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Visualizing 4 dimensional objects as thy appear in 3 dimensional space

Ever wondered how a 4-D cube or a 4-D sphere would look like? Even though I haven't seen one till date, I shall try to explain what a 4D cube or a 4D sphere would look like in our 3D space.

But before that I would like to make a noteworthy point. We cannot see a 4D object in its entirety in a 3D space for the simple reason that our 3D space is a sub-space of the 4D space. Hence, all we can see will be a partial view of the 4D object through our 3D plane! In order to understand this, consider a 2D space (i.e. a 2D plane). A cube in our 3D space would look like a square in a 2D plane. The entire 3D cube can never be visible in a 2D plane for the simple reason that the 3D cube extends outside the 2D space (i.e. the 2D plane) into the 3^rd dimension. (see below)

Using a similar analogy, now we shall see how a 4D cube would look like in our 3D space. 

Consider a 2D universe i.e. a 2D plane. Now pass a 3D cube through that 2D plane. Imagine a 2D man watching the 3D cube as it passes in his 2D universe. Initially he would see a square emerging from nowhere in his 2D universe. The square continues to be there as long as our 3D cube is present in the 2D plane and once our 3D cube moves out of the 2D plane, the square instantly vanishes from the 2D plane for the 2D man.

Now it's out turn to watch as a 4D cube enters our 3D universe. In this case, we would see a 3d cube appear in front of us from nowhere as the 4D cube enters our 3D subspace. The 3D cube will be in front of us as long as a part of the 4D cube is in our 3D subspace and vanishes instantly when the 4D cube completely moves out of our 3D subspace.

So, the next time you see a 3D cube appear in front of you, believe me it's a 4D cube.

That's about a 4D cube. What about a 4D sphere? How does it look like in our 3D universe?

Again we shall visualize by using what I call the known-to-unknown dimension analogy. Here we first see what it looks like in a lower (known) dimensional universe than ours as we can completely submerge lower dimensional universe in ours and then we use the same logic to see what happens in our universe when a similar situation arises into ours from a higher (unknown) dimensional universe. By this technique, we can visualize 4 dimensional things as they appear in our 3 dimensions, by our experience from how the same things in our 3 dimensions look like in 2 dimensional space.

If you see the above animation, and if you have understood the previous analogy, you can conclude that a 3D sphere looks like a circle of varying radius in a 2D space as it moves through the 2D space.

When a 3D sphere enters a 2D universe, it looks like a point circle in the beginning which grows in size as the sphere moves through the 2D universe. The circle reaches its maximum size when the center of the sphere enters the 2D space and after that the circle in the 2D universe starts to grow smaller as the 3D sphere moves out of the 2D space. Finally as the 3D sphere leave sthe 2D space, the circle reduces to a point circle and vanishes from the 2D space.

Similarly, when a 4D sphere enters a 3D subspace, it initially appears as a point sphere which grows in size as the 4D sphere moves through our 3D space and reaches its maximum radius when the center of the 4D sphere enters our 3D space and thereafter starts to shrink in size as the 4D sphere moves out of our 3D space and finally reduces to a point sphere and vanishes as the 4D sphere completely moves out of our 3D space.

So, the next time you see a small point sphere appear in front of you from nowhere, grow in size till reaches a maximum and then start reducing in size till it comes down to a point sphere and then vanish, then believe me you've just seen a 4D sphere move through your 3D space.   

Now, can you tell me how a 4D cone, a 4D pyramid, looks like as it moves through our 3D space??

-by Gurudev
MADE IN INDIA
gurudevp@vsnl.net
On 13 May 2003

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