Toratopes are a class of shapes that generalize cubes, cylinders, spheres, and tori into higher dimensions. Cylinders and tori in particular tend to split off into many different types.

Fundamentals

Toratopes are commonly represented by toratopic notation, which represents the shape by its formula. The notation uses parenthesis to denote the square root of a sum of squares (with a radius subtracte4d), while a vertical bar denotes a dimension/axis variable like x, y, z, w, v, etc. There must always be at least two elements under a pair of parenthesis, so (|), ((||)), etc. are not valid.

For example, ((|||)|) means sqrt((sqrt(x^2 + y^2 + z^2) - r_1)^2 + w^2) - r_2 = r_3

What people commonly miss from this description, though, is that in order for the formula to work properly so that the surface will appear on the graph where the value equals zero, multiple components at the top level must be put under the maximum function of their absolute values. If the entire thing is under one top-level pair of parenthesis, this is not necessary, but for example, (||)(|||)| means max(abs(sqrt(x^2 + y^2) - r_1), abs(sqrt(z^2 + w^2 + v^2) - r_2), abs(u)).

If there is a top-level pair of parenthesis, it is a closed toratope. These make spheres and tori. If there are multiple top-level elements, it is an open toratope. These make cubes and cylinders, and in general the top-level elements just get put into a Cartesian product operation.

Closed toratopes have the very helpful property that they're automatically signed distance fields. This makes raymarching them a breeze, and you can apply operations like union or domain repetition very easily. If you want to convert slices of them to triangle meshes in real time, the marching cubes algorithm will also work well.

Toratopes in their natural form as defined by equations are always symmetric on every orthogonal plane at the origin.

The number of open or closed toratopes in a given dimension is literally just the number of possible trees you can make with the notation. It follows sequence A000669 on OEIS The total number of toratopes is double that, since the presence of top-level parentheses is not accounted for by the sequence. The total number of tori (open toratopes other than the sphere) is the sequence number minus 1. Thus there is 1 torus in 3D, 4 tori in 4D, 11 tori in 5D, 32 tori in 6D, 89 tori in 7D, etc. As you can see, it grows rather fast.

Dimensions 1-3

In dimension 1, there is only one toratope: |. It's simply a line segment.

In dimension 2, there are two toratopes, the square || and the circle (||).

In dimension 3, there are four toratopes.

• The cube, |||. As you should probably know if you're human, it's a stack of squares.
• The cylinder, (||)|. The notation is easily recognizable as the Cartesian product of a circle (||) and a line segment |. One important thing is its slices, which split off generalization in higher dimensions. When you slice on the plane perpendicular to the extrusion, you get circles. When you slice on the extrusion axis and one other axis, you get growing and shrinking rectangles.
• The sphere, (|||).
• The torus, ((||)|), with its familiar donut shape. It's topologically a Euclidean surface and can be tiled like one, though it has extra curvature due to not having enough space in three dimensions to be flat.

Dimension 4: The Open Toratopes

There are five open toratopes in dimension 4.

• The tesseract, ||||, is a stack of cubes.
  
• The spherinder, (|||)|, is a stack of spheres, and is one possible generalization of the cylinder (as a stack of (n-2)-spheres).
  Its cross sections including the extrusion axis and two others is a cylinder that grows and shrinks.
  
• The cubinder, (||)||, is a stack of cylinders, and is another possible generalization of the cylinder (by a recursive definition).
  Its cross sections including the extrusion axis and two others is a rectangular prism that grows and shrinks in width.
  
• The torinder, ((||)|)|, is a stack of tori. It isn't really super important for anything as tori are typically not used as the base of a prism.
  It's effectively a spheritorus with cylindrical cross-sections.
  
• The duocylinder, (||)(||), is perhaps the weirdest one, and isn't similar to much of anything in dimension 3. It's the Cartesian product of two circles.
  On any orthogonal 3-plane, slicing it will result in cylinders that grow and shrink in width, but they face different directions.
  

Dimension 4: The Closed Toratopes

Four-dimensional space has some weird tori that are nothing short of magic. It has the best selection of them in my opinion; every one is unique, yet all of them are mostly simple and elegant in construction.

