiprism is one of my favorite shapes ever. It is an Archimedean polychoron, meaning it is a non-regular non-prismatic convex uniform polychoron not part of an infinite uniform family, of which there are only 41. It is actually part of an infinite family, the double antiprismatoids, but since none of the other ones are uniform the grand antiprism is considered Archimedean. This, along with the fact it has no similar uniform shapes in higher dimensions, makes me like it so much. Despite this, there are still many analogous non-uniform shapes in higher dimensions. Let's explore them.

Properties

As with any analog exploration, we must first define the properties of the shapes analyzed. The grand antiprism turns out to be the convex hull of two pentagonal-pentagonal duoantiprisms, with the two pentagons scaled differently. A duoantiprism of two shapes is the alternation of the Cartesian product of the unalternated version of these two shapes, although that makes it not applicable to every shape (e.g., a cubic duoantiprism does not exist as there is no polyhedron that only alternates to a cube). This construction can be generalized to any hull of multiple orthogonal multiantiprisms. They must also be abstractly uniform (recursively abstractly isogonal).

Analogs

Polygonal double antiprismatoids

The most trivial extension are the aforementioned double antiprismatoids. They are the hull of two polygonal duoantiprisms, with the two polygons scaled differently. They are all non-uniform, except of course for the grand antiprism and its conjugate, the grand retroprism (more commonly known as the pentagrammic double antiprismatoid). This results in this infinite family of shapes:

• Digonal double antiprismatoid;
• Triangular double antiprismatoid;
• Square double antiprismatoid;
• Pentagonal double antiprismatoid (a.k.a. grand antiprism);
• Pentagrammic double retroprismatoid (a.k.a. grand retroprism);
• Hexagonal double antiprismatoid, etc.

For a n-gonal double antiprismatoid, it has 4n n-gonal antiprisms, 4n² tetragonal disphenoids and 8n² sphenoids as cells and 4n² vertices. Their vertex figure is also a variant of the sphenocorona for n > 2.

Polytopal antiprismatoids

While polygonal double antiprismatoids are the simplest to visualize, it is also possible to make double antiprismatoids out of >2D polytopes. These polytopes, however, must be dealternable, as the final shape is the alternated hull of two duoprisms of differently scaled versions of the dealterated polytope. For example, there is no cubic double antiprismatoid because there's nothing that alternates solely to the cube. Attempting to use the rhombic dodecahedron, which does alternate to the cube and the octahedron, results in polytopes with cubic pseudantiprisms, which are not isogonal. There is, however, a pyritohedral icosahedral double antiprismatoid because a truncated octahedron alternates solely to that.

So, what do they look like? Well, due to the difficulty of creating these shapes, the best for now will have to be an educated guess. Let's take the snub dodecahedral double antiprismatoid for example. It has two spherical rings connected by various types of antiprisms. The spherical rings have digonal-snub dodecahedral duoantiprisms, triangular-snub dodecahedral duoantiprisms and pentagonal-snub dodecahedral duoantiprisms, corresponding respectively to the dealternated snub dodecahedron's square, hexagonal and decagonal faces.

Polygoltriates

As stated before, the double antiprismatoids can be seen as the hull of two orthogonal duoantiprisms, with two shapes scaled differently. This can be represented by the hull two perpendicular rectangles, with the edge lengths being the scaling factors, forming a ditetragon. This gives rise to the notion of powertopes, which can be seen as a form of exponentiation, e.g., a shape to the power of a cube is the shape times itself three times.

Just like the dimensionality of Cartesian products is the sum of the dimensionality of the operands, the dimensionality of powertopes is the multiplication of the two operands. The notation for such shapes is [base] [exponent]-oltriate. The dealternation of the grand antiprism, for example, is called the decagonal ditetragoltriate.

Double antiprismatoids with exponents other than the ditetragon are quite boring as they are not isogonal. For example, powertopes with a dodecagon exponent (dodecagoltriates) are simply ditetragoltriates with an additional layer of vertices between the two rings.

Multiple antiprismatoids

Another obvious generalization are the triple antiprismatoids and above. In theory, any brick symmetric polytope could work as the exponent, producing a 2nD shape, but only shapes isogonal under cubic symmetry become isogonal, which excludes non-cubic symmetric non-alternable exponents. To this end, the only allowed exponents then become the omnitruncated n-cube or the omnitruncated n-demicube (= subomnitruncated n-orthoplex) for any dimension. For simplicity, they will be named [n]-ple anti(demi)[n-cubi]prismatoids. This produces the following families: (There are links to the exponents themselves within the shape name, see the underline.)

• Triple antidemicubiprismatoids and triple anticubiprismatoids;
• Quadruple antidemitesseractiprismatoids and quadruple antitesseractiprismatoids;
• Quintuple antidemipenteractiprismatoids and quintuple antipenteractiprismatoids;
• Sextuple antidemihexeractiprismatoids and sextuple antihexeractiprismatoids;
• Septuple antidemihepteractiprismatoids and septuple antihepteractiprismatoids;
• Octuple antidemiocteractiprismatoids and octuple antiocteractiprismatoids.

The omnitruncated cubic exponents can be made by taking the hull of a shape exponentiated to all rotations of some cuboid with different edge lengths, as long as one of them is not zero. Otherwise, it results in a omnitruncated demicubic exponent.

Conclusion

Much like the icositetrachoron, which also has its fair share of analogs, the thing that makes it special is the fact that out of the multitude of analogs, it is the only one that's regular. The grand antiprism also has a gigantic and even infinite amount of analogs, but what makes it truly special is the fact that it is the only one out of that infinity of shapes to be uniform.