Description of a 4th dimensional sphere

Question

I have been wondering what a 4th dimensional "sphere" is like, because I already know what a tesseract looks like, and I am wondering if there is a description of a 4th dimensional "sphere".

Answer 1

First, as a commenter mentioned, as a technical term, "sphere" refers to the surface of a ball, not its bulk. More specifically the surface of a ball is the 2-sphere (note it's a 2d object) and the whole ball is the 3-ball. Thus you are referring to either the $4$-ball or the $3$-sphere.

There are quite a few tricks. One is to imagine projections and cross sections. Just like a sphere has disk cross-sections that start as points, get to a maximum size at the equator, and then taper off again, a 4-ball will have 3-ball cross-sections that start as a point, grow to the full radius at the "equator" (which is a 2-sphere) and then taper off.

If you like, you can imagine that the 4-ball is moving through our 3-space and we can only see the intersection. This is like a 3-ball moving through a flat plane. The people on the plane will see a disk expand and contract.

As a trick for imagining the 3-sphere, you can do much the same thing as above. However, in the comments achille hui suggested a nice alternative idea that might be a bit cryptic without further explanation. We can imagine cutting the 2-sphere along the equator and being left with two hemispheres. But the hemispheres are just curved disks. Thus, topologically, the 2-sphere is just two disks glued along their boundary.

Similarly, a 3-sphere is just two (curved) balls glued along their boundary. So there's a northern hyper-hemisphere and a southern hyper-hemisphere (both 3d bulks with a 2-sphere boundary) glued along their two-sphere boundary (the "equator", which we said before is a 2-sphere).

Answer 2

Most approaches give you the idea of the shape, but don't preserve lengths or anything geometric like that.

Method 1: Stereographic Projection

The stereographic projection takes a 1D circle (living in 2D space) minus the top point to a line. So a circle is like a line where there's a point "at infinity" connecting the "ends".

This projection takes a 2D sphere (living in 3D space) minus the north pole to the entire 2D plane. So a sphere is like a plane where there's a point "at infinity" that you approach by flying away from the origin in any direction.

Analogously, it takes a 3D hypersphere (living in 4D space) minus the north pole to the entire 3D space. So it's like space plus a point "at infinity" that you approach by flying away from the origin in any direction.

Method 2: Gluing two pieces

A circle is what you get when you take two closed line segments, bend them appropriately, and glue their endpoints together.

A sphere is what you get when you take two closed discs, bend them appropriately (to make the northern and southern hemispheres), and glue them together along their circle boundaries.

Analogously, as achille hui commented, A 3D hypersphere in 4D space is what you get when you take two closed 3D balls, bend them appropriately (in the fourth dimension), and glue them together along their sphere boundaries.

Method 3: Using time

For 4 dimensional things, it's common to use time for the fourth dimension. In some systems of natural units, you could in theory get the lengths in the different dimensions to match up correctly, which the other methods don't capture. And even if you don't bother to scale time with length properly, you can still get a representation of a hyperellipsoid at least.

You could see horizontal slices of a circle as a point, splitting into two that move apart, then move back together.

You could see horizontal slices of a sphere as a point, that grows into bigger and bigger circles, and then shrinks back down to a point again.

You could see "horizontal slices" of a 3D sphere (living in 4D) as a point that grows into bigger and bigger spheres, then shrinks back down to a point.

Method 4: Using color/shading

For 4 dimensional things, it's also common to use a color or shading gradient to represent the fourth dimension. It can help with things like the klein bottle, but isn't helpful here. Based on the descriptions in methods 2 and 3, you would get something like "a solid ball whose inside has two opposite colors simultaneously, with stronger colors towards the center". You can't really see something like that properly.