First, let's get the hypersphere out of the way because it's the most commonly mentioned shape in all of 4D discourse. Whenever anyone introduces 4D to newcomers, the ubiquitous growing and shrinking sphere explanation is used. It IS a valid toratope despite being boring, and it's notation is (|||). It is known more concisely as a glome.

Other than that, there are four types of tori:

• A spheritorus, ((||)||) is what you get when you revolve a sphere around a central point. You can move an entire, infinite 2D plane through its hole.
  In the same way that a torus is Euclidean, a spheritorus has geometry S2xE.
  
  
• A torisphere, ((|||)|), is what you get when you inflate a sphere into the fourth dimension with circular cross-sections.
  By some standards, it's a closer analogue to the 3D torus as only a rod can be sent through the hole.
  If a 4D person wore necklaces, rings, bracelets, etc., they would probably be more like torispheres than spheritori, since spheritori would fall off in those scenarios.
  
  
• The ditorus, (((||)|)|), is what you get when you revolve a torus around a central point. It can also be obtained by inflating a torus in a similar way to the construction of the torisphere.
  Weirdly, it kinda has two holes. One is the now exposed cavity of the torus that goes in a circle and has a gap in the middle, and the other is similar to the hole of a spheritorus, where you can
  stick a whole plane through it. This oddity allows it to chain with itself, as I'll discuss later.
  
  
• The tiger, ((||)(||)), is the most complicated. Start by taking the Cartesian product of two circles without their interiors. This results in a 2D surface in 4D space, which makes it infinitely thin. This 2D surface is also known as a Clifford torus or flat torus, and is a wrapped Euclidean plane with no weird curvature from stretching.
  Now, inflate it with circular cross-sections similarly to the torisphere.
  The result can be morphed into a ditorus continually (it's homeomorphic), but this specific form is really weird.
  It's also a torus with toric cross-sections, but the cross-sections are rotated differently. This shape also has symmetry where it retains its sliced appearance as two stacked tori even if you swap the axes around.
  

Links, Chains, and Mails

In 3D, two tori can link.

This property allows the construction of chains:

And chainmail, which is effectively a dimensional array equivalent to chains:

Can higher-dimensional tori chain the same way? The answer is yes, but it's more complicated. Not every type of torus can chain with itself, and some types of tori require an alternating pattern to chain. To understand the situation a bit better, I want to talk about holes.

Types of Holes

A standard 3D torus has a hole. Some shapes can have more than one hole, like this:

In 3D, you can stick a rod through it. If the rod is a little smaller than the hole, can't move through it, and must always touch the torus, its only option is to hug the wall. When it does this, it can still move around in a circle. You can think of this like the "boundary" of the hole.

In 4D, a similar type of hole occurs in the spheritorus, except that an entire plane can hug the wall and move around in a circle. Since a circle is a wrapped line, its surface has one dimension. I can say that the hole of a 3D torus or a spheritorus both have a 1-dimensional boundary. (Note: This is non-standard terminology, made to make this article a bit easier to write.)

A torisphere is a bit different. It can only fit a rod, but the rod can move around the surface of a sphere. Since a sphere is a 2-dimensional manifold, I will say the hole of a torisphere has a 2-dimensional boundary.

A ditorus is even weirder. Ignore the central hole and focus on the hole that goes around in a circle, corresponding to the hole in the cross sections. Again, in 4D this can only fit a rod.
However, this time when the rod hugs the wall, it can move on the surface of a 3D torus, another 2-dimensional manifold. It is a different type of 2-dimensional hole.

Back to Linking

The problem with linking two spheritori together is that there is too much space. They will easily fall off each other and unlink.

Meanwhile, two torispheres don't have enough space to link together at all.

So what's the solution? Use one spheritorus and one torisphere. The spheritorus is able to hook into the torisphere, but the extra boundary space of the hole blocks it from falling out. Meanwhile, the torisphere doesn't run into the same space issue it did before. Turning this into chains and chainmail requires an alternating pattern.
This Desmos 3D graph (quite laggy/RAM-intensive, so beware) illustrates a patch of this alternating chainmail structure that could be draped over a 4D object. Modify the w variable from about -1 to 1 to see the different slices.
Is there a more general way to check if two holes can link? It's actually quite simple given the definition of an n-dimensional hole earlier.
In dimension D, an N-dimensional hole and an M-dimensional hole can link if N + M = D - 1. That is, holes can link with their opposites. This also opens up odd dimensions to have more basic links. For example, in 5D, the equivalent of a torisphere (a spheritorisphere) can easily link with itself.

