Wrapping a 3D space wihtin a 4D space

[Sebastien77]

Let’s imagine a moving point bound to a 2D plane. If we wrap this plane on a sphere within a 3D space the point would now eventually end up at the same position while moving in a seemingly fixed direction (it is actually not fixed in the 3D space). I am now wondering: Can a 3D hyperplane be wrapped in such way inside a 4D space so that a point moving along a (seemingly) fixed direction inside the wrapped cube would also end up at the same position? What could be the resulting shape of this surface, a hypersphere?

The curved surface of the sphere is not a 2D hyperplane in the 3D space that can be defined by a 2D basis. It is "closed" and as a finite surface, still it is somehow two dimensional… is there a name and theory describing the geometry within this sort of space (distance, angle, projection...)? What about the wrapped 3D hyperplane?

We perceive the space around us as having 3 spatial dimensions but it is hard to picture it as infinite, it might just be wrapped to a closed space. Additionally the closed surface it lies on might be somehow inflating in this embedding 4D space that elude our perception.

 

[Simon Bridge]

Welcome to PF;
You can make a 4D hyperplane closed so that it is finite and unbounded in a manner analogous to making a 3D plane finite and unbounded by mapping it onto the surface of a sphere or other 3D volume. To understand how to work this, and how to communicate better in higher dimensions, it is a good idea to ground yourself in the maths.
https://www2.bc.edu/~reederma/Linalg18.pdf

The surface of a sphere has only 2 dimensions - though it is not a plane. There is nothing wrong with this.
I think the name you are looking for is "topology".

One of the popular theories of the overall topology of the Universe is that it is finite and bounded - if this is actually true, the evidence is that the overall curvature is very small. It is not generally considered as a 3D volume embedded in a 4D superspace ... that would just be begging the question of what the overall topology is: that way lies turtles all the way down. Instead, we consider the topology to be a 3+1D space-time with an intrinsic curvature ... that is, it is not curved in some other higher-D space-time whatsit.
https://www.physicsforums.com/threads/extrinsic-and-intrinsic-curvature.67724/
http://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds

 

[Sebastien77]

Hi Simon, thanks for your answer and references!

I am somehow familiar with vector field theory and the Euclidian geometry and understand that the dimension of an hyperplane is the minimum number of linearly independent vectors (of its embedding N-dimensional space) that are required to linearly decompose any of its elements. Now the surface of the sphere is not an hyperplane so how to compute the dimension of this subspace? I guess we could state that since there exists a function defined in R_2 --> R_3 to locate any point on the surface of the sphere then the dimension of this subspace is 2, but how can we prove than there is not any R_1 --> R_3 function doing the same job?

You said that some finite and unbounded 4D hyperplane exist, could you name some? Do some of them also have this property that moving in any fixed direction you end up at the initial position? Is there a way to represent their projections in a lower dimension space?

About the universe being 3 + 1D rather than 4D, I think that the representation of space-time is based on non Euclidian geometry, where time plays a very special role. Could this brings the same implication as for finite unbounded 4D hyperplane that moving in a fixed direction you would end up at the starting point?

 

[Philip]

There are numerous ways to fold a 3D space into 4D, with self-repeating curvature. But, we're not limited to just 4D, there are a few ways in 5D as well, and one more in 6D. Most of these closed surfaces can be made by folding a cube, and gluing its square-faces together. In addition to a 3-sphere , S³ (4D sphere) , there are a number of toroidal shapes as well. These are the four fundamental donut rings in 4D, with a self-repeating 3D surface.

Moving any direction on the surface of a sphere will yield a single value of repeating space: the circumference. Moving on the surface of a torus, however, will yield two different repeating distances. One around the small circle of the ring, and one around the big circle of the outside edge. Not to forget the inside edge of the hole, or top/bottom of the ring. You can move in any straight line you like, where you traverse two different sizes of repeating distance. These surfaces can be described with something called fiber bundles. Using the n-sphere terminology, we can represent how these spaces become expanded into each other. A torus can be built by revolving a circle S¹ along the outside edge of a bigger circle S¹ , making S¹ x S¹. The order reads left to right of increasing size. I provided the implicit surface definition as well:

Sⁿ - infinite repeating n-dimensional space on surface of sphere

2D

circle - S¹ = x² + y² - a²

3D

sphere - S² = x² + y² + z² - a²
torus - S¹ x S¹ = T² = (√(x² + y²) - a)² + z² - b²

To make things simpler when getting several S¹ 's together, we simply use Tⁿ to represent n-number of repeating S¹.

