Let’s think about what a thing would look like if it were to be imagined in four dimensions. Now before we ask that question let us take a step back and think about 2D shapes. A polygon or perhaps a circle – easy enough to imagine, right? Still that doesn’t change the fact that this is a 3D world and in terms of classical physics nothing extra dimensional, i.e. lower or higher than three dimensions can exist within it.
Now here is a bit of a technicality. Just because a thing can’t exist in the material landscape of our dimensional reality doesn’t means that it cannot exist at all. After acting like morons for past few centuries, we have started to talk about extra dimensional realities again. The easiest way of saying it is that if we can imagine it or define it (mathematically, intuitively, etc.) than it exists, just maybe not in the material form.
So coming back to circles. I was reading somewhere that a circle is a 2D representation of a 3D sphere. Although this might sound correct at first but it isn’t technically correct. Representation is a subjective term. Let me give you an example
Here is what you see from space:(Image by By Kevin Gill, License: Creative commons, Wikipedia)
These are a few ways it can be drawn for 2D:
(Image by By Strebe, License: Creative commons, Wikipedia)
(Image by By Strebe, License: Creative commons, Wikipedia)
(Image by By Strebe, License: Creative commons, Wikipedia)
I took earth as an example because these were the images I could most easily find and yet they conveyed the point very well. Geometrically speaking we use projections to convey 3D in 2D. We can talk about geometrical methods of projections like stereographic projections or isometric projections but that would only extend the post without adding much to the discussion at hand.
One of the most important things when discussing dimensions is to remember that we have been trained to visualize things in a certain manner. When we think of a sphere and then try to draw it, we automatically start with a circle because that’s how we are trained. Why don’t we start drawing a cylinder with a circle but rather with rectangles and ellipses?
In a similar way we assume due to our training that 3D sphere would become a 2D circle when it goes to pen and paper. Without realizing what paths and steps were taken for the sphere to be dumbed down to a circle. Personally I extrapolate this to say that if we are not able to extend our visualization into a realm beyond, than a true visualization of higher dimensions might not be possible at all.
The reality is that 3D is 3D and 2D is 2D. Our ability to pen down 2D and not 4D cannot be the criterion for an existence of a four dimensional reality.
An rough analogy can be derived through successive iterations of increasing the dimension:
The idea of 'stacking the cubes may not be immediately visualized since we lack the perspective of thinking in four dimensions. On the other hand the idea of stacking planes on top of each other sounds perfectly reasonable..... Here lies the crux of what we need to understand - How were we able to stack planes on top of each other? Because just thinking that a plane can be chopped in half and the put on the top of its other half gave birth to the idea of third dimension.
Here is a visualization of stacking cubes in 4D:
(Robert Webb created this image using Stella Software, License: Creative commons, Wikipedia)
Now just like you can rotate a cube to see all its sides, you can also rotate a tesseract. Check out this cool gif:
(Made by: Jason Hise, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1702289)
Okay I would end this post here and please leave comments below to
tell me your thoughts on what I wrote. Crappy, reasonable, should be
shot in the face or maybe we could do a bit more of geometry for a 4D
space next time. Actually not a bad idea. If you guys would like I could do a post on
the geometrical representation of 4D objects. So tell me about it in the
comments………………till next time.