This condition doesn't guarantee a link is possible since there could be an issue with more concrete aspects of the geometry that blocks other things from fitting in with it. For toratopes though, this is rarely a problem, and you mostly just have to find the right radii that gives enough space. Are there any other chains in 4D, and are there any less basic 4D toratopes that can link and chain with themselves? The answer to both is yes.

• The spheritorus has one hole. It is circular and has dimension 1.
• The torisphere has one hole. It is spherical and has dimension 2
• The ditorus has two holes. One is circular and has dimension 1, and the other is toric and has dimension 2.
• The tiger is homeomorphic to a ditorus, and has the same holes (1D and 2D).

• Spheritorus-Torisphere
• Spheritorus-Ditorus
• Spheritorus-Tiger
• Torisphere-Ditorus
• Torisphere-Tiger
• Ditorus-Tiger
• Ditorus-Ditorus
• Tiger-Tiger

5D Tori and Link Table

There are 11 5D tori.

• The glomitorus, ((||)|||), is obtained by revolving a glome around a central point. It has a 1D circular hole.
• The spheritorisphere, ((|||)||), is obtained by inflating a sphere into 5D with spherical cross-sections. It has a 2D spherical hole.
• The toriglome, ((||||)|), is obtained by inflating a glome into 5D with circular cross-sections. It has a 3D glomeral hole.
• The spheritiger, ((||)(||)|), is obtained as a Clifford torus, inflated into 5D with spherical cross sections. Just like a tiger, it has a 2D toric hole and a 1D circular hole.
• The toratiger, (((||)(||))|), is obtained as a Clifford torus, inflated into 5D with toric cross sections. Just like a tiger, it has a 2D toric hole and a 1D circular hole, but it also now has a 3D ditoric hole.
• The spheriditorus, (((||)|)||), is obtained by revolving a spheritorus around a central point, or inflating a torus into 5D with spherical cross sections. Just like a ditorus, it has a 2D toric hole and a 1D circular hole.
• The torispheritorus, (((||)||)|), is obtained by revolving a torisphere around a central point. It has a 1D circular hole and a 3D spheritoric hole.
• The ditorisphere, (((|||)|)|), is obtained by inflating a sphere into 5D with toric cross sections. It has a 2D spherical hole and a 3D torispherical hole.
• The tritorus, ((((||)|)|)|), is obtained by revolving a ditorus around a central point, inflating a torus into 5D with toric cross-sections, or inflating a ditorus into 5D with circular cross-sections. It has a 1D circular hole, a 2D toric hole, and a 3D ditoric hole.
• The cylspherintigroid, ((|||)(||)), is similar in construction to the tiger, but instead of starting with circle * circle, start with circle * sphere. In the same way that a tiger looks like two stacked tori when sliced in 3D, a cylspherintigroid looks like two stacked torispheres when sliced in 4D. Thus it should be homeomorphic to the torispheritorus, and have a 1D circular hole and a 3D spheritoric hole.
• The cyltorintigroid, (((||)|)(||)), is similar in construction to the tiger, but instead of starting with circle * circle, start with circle * torus. In the same way that a tiger looks like two stacked tori when sliced in 3D, a cylspherintigroid looks like two stacked ditori when sliced in 4D. Thus it should be homeomorphic to the tritorus, and have a 1D circular hole, a 2D toric hole, and a 3D ditoric hole.

       glomitorus - A
 spheritorisphere - B
        toriglome - C
      spheritiger - D
        toratiger - E
    spheriditorus - F
  torispheritorus - G
     ditorisphere - H
         tritorus - I
cylspherintigroid - J
  cyltorintigroid - K

  A B C D E F G H I J K
A N
B N Y
C Y N N
D N Y Y Y
E Y Y Y Y Y
F N Y Y Y Y Y
G Y N Y Y Y Y Y
H Y Y N Y Y Y N Y
I Y Y Y Y Y Y Y Y Y
J Y N Y Y Y Y Y N Y Y
K Y Y Y Y Y Y Y Y Y Y Y