## 4D Toratopes Explained

Discussion of shapes with curves and holes in various dimensions.

### 4D Toratopes Explained

Here's some new gifs I made recently of 4D shapes.

Torus ((II)I) What it looks like to pass a donut through a 2D world. The two angles of 0 and 90 degrees will make two circles, as side by side or one inside the other. We can also define the torus as a 'circle over circle', S1xS1 in something called fiber bundles. It's another way of saying circular ring S1 stretched over the surface of a circle S1. In their natural environment, 2D beings will not be able to see the 3D parts of the ring coming or going through. Only the immediate surface being scanned by the 2D world will be visible. This is the disadvantage we have to become familiar with, when looking at 4D donuts scanned in 3D. We will not be able to see the 4D parts of the ring, only the immediate 3D slices as it passes through our world. A much closer approximation of what 2D beings will see, if confined to the 2D world. In theory, this would be true 1D line segments, which a 2D creature could still tell are circles. Now compare this to the 2D scans we make of our 3D world. When we see 2D scans of a sphere, we can still tell it's a sphere from shading, instead of a circle.

Torisphere ((III)I) This 4D donut is made by circle over sphere, S1xS2. The ring is a circle, but stretching over the surface of a sphere. The w-axis threads through the hole, since a sphere is a flat circular object in 4D.

Spheritorus ((II)II) Made by sphere over circle, S2xS1, this donut is very similar to our 3D version. Except, this one has a spherical ring stretching over the circle. Just like the torus can make two circles side by side, a spheritorus can make two spheres, side by side. And also remember that the space between the two spheres is the donut hole that is sliced through.

Tiger ((II)(II)) This hyperdonut is truly bizarre. It's the most difficult to understand at first. Made by rotating a torus into 4D, we can define it as circle over Clifford torus, S1xC2. A clifford torus is the 90 degree edge of a duocylinder, a rolling shape in 4D. Expanding this 2D surface with a circular ring in every point makes this strange donut, which ends up having two sectioned off holes going through the same middle.

3-Torus (((II)I)I) A 3-torus can be built three different ways. We have the torus over circle, circle over torus, and circle over circle over circle, S1xS1xS1 , or simply T3. This one is also closely shaped to our 3D donut, like the spheritorus.

3-Torus (((II)I)I) There are three ways to rotate a 3-torus in 4D. This is number two. While looking at this, imagine how the rest of the ring is put together, into the mysterious and hidden 4th dimension. We cannot see the 4D parts coming or going, but only the immediate surface being scanned by our 3D world.

3-Torus (((II)I)I) Rotation type number three of the 3-torus. Sadly, after exploring all four distinct donuts in 4D, we have exhausted the list. So, in order to see anything new, we have to go into the fifth dimension. There, we will find 11 new types of toroidal ring.
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### Re: 4D Toratopes Explained

Very nice! Perhaps you should add the glome as well -- it only has one cut, but it's the "most basic" toratope in a way.

I especially like the 22.5 and 67.5 cuts which are very illuminating, as they have enough in common with 0 and 90 cuts to be familiar, yet they explore new ground.

I would add one thing to these, though: a specific plane.

Take the torisphere, ((III)I). Its two cuts are ((II)I) and ((III)), but these two cuts intersect in planar ((II)). Therefore, every one of your cuts of torisphere contains a plane which becomes two concentric circles in the midcut. If you can highlight this plane, people will see that all those cuts have one thing (an "anchor") that is the same.

Similarly for spheritorus ((II)II) where cuts are ((I)II) and ((II)I) with ((I)I) as a cross-plane.

For tiger ((II)(II)) the cuts are ((II)(I)) and ((I)(II)) with cross-plane ((I)(I)).

For ditorus (((II)I)I) you have (((II)I)) and (((II))I) with cross-plane (((II))), (((II)I)) and (((I)I)I) with cross-plane (((I)I)) and (((II))I) and (((I)I)I) with cross-plane (((I))I).

