Hi,
i was wondering...
How is a four dimensional space represented and what are geometrical analogs for thee common three dimensional geometrical Figures:

Cube
Sphere
Cuboid
Cylinder
Cone

like x y z Coordinate System...
What about Five-Dimensional representations?,Six-dimensional,n-dimensional?

thanks.
bye!

x, y, z suggests a 3d environement, right?

Please, can someone make a drawing of a 4D object ? :p

A Hypercube and things like that...

bye!

are you going to atempt to paint it?

Anyone who takes the challange to paint a 4D object? :D

You can represent higher dimensional space with more coordinates. Just don't excpect to be able to draw it on a piece of paper.

Look at how the objects are constructed in 1, 2 and 3 dimensional space and you just go from there.

Eg. A sphere is 3-d, it's just "all points 1 unit distant from the origin".
The part in quotes makes sense in any dimension, provided you have a notion of distance (the usual Euclidean distance works in all dimensions). So a 2-d sphere is the unit circle and the 1-d sphere is 2 points, {-1, 1}. The same construction works in 4, 5, etc. dimensions, but they aren't something you can draw very well.

You can draw cross sections of higher dimensional things. Look how the sphere in 2-d (the plain old circle) is just the intersection of a 3-d sphere with the x-y plane. The circle is a 2-d cross section of the 3-d sphere. In the same way, the 3-d sphere is a 3-d cross section of a 4-d sphere, and so on.

Schmoe put it perfectly. I just wanted to add something a Prof. said to me regarding the subject:

You know that from 3D, 2D looks flat. Same thing follows as you move up. From 8D, 7D appears flat.

Hi,
as i was looking through the internet,i found this :
http://www.geocities.com/CapeCanaveral/7997/hypercube.html
Please explain these figures if you understand them.Also how to extend the concept upto n-dimensions...

thanks.
bye!

Hi,also this site :http://www.hitxp.com/math/geo/euclid/130503.htm
explains that cube that we see in 3D space in part of a 4D cube actually and we will see it as waht it is :A cube and not a 4D cube.Can someone actually draw a 4D cube in a 3D space(Like Model it?) and go further? like 5D cube,further....?
And scale and generalize upto N-Dimensions?...

And what is this Hypercube actually? i dont get the figure that i mentioned in the site.Please explain,guys...

bye!

I understand Flatland analogy wesmorris...But i am in an imbroglio really.I dont know how to visualize a Hypercube on a 3D space,or how would you actually represent higher dimensional objects in 3D world?

bye!

Hi,
as i was looking through the internet,i found this :
http://www.geocities.com/CapeCanaveral/7997/hypercube.html
Please explain these figures if you understand them.Also how to extend the concept upto n-dimensions...

thanks.
bye!

I'll try to explain the 3-d analogue of the "slices" (what I called "cross-sections") in this link.

Imagine you have a 3-d cube that's balancing on it's point in a plane. The intersection of the plane and cube is a point. Now move the cube down through the plane slightly. The intersection (aka slice aka 2-d cross-section) is a small triangle. As you keep moving the cube down, you get larger and larger triangles. Eventually you'll get a 6-sided figure. The sides of this muck about as you move the cube down more, eventually turning back into a triangle, which shrinks in size until it's a point. You should get youself a piece of paper and try to draw these cross sections. Seriously, get a piece of paper and try to draw this.

Now forget where these cross sections came from and just stare at your piece of paper. Try to visualize how they can be put together and how this actually gets a cube. Pretend you're stuck in 2-d while you do this. Note that to actually draw or build a model of how these slices fit back together, you need to leave the 2-d confines of the paper, otherwise the slices will intersect.

Now go back to the slices in the link. Try to visualize how the slices go together. Note in this case, rather than a 2-d triangle as a slice, we get a 3-d tetrahedron (at the begining). You run into a quandry if you try to actually construct slices and stick them together since they are 3-dimensional and will interesect. In 4-dimensions, they are able to be next to each other without this intersection. This is why we are forced to restrict ourselves to these 3-d slices and can't actually assemble or even draw (without much intersection) a 4-d cube.

Simplest way to describe a hypercube is by it's coordinates. It's all points (x,y,z,w) where x, y, z, and w are anything from 0 to 1. note if you chop off the w coordinate (and the z coordinate), this describes a 3-d cube (or a 2-cube, i.e. a square). This might not help you visualize it though. You can do a more constructive approach, but work on understanding the slices above first.

http://www.geom.uiuc.edu/docs/outreach/4-cube/

might be of help.