Four-dimensional torii (ConceptTopic, 3)
From Hi.gher. Space
Unlike in 3D, where there is just one torus, there are four distinct four-dimensional torii. These, together with the glome, make up the full set of closed 4D toratopes. This page attempts to help the reader understand where these four different figures come from, and what they look like.
Construction
In 3D, we can start with a circle and extrude it into 3D to get an uncapped cylinder. Now there are two ways of making a 3D torus from this figure.
The first way is the more well-known way, which is done with a "long and thin" cylinder (e.g. a garden hose): we take the ends of the cylinder and pull them around in a loop to connect to each other. This forms a "seam" which runs through the hole in the torus.
The second way can be done with a "short and flat" cylinder (e.g. a strip of paper connected into a loop): we take the ends and roll them inside the original cylinder, to connect to each other on the inside. This forms a "seam" which runs around the hole in the torus, perpendicular to the seam we would form if we did it in the first way.
Now in 3D, obviously these both make the same figure. However, in 4D, when we start with a sphere and extrude it into 4D to get an uncapped spherinder, this is not the case! If you perform the first method, you get a spheritorus. If you perform the second method, you get a torisphere. This makes the spheritorus and torisphere toratopic duals to each other. Because they are toratopic duals, taking one of the pair and turning it "inside-out" gives you the other one – it follows that the two are topologically equivalent.
TODO: Add constructions for the tiger and ditorus.
Cross-sections
Four cross-sections of each torus can be read off from their surface equations by setting each coordinate in turn to zero.
The 3D torus
Before we take cross-sections of the 4D torii, we'll do it for the 3D torus for analogy's sake.
The surface equation of a 3D torus is:
(√(x^{2} + y^{2}) − R)^{2} + z^{2} = r^{2}
First, let's try setting z to zero. We get:
(√(x^{2} + y^{2}) − R)^{2} = r^{2}
If we ignore the -R in this equation, we can easily see we get the equation for a circle. The presence of this term however means we get two concentric circles, and the cross-section includes all the points between them. You can understand how this is the cross section you'd get if you were to place the torus flat on a table, then slice it with a plane parallel to the surface of the table.
Now, what happens if we set x or y to zero? Well, whichever of those we do, we end up with:
(√(t^{2}) − R)^{2} + z^{2} = r^{2}
where t is whichever coordinate we didn't delete. Obviously, √(t^{2}) is just t, so we can simplify:
(t − R)^{2} + z^{2} = r^{2}
Now again, if we ignore the -R, we get a circle! But this time the equation is arranged slightly differently, so the equation itself gives us two disjoint circles. This is what we'd get if we were to place the torus flat on a table, then slice down vertically through the middle.
Now we are ready to find the cross-sections of the 4D torii!
Torisphere
The surface equation of a torisphere is:
(√(x^{2} + y^{2} + z^{2}) − R)^{2} + w^{2} = r^{2}
If we set w to zero, we get:
(√(x^{2} + y^{2} + z^{2}) − R)^{2} = r^{2}
which, similarly to setting z to zero for the 3D torus above, gives us two concentric spheres; the cross-section includes all points between them.
If we set x, y or z to zero, we get:
(√(t_{1}^{2} + t_{2}^{2}) − R)^{2} + w^{2} = r^{2}
where t_{1} and t_{2} are the two coordinates we didn't delete. This is identical to the equation for the 3D torus.
Spheritorus
The surface equation of a spheritorus is:
(√(x^{2} + y^{2}) − R)^{2} + z^{2} + w^{2} = r^{2}
If we set w or z to zero, we get:
(√(x^{2} + y^{2}) − R)^{2} + t^{2} = r^{2}
which gives us the 3D torus.
If we set x or y to zero, we get:
(t − R)^{2} + z^{2} + w^{2} = r^{2}
which gives us two disjoint spheres.
Ditorus
The surface equation of a ditorus is:
(√((√(x^{2} + y^{2}) − ρ)^{2} + z^{2}) − r)^{2} + w^{2} = R^{2}
If we set w to zero, we get:
(√((√(x^{2} + y^{2}) − ρ)^{2} + z^{2}) − r)^{2} = R^{2}
which gives us two "concentric" torii, or more accurately, two torii with equal major radius but different minor radii.
If we set z to zero, we get:
(√(x^{2} + y^{2}) − ρ − r)^{2} + w^{2} = R^{2}
which gives us two concentric torii in a different way - two torii with equal minor radius but different major radii.
If we set x or y to zero, we get:
(√((t − ρ)^{2} + z^{2}) − r)^{2} + w^{2} = R^{2}
which gives us two disjoint torii.
Tiger
The surface equation of a tiger is:
(√(x^{2} + y^{2}) − a)^{2} + (√(z^{2} + w^{2}) − b)^{2} = r^{2}
Setting any coordinate to zero gives us:
(√(t_{1}^{2} + t_{2}^{2}) − c_{1})^{2} + (t_{3} − c_{2})^{2} = r^{2}
which gives us two disjoint torii.
Summary
Name | Toratopic notation | x | y | z | w |
---|---|---|---|---|---|
Torisphere | ((III)I) | ((II)I) | ((II)I) | ((II)I) | (III), C2 |
Spheritorus | ((II)II) | (III), D2 | (III), D2 | ((II)I) | ((II)I) |
Ditorus | (((II)I)I) | ((II)I), D2 | ((II)I), D2 | ((II)I), C2a | ((II)I), C2b |
Tiger | ((II)(II)) | ((II)I), D2 | ((II)I), D2 | ((II)I), D2 | ((II)I), D2 |
From this table we can deduce a way to find cross-sections from the toratope notation:
- instead of setting the k^{th} coordinate to zero, we delete the k^{th} "I" from the string
- if this leaves an "(I)" anywhere, we replace it with just "I" and note that we have 2 disjoint copies of it
- if it leaves the pattern "((φ))", we replace it with just "(φ)" and note that we have 2 concentric copies of it; the way in which they are concentric varies depending on the nesting level of the pattern we replaced.