So, how this gifs would translate in 5D?

In 2D, we have a cross-plane, but in 3D we have a cross-cut, so one type of similar gif would be one where the middle picture will be identical for every rotation.

1. Pentaglome (IIIII) - (III) crosscut (sphere) symmetrical, same as glome cut.
2. 41-torus ((IIII)I) - ((II)I) crosscut (torus) symmetrical, same as torisphere cut. ((III)) crosscut (two concentric spheres) rotating between torisphere ((III)I) and two concentric glomes ((IIII)).
3. 311-ditorus (((III)I)I) - (((I)I)I) (two toruses) crosscut symmetrical, same as ditorus cut. (((II))I) (major pair of toruses) crosscut rotating between major pair of torispheres (((III))I) and ditorus (((II)I)I). (((II)I)) (minor pair of toruses) crosscut rotating between minor pair of torispheres (((III)I)) and ditorus (((II)I)I). (((III))) (four concentric spheres) crosscut rotating between major pair of torispheres (((III))I) and minor pair of torispheres (((III)I)).
4. Tritorus ((((II)I)I)I) - (((()I)I)I) crosscut (empty) symmetrical, same as cut of two ditoruses. ((((I))I)I) crosscut (four toruses) rotating between major pair of ditoruses ((((II))I)I) and two ditoruses ((((I)I)I)I). ((((I)I))I) crosscut (two major pairs of toruses) rotating between medium pair of ditoruses ((((II)I))I) and two ditoruses ((((I)I)I)I). ((((I)I)I)) crosscut (two minor pairs of toruses) rotating between minor pair of ditoruses ((((II)I)I)) and two ditoruses ((((I)I)I)I). ((((II)))I) crosscut (major quartet of toruses) rotating between medium pair of ditoruses ((((II)I))I) and major pair of ditoruses ((((II))I)I). ((((II))I)) crosscut (major-minor quartet of toruses) rotating between minor pair of ditoruses ((((II)I)I)) and major pair of ditoruses ((((II))I)I). ((((II)I))) crosscut (minor quartet of toruses) rotating between minor pair of ditoruses ((((II)I)I)) and medium pair of ditoruses ((((II)I))I).
5. Tiger torus (((II)(II))I) - ((()(II))I) or (((II)())I) crosscut (empty) symmetrical, same as cut of medium stack of two ditoruses. (((I)(I))I) crosscut (2x2 array of toruses) rotating between two orientations of medium stack of two ditoruses (((II)(I))I) and (((I)(II))I). (((I)(II))) or (((II)(I))) crosscut (vertical stack of two minor pairs of toruses) rotating between minor pair of tigers (((II)(II))) and medium stack of two ditoruses (((I)(II))I).
6. 221-ditorus (((II)II)I) - ((()II)I) crosscut (empty) symmetrical, same as cut of two torispheres. (((I)I)I) crosscut (two toruses) rotating between ditorus (((II)I)I) and two torispheres (((I)II)I). (((I)II)) crosscut (two pairs of concentric spheres) rotating between minor pair of spheritoruses (((II)II)) and two torispheres (((I)II)I). (((II))I) crosscut (major pair of toruses) symmetrical, same as ditorus cut. (((II)I)) crosscut (minor pair of toruses) rotating between minor pair of spheritoruses (((II)II)) and ditorus (((II)I)I).
7. 320-tiger ((III)(II)) - ((I)(II)) crosscut (vertical stack of two toruses) symmetrical, same as tiger cut. ((II)(I)) crosscut (vertical stack of two toruses) rotating between vertical stack of two torispheres ((III)(I)) and tiger ((II)(II)). ((III)()) crosscut (empty) symmetrical, same as cut of vertical stack of two torispheres.
8. Torus tiger (((II)I)(II)) - ((()I)(II)) crosscut (empty) symmetrical, same as cut of two tigers. (((I))(II)) crosscut (vertical stack of four toruses) rotating between major pair of tigers (((II))(II)) and two tigers (((I)I)(II)). (((I)I)(I)) crosscut (two vertical stacks of two toruses) rotating between minor stack of two ditoruses (((II)I)(I)) and two tigers (((II)I)(I)). (((II))(I)) crosscut (vertical stack of two major pairs of toruses) rotating between minor stack of two ditoruses (((II)I)(I)) and major pair of tigers (((II))(II)).
9. 32-torus ((III)II) - ((I)II) crosscut (two spheres) symmetrical, same as spheritorus cut. ((II)I) crosscut (torus) rotating between torisphere ((III)I) and spheritorus ((II)II). ((III)) crosscut (two concentric spheres) symmetrical, same as torisphere cut.
10. 212-ditorus (((II)I)II) - ((()I)II) crosscut (empty) symmetrical, same as cut of two spheritoruses. (((I))II) crosscut (four spheres) rotating between major pair of spheritoruses (((II))II) and two spheritoruses (((I)I)II). (((I)I)I) crosscut (two toruses) rotating between ditorus (((II)I)I) and two spheritoruses (((I)I)II). (((II))I) crosscut (major pair of toruses) rotating between ditorus (((II)I)I) and major pair of spheritoruses (((II))II). (((II)I)) crosscut (minor pair of toruses) symmetrical, same as ditorus cut.
11. 221-tiger ((II)(II)I) - (()(II)I) or ((II)()I) crosscut (empty) symmetrical, same as cut of vertical stack of two spheritoruses. ((I)(I)I) crosscut (2x2 array of spheres) rotating between two orientations of vertical stack of two spheritoruses ((II)(I)I) and ((I)(II)I). ((I)(II)) or ((II)(I)) crosscut (vertical stack of two toruses) rotating betwee tiger ((II)(II)) and vertical stack of two spheritoruses ((I)(II)I).
12. 23-torus ((II)III) - (()III) crosscut (empty) symmetrical, same as cut of two glomes. ((I)II) crosscut (two spheres) rotating between spheritorus ((II)II) and two glomes ((I)III). ((II)I) crosscut (torus) symmetrical, same as spheritorus cut.

