Surfaces in 4D

Discussion of shapes with curves and holes in various dimensions.

Surfaces in 4D

Postby bsaucer » Sat Jan 07, 2006 4:42 am

How many ways can a surface "curve" or "bend" in 4D? In 3D a surface can be flat, rounded (like a bowl), saddle-shaped, or rolled up like a cylindar. I'm referring to the ways a surface can bend at one of its points, not the topology of the whole surface.
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Re: Surfaces in 4D

Postby Marek14 » Sat Jan 07, 2006 8:57 am

bsaucer wrote:How many ways can a surface "curve" or "bend" in 4D? In 3D a surface can be flat, rounded (like a bowl), saddle-shaped, or rolled up like a cylindar. I'm referring to the ways a surface can bend at one of its points, not the topology of the whole surface.


Well, imagine two perpendicular straight lines tangent to the surface in a certain point. Each of them can lie, say, in the surface, below, or above. Let's mark these three cases with 0, +, and -

Flat surface then corresponds to 00. Bowl is ++ or --, saddle is +- and cylinder is +0 or -0. All combinations are used now. This means that for 4D, where we have three mutually perpendicular tangent lines, the possibilities are:

000
+00 or -00
++0 or --0
+-0
+++ or ---
++- or +--

000 is a curvature of flat hyperplane. +00 can be found on cylinder based on another cylinder (cartesian product of circle and plane), ++0 is in spherical cylinder, +-0 is in cylinder based on one-piece hyperboloid (among others), +++ is glome, and ++- is on some of 4D hyperboloids.
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Re: Surfaces in 4D

Postby quickfur » Sat Jan 07, 2006 6:35 pm

bsaucer wrote:How many ways can a surface "curve" or "bend" in 4D? In 3D a surface can be flat, rounded (like a bowl), saddle-shaped, or rolled up like a cylindar. I'm referring to the ways a surface can bend at one of its points, not the topology of the whole surface.

What kind of "surface" are you speaking of? In 4D, you have 2D surfaces and 3D manifolds.

2D surfaces have a LOT more freedom in 4D, and can twist into bizarre shapes such as Klein bottles, or the Real Projective Plane (very pathological shape! I still can't quite visualize how it is connected). However, 2D surfaces do not bound a closed 4D region (they are as "thin" as strings, in 4D). Klein bottles cannot hold water, unfortunately.

3D manifolds are the 4D analogs of surfaces in our 3D world, and share many analogous properties. But because they have 3 dimensions, they also have more degrees of freedom than surfaces in 3D. Take a cylindrical 3-manifold, for example. You can either twist it around the 3-plane it lies in and join it end-to-end to make a 3D torus, or you can twist it perpendicular to the 3-plane, join it end-to-end, and make a different kind of torus: half the surface of a duocylinder. Or, you can fold it in 4 equal sections by 90-degree angles, to make the "flat" sides of a cubinder. Alternatively, you also have the round side of a cubinder, which you can get by rolling up a 3-manifold in one dimension. Rolling up a 3-manifold in two dimensions makes a spherindrical surface. Then you also have the surface of a 3-sphere, which is a 3-manifold rounded in 3 dimensions. You can also make spheri-conical surfaces by cutting a 3D "sector" from a 3-ball and rolling it up so that its cut faces join.

The possibilities are fascinating.
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Postby bsaucer » Sun Jan 08, 2006 2:48 am

I am referring to 2D surfaces. I'm not trying to make closed manifolds, but just want to know how many ways a small piece of 2D manifold can curve or bend in 4D.
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Postby wendy » Sun Jan 08, 2006 8:22 am

In 4D a surface is three-dimensional. A two-dimensional manifold is a hedrix, and as much divides (or is a surface), as a line in three-dimensions. Marek did indeed answer the question as asked.

For hedrices (2d manifolds), the curving is as in three-dimensions, really.

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Postby quickfur » Mon Jan 09, 2006 12:59 am

bsaucer wrote:I am referring to 2D surfaces. I'm not trying to make closed manifolds, but just want to know how many ways a small piece of 2D manifold can curve or bend in 4D.

2D surfaces in 4D have as much freedom as 1D strings in 3D. I.e., you can tie them into knots (along the entire perimeter!), you can coil them up, and do all sorts of weird things to them. (Think of a tangled mess of wires that usually sits behind a computer, except that the tangle is twice as complex, because 2D surfaces can be connected in 2 orthogonal directions.)

This is how you can fold 2D surfaces into Klein bottles, or get truly bizarre things like the Real Projective Plane, or RP<sup>2</sup> for short. (To get an idea of how bizarre RP<sup>2</sup> is, you might want to look at this and read this.)

The caveat here, of course, is that the "Klein bottle" is a misnomer, because it cannot hold any water.
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