Rotating Four Dimensional Donuts

A Rotating 3D Donut, while stuck in a 2D Plane

This gallery can be used in conjunction with the other 4D donut gallery, which shows the objects passing through the 3-plane, at different angles of the proceeding rotation animations : http://imgur.com/a/asnRZ .

Here are various views of a 3D donut, being sliced by a lesser two dimensional space. The important part of the concept is how the donut looks, inside the slicing 2D plane. We can describe the 3D donut as a solid circular ring that curves around in a circle.

In 2D, we'll get two circles : side by side (disjoint pair) , one inside the other (concentric pair), and many forms in between. The donut is 3D, so when looking at it in 2D, there will always be missing parts of it. These missing parts exist in the extra 3rd dimension of space.

Since we can't see all of the 3D donut at once in 2D, we set the slice in motion, by giving it a continuous 360 degree rotation. This lets us see how the separate objects in the 2D slice connect, into the extra 3rd dimension of space.

When we see the two disjoint circles side by side, the rest of it connects in big circular arches, above and below the 2-plane. At this angle, the circular arches are invisible from 2D view. When we see the concentric circles, the big arches are being sliced through sideways and fully revealed.

Other views: Passing through the 2-plane at different angles: http://i.imgur.com/4ViRGiB.gif and http://i.imgur.com/vkCYXbZ.gif

The /r/math link, which has some better descriptions and elaborations of this gallery : https://www.reddit.com/r/math/comments/3zilv4/rotating_four_dimensional_donuts/

A Rotating 4D Spheritorus

A 4D donut, shaped as a solid spherical ring that curves around in a circle. When we see the two disjoint spheres, side by side, they are connected by big circular arches, that exist above and below the 3-plane invisible from view.

Turning 90 degrees will slice through the big circular part, and reveal a 3D donut. The solid small ring of the donut-slice is connected spherically, in 4D space.

Passing through a 3-plane at different angles : http://i.imgur.com/cYzvzEl.gif

A Rotating 4D Torisphere

A 4D donut shaped as a solid circular ring that curves around, over a sphere. This donut has a different shape to its big arches. Instead og being a larger circle, it's the 2D sheet of a large sphere, with the solid ring stuck in it.

When we see the 3D donut-slice, the rest of the big arches curve around spherically into 4D. When we see the two concentric spheres (one inside the other), the solid ring is being sliced in half, and reveals the spherical shape of the 'arch'.

Passing through a 3-plane at different angles : http://i.imgur.com/aKueBYr.gif

A Rotating 4D Tiger

This is the strangest 4D donut, of them all. It's a solid, donut-shaped ring that curves around in a circle into 4D. Both coordinate 3-plane slices are this vertical column of two donuts. They join by the missing, circular arches, that exist in 4D space.

Since the solid ring has a hole to begin with, and the ring curves around in a circle to make a 4D ring-like object, the resulting shape has 2 holes that are independent of each other.

Passing through a 3-plane at different angles : http://i.imgur.com/j7LWsnP.gif

A Rotating 4D 3-torus Type A

This 4D donut is very similar to a 3D donut. Just like the tiger above, the 3-torus also has two holes. The solid, donut-shaped ring curves around in a big circle into 4D.

When we see the 2 disjoint donuts, side by side, the rest of it connects in big circular arches above and below the 3-plane, into 4D. Rotating back, and we'll see that big circular part of it.

Passing through a 3-plane at different angles : http://i.imgur.com/w4On50F.gif

Rotating 4D 3-torus Type B

Another way to rotate the 3-torus.

Passing through a 3-plane at different angles : http://i.imgur.com/lzNpNbu.gif

Rotating 4D 3-torus Type C

The third way to rotate a 3-torus.

Passing through a 3-plane at different angles : http://i.imgur.com/1J9dCQd.gif