2D as an Analogy?

[Paul777]

I think it is preposterous to use 2D as an analogy to prove the 4D. Why? No matter how small, there will always be the dimensions of length, width and height even when applied to microscopic objects. Some will say "think of a piece of paper" because it is flat or a "line drawn with an ink pen" but both the paper and the line of ink on the paper at the microscopic level have 3D properties. My wife even argued with me saying that a person on TV is in 2D because the screen is flat that there is only height and width to observe. But that image of the person on the TV is not the person, but a collection of pixels, protons, neutrons and electrons that are part of our 3D world. We like to revert to 2D and use 3D as if it were 4D to explain to ourselves how 4D works when no one has ever proved that there are 4 dimensions. How can we explain something that hasn't really been proven true? People say think outside of the box, but what if the box has infinite boundaries of height width and length.

[Klitzing]

There is nothing to prove about the existance of 4D! It is a missunderstanding that there might be some realworld experience of 4 spatial dimensions. Or even that one is able to see 4D. There are no such things. Even so, as abstract mathematical ideas, any higher dimensional spatial space is without internal contradictions, and therefore does exist.

Consider a fucntion of several variables. Volume e.g. is a function of 3 variables: height, width, depth. But the prizing of postal letters is a function of height, width, thickness, and weight. Or the pricing of cars depend on the model (which might be braught down again to the 3 dimensions of measure), on the motor (fuel, power), the color, several additional featurs, etc. - I.e. you might handle as many variables you want.

Or consider reaction equations of chemistry. Those are nothing but linear equation systems, where the variables represent the relative amounts of the involved elements, one variable per involved element. There is obviously no restriction, i.e. it would be pure nonsense to stop handling reaction equations with more than 3 involved elements, asking to "prove the existance" of a higher dimensional space first.

Our realworld experience is 3D, as you said. So to visualize 4D we couldn't do better than to embed that additional dimension somehow. There are different ways to do so. Out of my head I can think of 3 different ways I've seen so far:
A) One is passing a 3D sectioning hyperplane through the 4D shape. That is resulting in a sequence of sections. You might then take all these sectioning pictures and make therefrom a movie. So, in fact you have mapped 4 spatial dimensions onto 4D spacetime. -
B) You might use projection instead from 4D into 3D, this is best visualized with applying some perspectivical forshortening of the 4th direction. This is a good way to represent uniform polychora for instance, as there are only edges of a given size involved (per definition of uniformity), and so edges which look smaller are thus understood to be farer away in the additional dimension. -
C) For 4D shapes which don't have some given size-clues there is a 3rd way of representation - at least if those can be understood monochromatically. Then one can (without resorting to perspective re-scalings!) use the colorscale as a 4th dimension of representation, i.e. red and orange parts are very close to the 4D observer, yellow and green ones are farer away, and blue and violett parts are at the far end. -
Both latter ways ask for some translucent representations for sure (or even showing edge skelletons of polychora only)...

Even so we would never be able to walk within 4D space, and having said that higher dimensional spaces just are mathematical abstract ideas, there are lots of intrinsic restrictions for any such higher dimensional space. So we won't be able to measure such things. But we are able to calculate the relavant parameters.

--- rk