6D polyhedra-based duoprisms, or "duohyperprisms"

[개구리]

I have been spending a lot of time recently thinking about higher dimensional geometry, both flat and curved. I've been lurking this forum and wiki for many years on and off, but since I have fully come back to the topic, I made an account of my own. I was thinking about the duoprisms in six dimensions, that are constructed of polyhedral prisms. I looked around but could not find any content on the idea I had today, so I am posting it here as "duohyperprisms," because I don't know if it has a proper name. In 4D, the most basic duoprisms will be the simplex duoprisms, which are written as 3,3-duoprisms, the Cartesian product of two triangles. In 6D, you can have a 3,3,3,-trioprism, which is the Cartesian product of three triangles, but you can also have the simplex duohyperprism, which is the Cartesian product of two tetrahedra.
I guess because there is so much more possibility in 3D than there is in 2D, you can make a lot of weird but still regular duohyperprisms in 6D, such as the "triangular prism soccer ball" duoprism, which is the Cartesian product of a triangular prism and a truncated icosahedron. It becomes difficult to write out the names in an "m,n-duoprism" format, but I propose the usage of Schläfli symbols in place of m and n. For example, the simplex duohyperprism would be "{3,3},{3,3}-duoprism," and the triangular prism soccer ball duohyperprism would be "t{2,3},t{3,5}-duoprism." Kind of ugly to read, but it's rigorous.

The m,n limit of m,n-duoprisms in 4D is the duocylinder, and the m,n,o limit of m,n,o-trioprisms is the triocylinder. You can't really talk about limits with polyhedra without invoking the infamous apeirogon or apeirohedron, but I think the equivalent idea of m,n-duohyperprisms in 6D would be the "duospherinder." Maybe the real magic happens in 8D, where you can have duoprisms that are Cartesian products of lower dimensional duoprisms.

[wendy]

You could look at http://www.os2fan2.com/files/1wd-polytope.pdf .

Prism is derived from the greek word for 'off-cut', rather as if you had a stick of wood, and were cutting bits off the end.

A cylinder is a prism of a line and a circle. You could fill a cylinder with matches or pencils or the like, there would be a pencil for points on the circular base. Alternately, you can fill the cylinder with a stack of coins, there would be a coin (circle) for each base. This is because the prism product is a repetition of content. The content of the base is repeated for the height, and vice versa.

In higher dimensions, I would drop all the prefixes, and just use 'prism'. A tri-triangular prism is the prism-product of three lines.

The prism product is a mathematical product in two ways. The volume of the prism is the product of volumes of the base (1/4 sqrt(3)) ^3. Also, if you write the euler form with a leading 1, as 1a^2 + 3a^1 + 3a^0, this means 1 2d-body, 3 1d-parts (edges), 3 0-d parts. The prism product is simply the product of these, so a triangle-prism is this by a line 1a^1 + 2a^0, gives 1a^3 + 5a^2 +9a^1 + 6a^0.

The tritriangle prism is then (1a^2 + 3a^1 + 3a^0)^3 = 1a^6 + 9a^5 + 36a^4 + 81a^3 + 108a^2 + 81a^1 + a^0.

"hyper" is "above" and one might as easily suppose a square is a 'hyperpoint". Hyper-space is the dimension above the one you're looking at, so in 4d, hyperspace is 5d.

duo simply means two. I know that Bowers uses it, but I don't.

[Klitzing]

Hi - and welcome to hi.gher.space!

yes, you indeed can use any 2 polytopes (of whatever dimensionality each) and make up a duoprism of them. The edge,P-duoprisms (any polytope P) then just happens to be the usual P-prism.
You'll find lots of 6D examples eg. here: https://bendwavy.org/klitzing/dimensions/polypeta.htm (and further ones on the pages for appropr. dimensionality).

--- rk

[개구리]

Oh, I misunderstood the whole idea of prisms to begin with. How different is "prism product" from "Cartesian product"? Now I get why you don't use the duo- and trio- prefixes for 4+ dimensional prisms. I was led astray by the fact that all 3D prisms are necessarily just extrusions of 2D shapes (equivalent to multiplying by a line). So an n-gonal prism is really an n-gon×line prism, a duocylinder is a circle×circle prism, and the six dimensional shape I was thinking of was the sphere×cube prism. Thank you, wendy and Klitzing.

[quickfur]

The duoprisms are just the Cartesian product of two polygons. The Cartesian product is essentially the same thing as what Wendy calls the prism product. It can be applied to more than just polygons or polytopes; it can be applied to any shape, and in fact, any set, since the Cartesian product is a set operation.

The Cartesian product of two circles (well, more precisely, two disks) is the duocylinder, for example; the cartesian product of a disk and a polygon gives you a kind of cylindrical polygonal complex whose boundary consists of n cylinders and an n-gonal toroid that interlock each other. In dimensions higher than 4D, you can have more interesting combinations, like the "square" of polyhedron in 6D, such as icosahedron x icosahedron, which contains ridges in the shape of icosahedral prisms, facets in the shape of icosahedron x triangle prisms.

It's called a prism product because it produces elements which are prism-like.

There's a related product called the tegum product (diamond product), formed as the convex hull of the constituent shapes embedded in mutually-orthogonal subspaces of the product space. They are analogous to the prism products (Cartesian products), except that they have a diamond-like shape with diamond-like elements. A 4D example is the bicircular tegum, the convex hull of two disks placed respectively in the WX and YZ planes. Its boundary consist of two orthogonal non-intersecting circular ridges, and a toroidal surface with diamond-shaped cross-sections.

In 4D, the prism (cartesian) product of a circle and a square yields a cubinder (4 cylinders and a toroid with square cross-section); the tegum product of the same produces circle-square tegum (1 circular ridge, 4 edges, and 4 toroids with triangular cross-sections).

You can also have any combinations of either product in higher dimensions, e.g., the tegum product of a square-circular prism and a duocylinder, for example. All sorts of interesting shapes can be obtained this way.

[wendy]

The prism-product applies in geometries where the cartesian product does not exist, such as the spherical and hyperbolic geometries.

Also the cartesian product can be applied to tilings, eg a tiling of hexagon-prisms = {6,3}{U}. This is not a prism-product, because one should recall that a polytope is a surface (rather than a volume), and that "polyhedron" = many hedra (2-patches) made into a bag, and an apeirohedra = boundless 2d patches made into a carpet. So the tilings can be imagined to hold an 'interior' (eg {6,3} is a hexagonal-faced polytope covering half-space), and this interior is not repeated in the hexagon-prism tiling. That tiling is a repetition of surface (ie there is a column of hexagon-prisms for each hexagon, and a layer of hexagon-tilings for each prism in the column). But this is a surface of a 4d polytope, not a five-d polytope (as 3+2 would suggest).

The prism-product would involve interiors of the tilings, which would produce a ledge or shelf, with two walls at right-angles.