Here, the two dimensions we use for rotation are the two dimensions we don't use for cuts. Using two dimensions we DO use for cuts is meaningless as that would just rotate the 3D view. But we can also use 1 dimension of cut and 1 outside dimension. This would fix a plane at mid-cut.

Possibilities are:
1. Pentaglome (IIIII):
No new possibilities because of symmetry.

2. 41-torus ((IIII)I):
Rotation between two glomes ((IIII)) and torisphere ((III)I) with mid-cut changing from two spheres ((III)) to torus ((II)I).

3. 311-ditorus (((III)I)I):
Rotation between ditorus (((II)I)I) and major pair of torispheres (((III))I) with mid-cut changing from two toruses (((I)I)I) to major pair of toruses (((II))I).
Rotation between ditorus (((II)I)I) and major pair of torispheres (((III))I) with mid-cut changing from minor pair of toruses (((II)I)) to pair of concentric spheres (((III))).
Rotation between ditorus (((II)I)I) and minor pair of torispheres (((III)I)) with mid-cut changing from two toruses (((I)I)I) to minor pair of toruses (((II)I)).
Rotation between ditorus (((II)I)I) and minor pair of torispheres (((III)I)) with mid-cut changing from major pair of toruses (((II))I) to pair of concentric spheres (((III))).
Rotation between major pair of torispheres (((III))I) and minor pair of torispheres (((III)I)) with mid-cut changing from major pair of toruses (((II))I) to minor pair of toruses (((II)I)).

