What is the difference between a torisphere and a spheritorus?
I don't understand the explanation in wiki
Marek14 wrote:Mathematically, a normal 3D torus is set of points in 3D with a given distance from a circle.
Marek14 wrote:You can also make both spheritorus and torisphere by rotating a torus along a bisecting plane.
The torisphere, previously known as the toraspherinder, is a four-dimensional torus formed by taking an uncapped spherinder and connecting its ends through its inside.
wendy wrote:"Spheration" is a surface-finish. It converts thin lines into tubes, points into spheres, etc.
Prashantkrishnan wrote:wendy wrote:"Spheration" is a surface-finish. It converts thin lines into tubes, points into spheres, etc.
In tetraspace, does this mean converting thin lines into spherinders, points into glomes and planes into cubinders?
This is how I would have to take it to interpret spheration of circle in 3D and in 4D differently.
Marek14 wrote:Prashantkrishnan wrote:wendy wrote:"Spheration" is a surface-finish. It converts thin lines into tubes, points into spheres, etc.
In tetraspace, does this mean converting thin lines into spherinders, points into glomes and planes into cubinders?
This is how I would have to take it to interpret spheration of circle in 3D and in 4D differently.
Yes, basically. You can imagine it as adding missing dimensions to the lower-dimensional entities.
wendy wrote:In 4d, a torisphere, is a 'spherated sphere' . This means you take a hollow sphere in the XYZ space, where R is a radius, and replace the points on the R surface by a circle in the WR plane. In other words, you replace the thin edge of the sphere with a much thicker boundary, such as formed by a clay model.
A spheritorus is made by 'spherating a circle'. This means that you take a circle in the WX plane, and fatten it up to make a solid thing. This makes a point into a sphere in the RYZ space, so you get a lot of sphere disks, rather like disks in a juke box. The way you form it from the sperinder is to imagine a spheric prism, (in the XYZ plane, a sphere, in the W space, a line), and then convert the thing into a loop, by bending the spherinder into a circle in the XW space.
Marek14 wrote:Distance from a circle is pretty simple -- you must take the circle as only the edge, not the full disc. If you take a circle in 2D and take points that have a certain small distance from this circle, you'll get two more circles; one outside, one inside. The original circle of radius R will give you two additional circles of radii R+r and R-r.
This is a cut of torus. If you look at the parallel slices of torus in this direction, they are all pairs of concentric circles. The further you are from the mid-cut plane, the closer the circles will be, until, at the distance of r, they merge into a single circle and vanish.
Both torisphere and spheritorus have analogical sets of 3D slices. The torisphere slices all look like pairs of concentric spheric shells, which come closer together as you move from the mid-cut hyperplane, merge into a single sphere and vanish. The spheritorus slices look like normal 3D torus whose major radius stays the same, but the minor radius shrinks as you move from the mid-cut hyperplane, until it's reduced to zero, making the torus into a circle, and vanish.
A torus can be imagined as an "inflated circle", with every point of a circle replaced with another circle perpendicular to the original circle. In this analogy, spheritorus would replace each point of a circle with spherical shell and torisphere would replace each point of sphere with a circle.
Your rotation ideas for torus are correct.
I don't think spheritorus looks like a duocylinder, but there is a toratope (the tiger) that is derived from it.
There are multiple ways of deriving the toratopes; one of them is that each toratope is a particular rotation of a lower-dimensional one. The rotations are very easy to see in the toratopic notation, as they can be mechanically derived by character manipulation.
In 3D, you can only rotate a circle, written as (II) in the toratopic notation.
You can either make a bisecting rotation, which takes a I and replaces it with II, and you'll get (III), which is a sphere.
Or you can make a nonbisecting rotation, around a line that doesn't intersect the circle, and that replaces I with (II), getting ((II)I), the torus. This is why a torus has a mid-cut that looks like two separated circles.
In 4D, you can rotate two 3D things: a sphere (III) or a torus ((II)I).
Bisecting rotation of sphere is (IIII), a glome.
Nonbisecting rotation of sphere is ((II)II), a spheritorus. That's where the spheritorus mid-cut of two separated spheres comes from.
Rotation of torus is more complicated because torus has two non-equivalent positions of I symbols, which is another way to say that it has two distinct types of coordinate planes.
Bisecting rotation of torus can be ((III)I), a torisphere, or ((II)II), a spheritorus. Both of these have a mid-cut that is a single torus.
Nonbisecting rotation of torus adds additional complexity to the shape. It results in (((II)I)I), a ditorus, or ((II)(II)), a tiger.
Prashantkrishnan wrote:This toratopic notation is one of the most confusing things here for me
Does this indirectly mean that getting a cylinder by rotating a square is nonbisecting rotation of II to (II)I or does the question of bisecting or nonbisecting rotation arise only for closed polytopes? And what is nonbisecting rotation? Is it the same as taking a prism of a toratope of n dimensions and making its ends meet in n + 1 dimensions?
I would be grateful to have clarifications of these.
Prashantkrishnan wrote:Is there a relation between the open toratopes and the corresponding closed toratopes?
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