In 4D, we get the 3-sphere, and four ring-like shapes with holes. Now, bear with me here on this one. In order to describe one of these properly, I have to get inventive. I'm adopting the notation C2 to represent the duocylinder ridge, a.k.a. the Clifford Torus: http://en.wikipedia.org/wiki/Duocylinder#The_ridge . This is a smoothly curving 2-manifold that bends into 3 and 4D. It can be made by folding a square into a hollow tube, then instead of bending and joining the ends in 3D, we do this into 4D. It's identical to the shape of a regular 3D donut. There are many more types of flat tori in higher dimensions, that are based on this unique and fundamental structure.

4D

glome : S³ = x² + y² + z² + w² - a²

torisphere : S¹ x S² = (√(x² + y² + z²) - a)² + w² - b²

spheritorus : S² x S¹ = (√(x² + y²) - a)² + z² + w² - b²

3-torus : S¹ x S¹ x S¹ = T³ = (√((√(x² + y²) - a)² + z²) - b)² + w² - c²

tiger : S¹ x C2 = (√(x² + y²) - a)² + (√(z² + w²) - b)² - c²

( Some names are not my own, they were developed on the HDDB, http://hddb.teamikaria.com/forum/ )
Moving in any combination of X, Y, Z on a glome will yield a single repeating distance. The four toric shapes will have different lengths of repetition, as you encircle the ring or any of its holes. The 3-torus has three different repeat lengths, one for each cardinal direction. The tiger has two independent large, and one small. For all toroidal shapes, the shortest distance around the ring is accessible from any location on the larger diameter frames.

As mentioned above, there are two ways to fold a cube into 5D, making flat-torus analogs of the Clifford torus. Both of them are the edges of 5D hyperprisms. These prisms are duo-cylindrical, with 2 separate, curved 'rolling' sides only. Where these rolling sides join is a 90 degree edge, that is a smoothly curving, self repeating 3-space.

Ridge of 5D Cylspherinder : S² * S¹
--- Implicit of cylspherinder : |√(x² + y² + z²) - √(w² + v²)| + |√(x² + y² + z²) + √(w² + v²)| - a²
--- ridge is cartesian product of 2-sphere and a 1-sphere
--- Prism is made by extending a 3D sphere along W into 4D, then bisecting rotate around plane XYZ into 5DRidge of 5D Cyltorinder : C2 x S¹
--- Implicit of cyltorinder : |√((√(x² + y²) - a)² + z²) - √(w² + v²)| + |√((√(x² + y²) - a)² + z²) + √(w² + v²)| - a²
--- ridge is clifford torus C2 bundle over the circle, a torus of a Clifford torus
--- Prism is made by extending a 3D torus along W into 4D, then bisecting rotate around plane XYZ into 5D
--- also made by duocylinder bundle over a circle, a 5D torus with duocylinder cross cut

And, finally, the six dimensional way to fold a cube is simply the ridge of a 6D triocylinder. This is the cartesian product of three solid disks. Also made by extending a 4D duocylinder into 5D, then bisecting rotate around plane XYZW into 6D.

Ridge of 6D Triocylinder : C3 (again, getting inventive with new notations)
--- Implicit definition of triocylinder : ||√(x² + y²) - √(z² + w²)| + |√(x² + y²) + √(z² + w²)| - 2√(v² + u²)| + ||√(x² + y²) - √(z² + w²)| + |√(x² + y²) + √(z² + w²)| + 2√(v² + u²)| - a²
--- ridge is the 'Clifford 3-torus' , a closed 3-space that curves into 4, 5, and 6D.

As for visualizing toroidal shapes, or anything curved, projections don't do too well with showing structure. Not as well as cross sections. Best way is to pass a hypertoric shape through a 3D plane. This will make what we see in 3D evolve and morph, with fantastic cassini deformations. Four dimensional toric shapes are cool and all, but the real awesomeness begins in 6 and 7D.

Hope that helps!

 

[phinds]

One of the popular theories of the overall topology of the Universe is that it is finite and bounded

I think you meant UNbounded here, yes?