What is a hypersphere and how can we comprehend its time dimensions?

[billy_boy_999]

i would like for once to find a non-analogous, qualitative description of spacetime...i can readily picture the 2-dimensional analogies of curved space but i want to be able to, not picture, but comprehend using something like my normal ideas - or even protracted but logical ideas - about space and time...

start with a 4-dimensional sphere ('hypersphere')...what are the time dimensions of a hypersphere that differentiate it from a 3-dimensional sphere? can we describe such a sphere using 3-dimensions and adding on, qualitatively, the variables of its dimensions in time in order to gain a qualitative comprehension?

i understand this is a very difficult thing to ask, a qualitative, non-mathematical description of 4-dimensions without using 2-dimensional analogies, but I'm wondering if there is still a way to do it?

 

[yogi]

Hi BB: Humans cannot visualize 4 spatial dimensions - but we can treat them mathematically. At present it cannot be said with certainty whether the universe is a hypersphere - the metric equation enables any point on a curved three dimensional space to be described without resort to extra dimensions - that is, it is not necessary to consider a curved three dimensional universe as being embedded in a four dimensional space.

 

[turin]

Try this:

Imagine blowing up a balloon of infinite capacity. You see the balloon in front of your face. You decide that you are clearly on the outside of the balloon. As you blow into the balloon, you observe the surface of the balloon just in front of you to steepen as the surface approaches flatness. You note that at some point, the balloon is so large that all you can see is a flat plane of rubber in front of you. You keep blowing. To your surprise, the balloon begins to curve slightly backwards around you. Your first thought is possibly that the other side of the balloon opposite from you has hit a wall, and the rest of the expansion of the balloon has nowhere else to go but to continue expanding behind you. Later, you find that this is not the case; as you continue blowing you find that you are inside the balloon and it gets smaller for every breath. Physically, the fact that the balloon gets smaller as you blow through the hole should not surprise you. By blowing through the hole, you are essnetially transfering the air from within the balloon into your nose out through you mouth and into the environment. The startling fact is that you began on the outside but you somehow wound up on the inside, even though you continued to blow up the balloon. This is one consequence of a spherical 3-D space. At some point, not only did the balloon appear to be a flat plane sheet of rubber, but it literally was a flat plane sheet of rubber. You were outside the balloon before this point; you became inside the balloon afterwards.

It should be mentioned that you and the balloon are the only two objects in this 3-sphere universe. If you imagine another object that began outside the balloon, then this will emphasize the spherical nature of the 3-D space by appearing insider the balloon along with you. This shows that the balloon did not simply hit a wall, nor did it get forced inside out, but that it expanded to a point at which space itself folded back on itself. In other words, outside and inside cannot be strictly distinguished, at least, not in the usual way.

Deepening the level of abstraction a bit, this amounts to the volume of 2-spheres and the area of circles increasing in the same sense as the radius until this radius reaches the radius of the 3-sphere. After this point, an increase in radius will cause a decrease in the volume or area of a 2-sphere or circle.

 

[LURCH]

For my first intuuitive step into higher dimensions, I found a hypercube to be more usefull to think about then a hypersphere. If we take a line (one dimension) and expanded so it becomes a plane (2 dimentions), we can form a square. At the corner of a square two lines meet at 90^o. Expanding our square into three dimensions, we'll find that each corner has three lines that meet, each at 90^o to the other two. Not we have proven that two lines can meet at right angles, and three lines can meet, all at right angles to one another, so why not four? In this way, we can logically deduce the existence of a fourth dimension to the cube, adding a fourth line at right angles to the other three, even if we cannot picture it.