Visualizing the hyperspheres (only the hyperspheres, no tesseracts etc. please)

[Secret]

Hi I'm new to the forums

I have a great interest in dimensions since i was about 7
But only started to search for dimension stuffs when i was about 15 (as I failed to understand the concept of dimensions when i was 7)

Like many users in the 4d topic, i also have a hard time visualizing higher dimensions

I think the key to understand higher dimensions is the ability to visualize the corresponding hyperspheres

But i can't find any 3D projection of them on the internet (unlike other higher dimension objects)

Therefore can you provide me guides to visulize them? (no other objects please, only hypershperes)

thanks!

 

[K^2]

Yeah... The problem is that projection of 4-sphere onto 3-space is a 3-sphere. So visualization of a 4-sphere on a flat screen is identical to visualization of a 3-sphere.

Of course, the same problem is encountered when you try to distinguish between 3-sphere and a circle on a 2D projection. The solution is usually to draw an "equator" at an angle to the perspective. So basically, you draw half of an ellipse inside a circle and call that a sphere.

Corresponding effect for projecting from 4D to 3D would call for an ellipsoid drawn inside the sphere. The way it fills the sphere depends on the projection angle, but it will touch the sphere at two points. You can then project that onto a screen by drawing "equators" on sphere and ellipsoid.

What you'll get is a mess of ellipses embedded in a circle. It is not terribly useful, and I'm not surprised you haven't found any good depictions.

If there is a better way to go about it, I'm not aware of it.

Edit: Here is a logo for a company called HyperSphere. Appropriately enough, it features a schematic representation of what I was just talking about.
[PLAIN]http://www.hypersphere.com.au/images/hs-logo.gif

 

[Secret]

Thanks for the solution

(now to see whether it is possible to inflat the tesseract onto the 3 sphere and try to make sense of the 4 volume)

New edit: I often heard from people that the 3 sphere is shaped like a donut
I think it's weird because n spheres are defined as the locus of a point which always keep a fixed distance away from a fixed point with n coodrdinates

Edit2: Does a n torus also obey the rules of hyperspheres? (Not off topic because it is something in connection with the hyperspheres)

 

[K^2]

No. An N-sphere has a different topology than an N-torus. An N-torus can be mapped onto an open region of R^N space by a continuous 1-to-1 function. An N-sphere cannot. You'll always have at least one point that cannot be mapped left over.

An N-torus is going to be defined very different from an N-sphere.

I know a professor at our math department that studies multi-dimensional geometry. I can try and ask him if there are any good introductory books on the subject next time I run into him.

 

[Secret]

I'm having problem to obtain a corner slice of the hyperspheres

what would the hyperslices look like when projected into lower dimensions?

 

[K^2]

All possible cross-sections of a hypersphere are spheres. Though, projections will be ellipsoidal. Just like sphere's cross-sections are circles, but appear as ellipses on projection.

Or do you mean something different?

 

[Secret]

No I'm not talking about cross sections

I'm talking about the slice you get when

e.g. for example the shape you get when you cut the circle into quarters

So how does the projections of the higher dimensional anologes of (1/4 of a circle) look like

 

[Glen Bartusch]

The only way one can visualize a hypersphere is to visualize many spheres (which are infinitely-thin slices of hypersphere) beginning with the smallest of spheres; the spheres gradually get larger until a max diameter is reached, then back down in diameter. Oh, and you must visualize all these spheres extending in a direction that is not considered to be length, width, or depth.
It theoretically can be visualized in the mind's eye, provided you have sufficient memory (which you don't--only hi-power computers have such memories...)

 

[Secret]

The only way one can visualize a hypersphere is to visualize many spheres (which are infinitely-thin slices of hypersphere) beginning with the smallest of spheres; the spheres gradually get larger until a max diameter is reached, then back down in diameter. Oh, and you must visualize all these spheres extending in a direction that is not considered to be length, width, or depth.
It theoretically can be visualized in the mind's eye, provided you have sufficient memory (which you don't--only hi-power computers have such memories...)

Because of this fact
I'm trying to look for projections of the hyperspheres into lower dimensions insteadof using cross sections

Currently I'm frustrated on how identical sectors of a hypersphere fit together to form a complete hypersphere, just as 1/8 sectors of a sphere stick together to form a complete sphere and 1/4 sectors of a circle into a complete circle.

And what I'm currently looking for are the projections of these sectors in 3D and 2D space
which still failed to find any and that's why i need your help

Edit: Hyperspheres are spheres with surface > 2 and dimensions > 3

 

[TubbaBlubba]

The best way to understand a hypersphere I could find would be two spheres, the surfaces of which are attached orthagonally to one another at every point of the sphere, and are rotated about the fourth dimension by "wrapping" around one another.

https://www.physicsforums.com/showthread.php?t=407154 - I made a topic on this in the Topology forum some time ago, you might get something out of it.

 

[JDługosz]

Therefore can you provide me guides to visulize them? (no other objects please, only hypershperes)

thanks!

I think it would be most helpful if you learn to do the math and create your own projections. Doing it will give you more insight then the resulting picture!

 

[broean01]

The trouble with higher dimensions is that we can't visualize them. Ultimately, you have to let the math do the seeing for you.

 

[Time02112]

http://sphotos.xx.fbcdn.net/hphotos-snc6/8733_101004519919775_100000306818475_22662_7600549_n.jpg
I drew this image of Tesseracting 'Hypersphere' over a decade ago. Looking inside the cutaways, you will notice the front & back vortexes were removed, in order to allow transparent viewing of the inner construct.

 

[Time02112]

Incidently; the Sphere, or 'Hypersphere' is the only shape that 'can' survive passing through the menassic ring of a wormhole!