4. Tritorus (((II)I)I)I):
Rotation between two ditoruses (((I)I)I)I) and major pair of ditoruses (((II))I)I) with mid-cut changing from empty ((()I)I)I) to four toruses (((I))I)I).
Rotation between two ditoruses (((I)I)I)I) and major pair of ditoruses (((II))I)I) with mid-cut changing from two major pairs of toruses (((I)I))I) to major quartet of toruses (((II)))I).
Rotation between two ditoruses (((I)I)I)I) and major pair of ditoruses (((II))I)I) with mid-cut changing from two minor pairs of toruses (((I)I)I)) to major/minor quartet of toruses (((II))I)).
Rotation between two ditoruses (((I)I)I)I) and medium pair of ditoruses (((II)I))I) with mid-cut changing from empty ((()I)I)I) to two major pairs of toruses (((I)I))I).
Rotation between two ditoruses (((I)I)I)I) and medium pair of ditoruses (((II)I))I) with mid-cut changing from four toruses (((I))I)I) to major quartet of toruses (((II)))I).
Rotation between two ditoruses (((I)I)I)I) and medium pair of ditoruses (((II)I))I) with mid-cut changing from two minor pairs of toruses (((I)I)I)) to minor quartet of toruses (((II)I))).
Rotation between two ditoruses (((I)I)I)I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from empty ((()I)I)I) to two minor pairs of toruses (((I)I)I)).
Rotation between two ditoruses (((I)I)I)I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from four toruses (((I))I)I) to major/minor quartet of toruses (((II))I)).
Rotation between two ditoruses (((I)I)I)I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from two major pairs of toruses (((I)I))I) to minor quartet of toruses (((II)I))).
Rotation between major pair of ditoruses (((II))I)I) and medium pair of ditoruses (((II)I))I) with mid-cut changing from four toruses (((I))I)I) to two major pairs of toruses (((I)I))I).
Rotation between major pair of ditoruses (((II))I)I) and medium pair of ditoruses (((II)I))I) with mid-cut changing from major/minor quartet of toruses (((II))I)) to minor quartet of toruses (((II)I))).
Rotation between major pair of ditoruses (((II))I)I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from four toruses (((I))I)I) to two minor pairs of toruses (((I)I)I)).
Rotation between major pair of ditoruses (((II))I)I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from major quartet of toruses (((II)))I) to minor quartet of toruses (((II)))I).
Rotation between medium pair of ditoruses (((II)I))I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from two major pairs of toruses (((I)I))I) to two minor pairs of toruses (((I)I)I)).
Rotation between medium pair of ditoruses (((II)I))I) and minor pair of ditoruses (((II)I)I)) with mid-cut changing from major quartet of toruses (((II)))I) to major/minor quartet of toruses (((II))I)).

5. Tiger torus (((II)(II))I)
Rotation between medium stack of two ditoruses (((I)(II))I) and medium stack of two ditoruses (((II)(I))I) with mid-cut changing from empty ((()(II))I) to 2x2 array of toruses (((I)(I))I).
OR: Rotation between medium stack of two ditoruses (((I)(II))I) and medium stack of two ditoruses (((II)(I))I) with mid-cut changing from 2x2 array of toruses (((I)(I))I) to empty (((II)())I).
Rotation between medium stack of two ditoruses (((I)(II))I) and medium stack of two ditoruses (((II)(I))I) with mid-cut changing from vertical stack of two minor pairs of toruses (((I)(II))) to vertical stack of two minor pairs of toruses (((II)(I))).
Rotation between medium stack of two ditoruses (((I)(II))I) and minor pair of tigers (((II)(II))) with mid-cut changing from empty ((()(II))I) to vertical stack of two minor pairs of toruses (((I)(II))).
OR: Rotation between medium stack of two ditoruses (((II)(I))I) and minor pair of tigers (((II)(II))) with mid-cut changing from empty (((II)())I) to vertical stack of two minor pairs of toruses (((II)(I))).
Rotation between medium stack of two ditoruses (((I)(II))I) and minor pair of tigers (((II)(II))) with mid-cut changing from 2x2 array of toruses (((I)(I))I) to vertical stack of two minor pairs of toruses (((II)(I))).
OR: Rotation between medium stack of two ditoruses (((II)(I))I) and minor pair of tigers (((II)(II))) with mid-cut changing from 2x2 array of toruses (((I)(I))I) to vertical stack of two minor pairs of toruses (((I)(II))).

