I have had a little exposure to mathematics here and there. But I have always found it a fascinating topic. One thing I have always found interesting is the graphing system, and its relationship to other dimensions besides the 3 known ones. As I understand it, most scientists assume, time is the fourth dimension.
My question is simply this: what does a 4th-dimensional hypersphere look like? I have tried in the past graphing it as well as I could on two-dimensional graphing paper. And I suspect it looks like a balloon suddenly blowing up, then disappearing. But two-dimensional graphing paper is all I have to go by, so I am still not sure.
So what does it look like? And while we’re at it, is there any website or free (or not free) software on the net that would allow me to see fourth-dimensional graphs?
A 4-d hypersphere looks like a 4-d hypersphere. Nothing in any lower dimension can reproduce it exactly, and we can only perceive three dimensions.
If you want to use time as the fourth axis, you can get some idea of what the unit 4-sphere centered at the origin is like. Imagine starting at t = -1 with a point at the origin, and growing a sphere until t = 0, when you have the unit 3-sphere centered at the origin. Then you start shrinking back down to the point at the origin with t = 1. More generally, the sphere at time t is the set of points (x, y, z) satisfying x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = 1 - t[sup]2[/sup].
You can try to "visualize 4D (and higher) objects by contemplating their projections in a lower dimension, or by unfolding them in a lower dimension, or by looking at their intersection with a lower dimension.
Hyperspheres aren’t very interesting in this regard – all their projections and intersections are spheres and circles. In Edwin Abbott’s clasic Flatland, a 3D sphere interacts by intersecting with a 2-D world, which sees him as a circle – but a circle that changes size as he moves his plane of intersection through his third dimension. The 2D beings of Flatland cannot comprehend the shape of the sphere as he moves in a direction perpendicular to their world, but they see his circular symmetry. In just the same way, the intersection of a 4D hypersphere through our 3D world would be a sphere of changing size that finally disappears when it’s moved more than a disance equal to its radius from the “space” where its center interscets our world.
You might have more luck – or gain more insight – by contemplating a 4D cube (tesseract), which you can project, unfold, or intersect with our world. Or a 4D triangle/tetrahedron figure, the other logical extension of extending a figure to higher dimensions.
This is why time is considered the fourth dimension. A sphere exists over a period of time, and each “cross-section” of its 4D existence is a momentary 3D sphere.
nitpick: everyone here is describing a three-dimensional sphere, not a 4-D one. The sphere we’re used to – the surface of a ball – is a 2-sphere. We name geometric objects by their own dimensions, not by the dimension of a common embedding space.
A 4-sphere would be the set of 5-tuples (a,b,c,d,e) so that
If you want to know what a 3-sphere (thanks Mathochist) in 4-d space* looks like (or a 4-sphere in 5-d space), imagining time as one of the dimensions doesn’t help.
*“In 4-d space” here means “drawn on 4-d graph paper” because you only need 3 dimensions for a 3-sphere – just not our boring 3 Euclidean dimensions. You need a nice round non-euclidean universe for that.
It sounds like you’re saying a sphere is a surface, and therefore is two dimensional. I get that (but they never told me that in school). What is the name of the corresponding solid? Wouldn’t that solid be truly a 3D object?
I believe you need 3 dimensions to describe a point on the surface, too, but that surface has no thickness so is 2-dimensional (but curves within 3-space).
Okay, Hamsters, that was cute: register that I edited the post, but don’t register the text I edited to add. Don’t try it again. I have my eyes on you!
Anyway, as I was saying (and I’m sure a topologist will correct me if I am mistaken) I don’t believe one can say a sphere is “hollow” because the sphere isn’t. the “hollow part” isn’t part of the sphere itself. I can accept a “hollow ball” (marginally) as a spheroidal shell of nonzero thickness (and hence, not a sphere)
It’s the same relationship shared between a circle (a 1 D line embedded in a 2-D space), a disk (a 2-D surface bounded by a circle) and a ring (a 2-D surface bounded by a circle on the “outside” and a smaller circle “inside”) – though of course, since we are accustomed to conceptualizing/living in 3-space plus time, we commonly use the “disk” and “ring” to mean short cylinders with a disk or ring cross-section.
Yes, when you talk about a sphere you have a fixed radius in mind.
Or topologically it doesn’t matter, 'cause you’re talking about the sphere itself, independently of any particular embedding into space. Roughly speaking, in that view you say it’s 2-dimensional 'cause when you zoom in really close it looks like it’s a 2-dimensional plane. This is just the mathematical version of the way the Earth looks flat until you get far enough away from it.
Do you mean a ball (all points with less than a certain distance from a fixed point)? Or do you mean something like a 2-sphere with a finite thickness – what’s left after you take a ball and scoop out a ball with the same center but a smaller radius?
In the latter case, I’ve seen various terms, and there is no standard. “Spherical annulus” and “spherical shell” get the connotation of a ball-with-ball-shaped-hole. “Thickened sphere” gets the connotation of all the points within some small distance from a reference sphere.
The clarification still holds. Mathematicians describe shapes (like spheres) in terms of intrinsic properties (like how many degrees of freedom one has to move around on them), not in terms of extrinsic properties like the dimension of the embedding space.
I’m sure that the OP knew what object he was thinking of, but it’s still worthwhile to know the proper terminology.
