Prime Eval, Part 2


Starting from the basics. A “point” in mathematics is an imaginary concept (drawing on what I remember from my high school classes, and “imaginary” in the sense that there’s no physical representation of a “point” that we can pick up, examine and measure). It has no length, depth or height. It is “dimensionless”. Since there are no dimensions, the point doesn’t move and therefore is unchanging. It just “is”.

If we give it a degree of freedom, or if we introduce “time” and allow the dot to move in something we arbitrarily declare is “forward” and “backward”, then we get movement, and we get change. If we attach a pencil to this point and have it draw on a piece of paper as it moves, we will get a straight line. The point is unable to change directions beyond simply marching forward or backward. We have a line that, if we wait long enough, will extend to infinity at both ends.

A line has one dimension, which due to tradition we call “length”. Otherwise it has no height and no depth. If we go back to our point and let it turn around in a full 360 degree circle, and not just in 180 degree increments, we add a second dimension, while keeping the concept of “time”. Then, if we take that solid line and add an infinitely wide pencil to it, telling the line to move in what it thinks is forward and back (perpendicular to its current position), we’re going to trace out a plane that again extends out to infinity at both sides.

A plane has two dimensions – length and “width” (or “depth”). But no height (it’s an infinitely thin sheet of paper). Our plane is going to have what it considers “forward and backward” (“up” and “down”). If we give it one more degree of freedom, and if we stock up on more pencils, it’s going to draw out a cube which is going to envelope all of space. This cube will have length, width and “height”.

Can we go one more step? The cube has 3 dimensions, do we have to stop here? Theoretically, there is a 4-dimensional object called the tesseract, but since we as human observers are dwelling in 3-dimensional space, it’s kind of hard to go to the store to buy a box of tesseracts to hide in the back of the closet to pull out during show-and-tell at school. But, are tesseracts really real?

If you look at the wiki page for dimensions, it uses latitude and longitude instead of length and width. Regardless, going back to my premise in part 1, language colors perception. We think and live (we think) in 3D space, so we have names for our three dimensions. We could switch over to “x”, “y” and “z” axes, but again, there’s a hidden bias at work. What do we call the 4th axis? “hyper-length”? “w”? “Poly” (want a cracker)? Maybe it would be better to scrap the system and start over with D0, D1, D2, D3 and D4. D0 would be a dimensionless coordinate system that is used only for representing points. D4 would give us tesseracts, and doctors have laser surgery procedures for correcting those…

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