6. 221-ditorus (((II)II)I)
Rotation between two torispheres (((I)II)I) and ditorus (((II)I)I) with mid-cut changing from empty ((()II)I) to two toruses (((I)I)I).
Rotation between two torispheres (((I)II)I) and ditorus (((II)I)I) with mid-cut changing from two toruses (((I)I)I) to major pair of toruses (((II))I).
Rotation between two torispheres (((I)II)I) and ditorus (((II)I)I) with mid-cut changing from two pairs of concentric spheres (((I)II)) to minor pair of toruses (((II)I)).
Rotation between two torispheres (((I)II)I) and minor pair of spheritoruses (((II)II)) with mid-cut changing from empty ((()II)I) to two pairs of concentric spheres (((I)II)).
Rotation between two torispheres (((I)II)I) and minor pair of spheritoruses (((II)II)) with mid-cut changing from two toruses (((I)I)I) to minor pair of toruses (((II)I)).
Rotation between ditorus (((II)I)I) and minor pair of spheritoruses (((II)II)) with mid-cut changing from two toruses (((I)I)I) to two pairs of concentric spheres (((I)II)).
Rotation between ditorus (((II)I)I) and minor pair of spheritoruses (((II)II)) with mid-cut changing from major pair of toruses (((II))I) to minor pair of toruses (((II)I)).

7. 320-tiger ((III)(II)):
Rotation between tiger ((II)(II)) and vertical stack of two torispheres ((III)(I)) with mid-cut changing from vertical stack of two toruses ((I)(II)) to vertical stack of two toruses ((II)(I)).
Rotation between tiger ((II)(II)) and vertical stack of two torispheres ((III)(I)) with mid-cut changing from vertical stack of two toruses ((II)(I)) to empty ((III)()).

8. Torus tiger (((II)I)(II)):
Rotation between two tigers (((I)I)(II)) and major pair of tigers (((II))(II)) with mid-cut changing from empty ((()I)(II)) to vertical stack of four toruses (((I))(II)).
Rotation between two tigers (((I)I)(II)) and major pair of tigers (((II))(II)) with mid-cut changing from two vertical stacks of two toruses (((I)I)(I)) to vertical stack of two major pairs of toruses (((II))(I)).
Rotation between two tigers (((I)I)(II)) and minor stack of two ditoruses (((II)I)(I)) with mid-cut changing from empty ((()I)(II)) to two vertical stacks of two toruses (((I)I)(I)).
Rotation between two tigers (((I)I)(II)) and minor stack of two ditoruses (((II)I)(I)) with mid-cut changing from vertical stack of four toruses (((I))(II)) to vertical stack of two major pairs of toruses (((II))(I)).
Rotation between two tigers (((I)I)(II)) and minor stack of two ditoruses (((II)I)(I)) with mid-cut changing from two vertical stacks of two toruses (((I)I)(I)) to empty (((II)I)()).
Rotation between major pair of tigers (((II))(II)) and minor stack of two ditoruses (((II)I)(I)) with mid-cut changing from vertical stack of four toruses (((I))(II)) to two vertical stacks of two toruses (((I)I)(I)).
Rotation between major pair of tigers (((II))(II)) and minor stack of two ditoruses (((II)I)(I)) with mid-cut changing from vertical stack of two major pairs of toruses (((II))(I)) to empty (((II)I)()).

9. 32-torus ((III)II):
Rotation between spheritorus ((II)II) and torisphere ((III)I) with mid-cut changing from two spheres ((I)II) to torus ((II)I).
Rotation between spheritorus ((II)II) and torisphere ((III)I) with mid-cut changing from torus ((II)I) to pair of concentric spheres ((III)).

