It’s been proven that it is impossible to measure 3D and higher dimensional space the same way we measure 2D-space using the Pythagorean theorem. Fermat’s Last Theorem and subsequent proofs in the 20th Century affirm Fermat’s 17th Century conjecture. To me, this demonstrates the primary difference between the 2D and 3D worlds. Lately, I’ve been pondering about the correlations between Fermat’s theorem (about not being able to measure the same way in different dimensions) and the 4th dimension in particular, better known as time.
So, let’s review the dimensions and work our way back… perhaps in time.
There’s the 1st dimension, which can be represented with a single line or dot. Technically anything being measured within the confines of a single dimension cannot have a length because there is nothing to compare a lone line or dot to, if we think in terms of the line/dot itself and not of anything else (1 dimension). I don’t consider the line/dot to be infinite nor finite, as both those descriptions require some form of measurement. If you have two points on a line in the 1st dimension, you can measure the distance between the two dots by finding the difference between the ratios of the points’ position to any defined starting point. That can tell you about differences between points but not about the unit itself, though, which disallows measurement because it can’t be compared.
Until you enter the 2nd dimension: which can be represented with 2 lines, intersecting at some point. Measurement is allowed in 2D space, because the width and height of the two intersecting lines can be compared and thus measured. In 2D space the shortest distance between point A and B is to bend the dimension itself (like a sheet of paper), and connect the two points (making the distance 0).
Then, there’s the 3rd dimension, which some liken to the 2nd dimension, calling it a “spacial dimension”. My interpretation of Fermat’s proof is that this is not a good comparison, as the 2nd and the 3rd dimension differ in the way they can/cannot be measured. To measure a 3D object, we take the size of 3 planes and add up all the units that arise. But what if we represented a 3D object in the same regard we measure a 1D object, considering it immeasurable and describing it with only the relationship between all the points. We would then have to throw out a notion of volume or size for the overall mass and think of all 3D masses as ratios between individual XYZ points and 3 corresponding, defined starting points of the object (the total mass not being finite or infinite).
Then the fourth dimension, which can be represented as different phases of a 3D object. Measurement is allowed in 4D space like it is in 2D space, because the change in state of the object can be compared and thus the amount of phases between the two can be calculated. The amount of phases between two phases is what we describes as the amount of time something took. What if the shortest distance between now and tomorrow was to bend the dimension itself and connect the two events? Einstein liked this idea, from what I understand.
Perhaps this could be accomplished by determining the amount of entropy (atomic disorder) at one moment in time and determing how mugh entropy was existant at a past or future point in time, and then manipulating the atomic entopy of the environment to that state, since entropy is pretty much the governing vehicle of the motion of time according to the 2nd Law of Thermodynamics.
This could be potentially accomplished by speeding up atoms and giving them specified velocities (to potentially cause them to travel into the future) or by slowing atom energy down to negative levels (considered by some to be impossible) as to reduce their entropy and inhibit past-time travel. Proving this wrong is half the battle to figuring out a way to make it work. In other words, we might need more entropy.