About the 4D-hypersphere...?

Question

If you were to draw a hypercube on a piece of paper, you would draw two cubes and then connect the corresponding points in the two cubes.

But how would you draw a hypersphere on a piece of paper?

Answer 1

Hi!!



They already explained very good above, but I've got something to add:



Imagine a two dimensional being imagining a cube... he might think of two squares, one inside each other joined by the edges... that's a vague idea of a cube :D.



What you think of a hypercube is really the 2D visualization of the 3D cross-section of a 4D object... but that's hard to do with a glome (hypersphere).



Imagine you show a sphere to a two dimensional being... he would just see a dot that increases to a big circle and then decreases again to a dot to dissapear. That's the succesive show of 2D cross-sections of a sphere. The sphere is intersected by the his plane (his 2D world is a plane).



The same happens for us. Any visualization you can do of a glome is really a sphere, of smaller or bigger size depending which cell, volume or 3D space is crossing ours (our space). That is, we can only see the parts of the glome intersecting our space. But like a sphere is volume inside a curved plane, a glome is hyper-volume (a part of tetraspace, hyperspace or 4-space) enclosed by a curved space, which can be only PARTIALLY intersected by our 3D space. It isn't hard, but instead impossible to imagine different spaces existing.



You can possibly imagine two parallel lines, two crossing planes, a curved line, ... but you simply can't imagine two paralell spaces, or two spaces intersecting in a plane... you haven't that power. Neither do I.



Well, bye hope it helped ^.^ !!



Edit:



To the person below:



To represent a hypersphere as you represent a sphere, you would need a 3D volume or a cube of paper or whatever xD.



For the sphere, you draw a circle, that is, 3D to 2D.



For the glome you should draw a sphere, that would be, 4D to 3D.



The same happens with Circle-Ellipse to Sphere-Ellipsoid (again crossing dimensions).



Now, if you could see that 3D object we just created in space, and put it into a paper, would look exactly like a sphere ;). Glome = Sphere in a paper.



To see what fails in our glome, see firsts what fails in the sphere... The sphere is draw on just 1 plane, no depth for it :(... same for glome, it is drawn in just 1 space... no 4D orthogonal extra direction for it :(...



If the diameter of the glome was marked, when it crosed our spaces we would see that sphere within ours, going smaller as the bigger sphere does :P.



Hard to imagine :D...

Answer 2

As others have pointed out, with the hypercube you are drawing a 2 dimensional projection of a 3 dimensional projection of a 4 dimensional object. That loses a lot of information.



That said, think about how you draw a 3D sphere on a piece of paper. You will draw a circle for the projection of the whole thing, an ellipse for the equator and probably some type of circumpolar ellipse. The two ellipses represent circles on the actual sphere. Alternatively, you could draw several ellipses representing 'latitude' lines on the sphere.



The second turns out to be slightly better for the projection of the 4D sphere. Draw a large circle. Now, at a diameter, draw an ellipsoid (a flattened sphere) to represent the equator of the 4D sphere. Now move up a bit and draw another ellipsoid with the same proportions but smaller representing a 'latitude sphere'. Do this a couple of times above and below the equator to get your picture.



If you want to go the first route, draw your large circle and the ellipsoid for the equator. Now draw a perpendicular ellipsoid representing a 'longitude sphere' but make sure that the two spheres have a common circle/ellipse in the center of your diagram. This represents a circle that is actually a diameter of both the equator sphere and the longitude sphere, which are perpendicular to each other.



Neat huh?

Answer 3

Keep in mind that the hypercube drawing you're referring to is a 2-dimensional shadow of a 4-dimensional figure. Not sure that you could be successful trying to draw a 2D shadow of a 4D figure that contains no straight lines or corners. And it is certainly a difficult proposition to sculpt a 3D shadow of a 4D figure.



My mind can't even wrap around what a hypersphere's shadow would begin to look like, and I'm not sure too many human minds would truly be able to either.

Answer 4

You cannot draw a four dimensional figure on two-dimensional piece of paper. Okay? You cant even construct a 4-D figure if you had all three dimensions to work with. You cant even imagine a four dimensional object in your mind. We have a hard enough time imagining a three dimensional object in our minds.

Answer 5

draw three intersecting circles and spherically shade then in different colors on the same axis



R^2 = x^2 + y^2 +z^2 +w^2



so R^2-(z^2 + w^2) = x^2 + y^2

and R^2-(w^2 + x^2) = y^2 + z^2

and R^2-(x^2 + y^2) = z^2 + w^2



and R^2-(y^2 + z^2) = w^2 + x^2 is hidden

Answer 6

I think it would look like a sphere combined with another sphere combined with another sphere combined with another another sphere