Eli5: 4th dimensional bodies, specifically the hypersphere



I can’t visualize them right now and I need someone to explain them to me simply

In: Mathematics


Specifically for the hypersphere:

Imagine a circle. It is essentially a closed 1d space. A line that closes in on itself. You can move in any direction on the line and you will end up where you started. All points are connected. It is the collection of points at an equal distance from a set center. You could call a circle a ‘2d sphere’.

A sphere is a closed 2d space. A surface that closes in on itself. You can go in any direction on the surface of a sphere and end up where you started. It is a collection of points at an equal distance from a set center, like a circle, but in 3d. You could call a sphere a ‘3d circle’. You can also imagine it as an infinite collection of same-sized circles with the same center, rotated in 3d space.

A hypersphere, then, is a closed 3d space. You can move in any direction on the three dimensional axes and you will end up in the same spot. It is also an infinite collection of spheres with the same center, rotated in 4d space.

We do not live in 4d space so we have a hard time imagining this kind of shape. We are very used to 3d shapes being represented in 2d, so much so that we hardly realise it. But these are optical illusions, like any hypersphere or tesseract video you’ve seen, but less obvious.

This video helps to make it clear, imho. French accent is an added bonus: https://youtu.be/RFK2_dAZvLo

Let’s try to play with cross sections and use time as a tool for representing one of the axes.

Imagine a 3D sphere and three axes: X, Y and Z. Let’s say we have an ability to move along Z and see the cross section of this sphere parallel to XY. While you move evenly, you would see nothing first, then a rapidly growing circle would appear reaching the diameter of the sphere and then collapsing back to nothing. Pretty easy, right? This was a visualization of a 3D object on a 2D surface stretched in time.

Now let’s do the same trick with a 2D circle, moving along Y and looking at its cross section parallel to X. Same here, nothing first, then a line would appear, grow, shrink and disappear in the same manner as a circle in the previous example. This was a visualization of a 2D circle on a 1D line.

You can see how these two examples share very similar behavior, and I think it’s acceptable to apply the same principle to a 3D cross section of a 4D sphere existing in a XYZU world. While moving along axis U, you would see a growing-and-shrinking 3D sphere.

Like Traditional_Dinner16 said, we live in a 3D world and can only visualize all 4D shapes as 3D cross sections using some tools like time to get the information about the dimension impossible for perception. Like a 2D person living in a 2D world would visualize all 3D spheres as 2D circles, getting information about third dimension using time or so.

Actually it’s possible to go even deeper and get some idea of 5 and more dimensional shapes applying this model.

Imagine a 2D universe. Totally flat. When a being or object (Benny) from a 3D universe interacts with the 2D universe, the beings in 2D will only ever be aware of the parts of Benny that are currently in their dimension. A very thin cross section of Benny. And as Benny moves, that cross section will change as different parts of Benny move in and out of the 2D space. 2D can’t comprehend what Benny is really like because they only ever see one tiny incomprehensible part at any given time. They might understand Benny’s shadow as a 2D object but little more than that.

Now see if you can translate that in your head to a 4D object interacting with our 3D universe. We would only ever comprehend part of the shape. Its shadow would be a 3D object we could probably understand.

A 1-sphere is a circle, a 2-sphere is the sphere in the usual vernacular.

There is a simple way to visualize a 3-sphere. What is it and how does it work?

Let’s start with a 2-sphere, like the surface of Earth. If you have ever seen the United Nations logo, then you have an idea what the [Azimuthal equidistant projection](https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection#/media/File:Azimuthal_equidistant_projection_SW.jpg) is. It’s not the Earth’s surface, but it’s its *map.*

As you would expect from a map, each point on the map corresponds to a real point on the Earth. Also the map preserves “nearness” in the sense that points that are near in real world are also near on the map, but with one crucial exception. The south pole does not appear on this map as one point, rather it’s the whole boundary circle.

To get the original sphere from the map, you would need to replace the bounding circle with a single point. Imagine cutting off this boundary circle and enclosing the remaining interior by gluing it to a point representing the south pole. You get the shape of a sphere.

Now, repeating this process by starting with a solid ball, being a *map* of a 3-sphere, where the boundary of this ball, the 2-sphere, represents the south pole, you can get a good idea what kind of shape a 3-sphere is.