10. 212-ditorus (((II)I)II):
Rotation between two spheritoruses (((I)I)II) and major pair of spheritoruses (((II))II) with mid-cut changing from empty ((()I)II) to four spheres (((I))II).
Rotation between two spheritoruses (((I)I)II) and major pair of spheritoruses (((II))II) with mid-cut changing from two toruses (((I)I)I) to major pair of toruses (((II))I).
Rotation between two spheritoruses (((I)I)II) and ditorus (((II)I)I) with mid-cut changing from empty ((()I)II) to two toruses (((I)I)I).
Rotation between two spheritoruses (((I)I)II) and ditorus (((II)I)I) with mid-cut changing from four spheres (((I))II) to major pair of toruses (((II))I).
Rotation between two spheritoruses (((I)I)II) and ditorus (((II)I)I) with mid-cut changing from two toruses (((I)I)I) to minor pair of toruses (((II)I)).
Rotation between major pair of spheritoruses (((II))II) and ditorus (((II)I)I) with mid-cut changing from four spheres (((I))II) to two toruses (((I)I)I).
Rotation between major pair of spheritoruses (((II))II) and ditorus (((II)I)I) with mid-cut changing from major pair of toruses (((II))I) to minor pair of toruses (((II)I)).

11. 221-tiger ((II)(II)I):
Rotation between vertical stack of two spheritoruses ((I)(II)I) and vertical stack of two spheritoruses ((II)(I)I) with mid-cut changing from empty (()(II)I) and 2x2 array of spheres ((I)(I)I).
OR: Rotation between vertical stack of two spheritoruses ((I)(II)I) and vertical stack of two spheritoruses ((II)(I)I) with mid-cut changing from 2x2 array of spheres ((I)(I)I) to empty ((II)()I).
Rotation between vertical stack of two spheritoruses ((I)(II)I) and vertical stack of two spheritoruses ((II)(I)I) with mid-cut changing from vertical stack of two toruses ((I)(II)) to vertical stack of two toruses ((II)(I))
Rotation between vertical stack of two spheritoruses ((I)(II)I) and tiger ((II)(II)) with mid-cut changing from empty (()(II)I) to vertical stack of two toruses ((I)(II)).
OR: Rotation between vertical stack of two spheritoruses ((II)(I)I) and tiger ((II)(II)) with mid-cut changing from empty ((II)()I) to vertical stack of two toruses ((II)(I)).
Rotation between vertical stack of two spheritoruses ((I)(II)I) and tiger ((II)(II)) with mid-cut changing from 2x2 array of spheres ((I)(I)I) to vertical stack of two toruses ((II)(I)).
OR: Rotation between vertical stack of two spheritoruses ((II)(I)I) and tiger ((II)(II)) with mid-cut changing from 2x2 array of spheres ((I)(I)I) to vertical stack of two toruses ((I)(II)).

12. 23-torus ((II)III):
Rotation between two glomes ((I)III) and spheritorus ((II)II) with mid-cut changing from empty (()III) to two spheres ((I)II).
Rotation between two glomes ((I)III) and spheritorus ((II)II) with mid-cut changing from two spheres ((I)II) to torus ((II)I).
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### Re: 4D Toratopes Explained

Awesome, thanks for the information! Yeah, I've been thinking about the 5D toratopes explored the same way. It'd be neat to see all possible rotations shown with the 0, 22.5, 45, 67.5, 90 degree scanning. It would be important to show the progression, too, starting with 3D, then the 4D toratopes that are present, and finally the full 5D shape, making passes at different angles.

I would add one thing to these, though: a specific plane.

That would be good, too, but I'm not sure how to do that in CalcPlot right now. It wouldn't be realistic to edit each frame, since one gif is about 180 frames. But, there may be some tool in the program that does it.

Perhaps you should add the glome as well -- it only has one cut, but it's the "most basic" toratope in a way.

I have an old one, but no decent 3D analogy gif. Not like the new one of the torus, above. So, that'll be something to do.
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