4D space? 4D Cubes?

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Ah, yes, higher dimensional objects. These entities are strange, aren't they? So unimaginable, they defy intuition and common 3D sense. I know the feeling. Even though it's impossible to represent a true 4D object in it's full 4D glory, we can still cancel out a dimension and view it in 3D. Either cross sections or shadow projections can do the trick. Depending on the type of shape you're working with, one works better than the other.

But, before one goes exploring 4D shapes in 3D slices, it's good to get your bearings with what you're going to be seeing. The problem is, we have to break our 3D sense and start thinking in 4 dimensions. Since we have a natural grasp on basic 3D shapes, viewing them in 2D slices will make more sense to us, with how it comes out. We understand quite well how the surface of the 3D object is a way of predetermining all 2D slices.

The trick is to imagine the whole 3D shape, while exploring in 2D. While imagining the 3D whole, try to place where the 2D plane is slicing. It's important to understand how we are unable to see the rest of the object, even though we know it's still there. So, when we move on to exploring 4D objects in 3D slices, our minds will be better prepared for this method of aided visualization.

Watching the 3D slices morph by moving a 4D shape will make more sense this time, instead of only thinking in 3D. We have to consider how we are unable to see the 4D whole, even though it is still there. And how the shape of this 4D object predetermines what all the 3D slices will be, depending on angle and depth.

Exploring 3D Torus in 2D: https://www.desmos.com/calculator/otpsrtlnww
-- adjust 'm' to slide, 'n' to rotate in 3D

Exploring 3D Cylinder in 2D: https://www.desmos.com/calculator/otpeykbx8g
-- adjust 'm' to slide, 'n' to rotate in 3D

Exploring 3D Cone in 2D: https://www.desmos.com/calculator/w3xptfnyhb
-- adjust 'a' to slide, 'b' to rotate in 3D

Exploring 3D Tetrahedron in 2D: https://www.desmos.com/calculator/ymijhsdgxc
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Triangle Prism in 2D: https://www.desmos.com/calculator/1mmbj9339n
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Square Pyramid in 2D: https://www.desmos.com/calculator/xupqv2qyq7
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Cube in 2D: https://www.desmos.com/calculator/fgqzkxuntu
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

And, since we can write an equation for a discrete three dimensional object, we can also write them for 4D shapes and beyond. These are objects of finite extent, having four true dimensions of space. Instead of thinking " how is that possible, it's only imaginary" , it's more to do with allowing it to be possible, and writing an equation that uses four dimensions:

http://www.reddit.com/r/hypershape/comments/2p1q3l/the_math_of_shapes_the_shape_of_math/

Here's a gallery of passing several types of 4D hyperdonut shapes through 3D. These are the four fundamental toroidal rings in 4D:

http://imgur.com/a/asnRZ

For some other types of shapes, see here:

Cyltrianglinder, cartesian product of triangle and circle : http://imgur.com/a/DvMEj

Tesseract, 4D hypercube : http://imgur.com/a/9rdLp

But, of course, there's no guarantee in wrapping one's mind around the concept. Usually not on the first couple times you try. It does actually take time, to the point where you have memorized the visuals. The main point is to compare what's happening in a lower dimension when an object gets scanned. We tend to see some of the same shape morphing patterns when going from 2D -> 3D -> 4D, especially with the toroidal shapes. We have to make the connection that the 3D things morphing is just a scan of an object that extends further than 3D. Even though we can't see all of it, the shape still has a precise physical structure, of finite extent.
 

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