The Curse of Dimensionality

Picture this..
say you have a square.. Perfectly flat.
2D cube


Now say you take that square and make it 3D perhaps not a cube, but just a tiny bit of height too it.. maybe a paper cut out of a square.
3D Cube projected in wire lay out


That square now has more weight (relative) and mass than the 2D square.

The same happens going from 1D to 2D and 3D to 4D, every time you add a dimension. You add mass to the object.
4D cube in wire layout

5D cube in wire layout

This is problematic because when you are talking about a multidimensional existence (parallel universes and such)
Different universes will have different perceived weight due to their increased mass.
11D cube


Now say all dimensions appear universally but we cannot perceive any but 2 and 3 then the universe would have infinite mass as mass increases every time you add on a new dimension.

and it would have virtually no middle as 1D and 0D are imperceivable too the human mind.. BUT HANG ON, I hear you say.. You can perceive 1D..

Well no you can't sorry to say..

Imagine the smallest thing you can think of, thats 3D..
Tiniest flattest thing... Well its 3D too, everything we can perceive is 3D, accept when we look at something displayed upon something..

For example the image on your computer monitor is 2D.. The monitor is 3D the pixels are 3D but what you are viewing is a 2D image..
We can perceive this because it has length and height..

But if you take away either of these it would not be a straight line, because even a straight line has some height or some width, it would be an infinitly receding point in existence, being ever smaller than is capable.. Thus it is out of our knowledge to perceive it.

Thought you might find some geometry interesting on this the day of the NSW State election!

K-Nex, Tesseract, The Four Dimensional HyperCube

I have after much trial and tribulation, discovered that it is impossible to construct a 3d representation of a tesseract (4 dimensional hyper cube) out of knex,
I have been trying for over an hour however there is no way to build two cubes that have the correct size ratio, so now I am stuck with a cube inside a cube that is not a tesseract...

I called my friend for help with this and his response was "what is a tesseract" as many people may have first questioned so I figured, time to enlighten the blog.

The most common way of explaining it is flatland as told by the legendary Carl Sagan, Youtube link after the break.

It is a good example to explain alternate dimensions.

A tesseract is a 4 Dimensional version of a cube, So Imagine you have a straight line, to make a square you extend the line by right angles from itself, furthermore to make a cube you extend a square at right angles too itself, to make a 4d hypercube you extend a cube at right angles from itself, however since we ourselves are 3D we cannot see the new direction that a 4d cube would take, however we can see a projection of it in 3d much like we can see a projection of a cube in 2d.

Click the link for the animated version.


Originally when you see this image you may be very confused as too what is going on, well this is what a 4D image looks like when it rotates in 3D, of course even more confusing is because it is rotating in 2D but it is similar.
Here is it rotating from another perspective.


The interesting thing is that if you could see in 4 dimensions but be in a 3D world, you would have a sort of X-Ray vision, you could see inside everything at a level we cannot understand..

Unfortunatly, I cannot make a tesseract out of K-Nex too show off to my science buddies.. but I have a set of Knex cubes that I can hold in the right shape.

Hit the video to see Carl Sagans explanation!