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', S

^{1}xS

^{1}in something called fiber bundles. It's another way of saying circular ring S

^{1}stretched over the surface of a circle S

^{1}.

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, S

^{1}xS

^{2}. 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, S

^{2}xS

^{1}, 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, S

^{1}xC2. 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, S

^{1}xS

^{1}xS

^{1}, or simply T

^{3}. 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.