Wow, that's an amazing looking figure! Thank you for posting it

I haven't really been following this thread, just happened to notice the image.

And that got me really interested in the possibilities...

Looking back at Marek's list:

Marek14 wrote:2D:

Circle (II)

3D:

Sphere (III)

Torus ((II)I)

Multitoruses ([II]n I) based on multiple semicircles attached to a common line.

4D:

Glome (IIII)

Torisphere ((III)I)

Multitorispheres ([III]n I) based on multiple half-spheres attached to a common circle.

Spheritorus ((II)II)

Multispheritoruses ([II]n II) based on multiple semicircles attached to a common line. This may include both 2D and 3D arrangements of semicircles, creating a mid-cut with circular or polyhedral arrangement of spheres.

Ditorus (((II)I)I)

Dimultitoruses (([II]n I)I) based on multitoruses.

Multiditoruses ([(II)I]n I) based on multiple half-toruses attached to common pair of concentric circles.

Multidimultitoruses ([[II]m I]n I) based on multiple halves of multitoruses attached to common group of circularly arranged circles.

Tiger ((II)(II))

Multitigers ([II]n (II)) based on multiple halves of duocylinder margins (circle x semicircle) attached to a common pair of parallel circles. Mantis would belong here.

Hypertigers ([II]m [II]n) based on multiple quarters of duocylinder margins (semicircle x semicircle). Their attachment seems more vague to understand.

I suppose we could say that sequence ([II]n I) = {(III), ((II)I), ([II]3 I), ([II]4 I), ([II]5 I), ...} follows the sequence of regular polygons including the initial degenerate cases Gn = {point, digon, trigon, square, pentagon, ...}

As I see it, the most direct extension of the 3D torus into higher dimensions is the sequence ((II)I), ((III)I), ((IIII)I), ... - so the most direct extension of this new sequence of multitoruses into higher dimensions is as ([II]n I), ([III]n I), ([IIII]n I), ... - this case seems simple enough to understand.

If we now look at the opposite case - the spheritorus instead of the torisphere, and the related sequence ((II)I), ((II)II), ((II)III), ... - I believe these would be arrangements of semicircles around a common

something - not necessarily a line (digon) as Marek states. For the ((II)II) case I imagine any of the Gn sequence could replace the line. Therefore ([II]n II) isn't specific enough to represent the full array of possibilities. I believe ([II]n II) would represent the n-prisms, while other possibilities would be non-prismatic 3D figures, such as a dodecahedron, an Archimedean solid, or maybe even some (all?) of the Johnson solids.

If the above is true then the multitoruses and multitorispheres (and their analogues in higher dimensions) are perhaps better represented by replacing "n I" with "Gn" - and then the multispheritoruses can be represented by replacing "n II" with "X" - any qualifying 3D polytope. So ([II]5 I) is now ([II] G5), and potential multispheritoruses might be ([II] +G5) (pentagonal prism arrangement), ([II] Ko6) (truncated octahedral arrangement), etc.

Now, I'm a fan of powertopes, so you know where I'm going with this...

For the ((II)III) case, the family elements are identified by a 4D polytope. I imagine a duoprism would qualify. Would a duotegum also qualify? How about a

square octagoltriate?

In general, what defines whether a multitorus exists for a given polytope? Or do they all exist?

Are there any restrictions on combining different qualifying polytopes at different nesting levels? For example, does ([[II] X] Y) exist for all valid ([II] X) and ([II] Y)?

As if that didn't raise enough questions already, I notice that my replacement of "n I..." with "X" doesn't work for tigers, because the "I" is not there. So, is ([II] X [II] Y) a construction that even makes sense? Or is some even more elaborate notation required to fully describe the higher-dimensional hypertigers?