Squares in nature

[Andrew]

I'm trying to understand where squares in mathematics represent a function in nature. Like where acceleration = s / t^2 in the Reciprocal system. I can't find how this compounding effect is a reflection of something natural. Where does it come from?? And I can't for the life of me think if this was ever brought up in any arithmetic class I've ever taken. It was just, a square is the number multiplied by itself by x many of times, x being the exponent. Can someone enlighten me as to the nature of the square?

[animus]

Instead of your arithmetic class try to remember your physics class, in particular the ones where you have discussed the inverse-square law. As to the nature of the square, I bet Gopi and Miles Mathis could give you a straight answer right away. The former will probably give you a more comprehensive answer than the expected one-line from the latter. Personally, I would start with Euclid and see where he got his ideas from.

[SpaceMan]

I'm trying to understand where squares in mathematics represent a function in nature. Like where acceleration = s / t^2 in the Reciprocal system. I can't find how this compounding effect is a reflection of something natural. Where does it come from?? And I can't for the life of me think if this was ever brought up in any arithmetic class I've ever taken. It was just, a square is the number multiplied by itself by x many of times, x being the exponent. Can someone enlighten me as to the nature of the square?

Exponents are just a way of representing the number of dimensions that are in the expression. A geometric perspective makes x^0 a point, x^1 a line, x^2 an area, and x^3 a volume. You are essentially just taking the point and doing a linear translation to get a line, then sliding the line to get an area, and finally doing the same to the area to get a volume. This process gives you a square for x^2 and a cube for x^3 which is why we say squaring or cubing an expression respectively. One of the manifestations of this in nature is actually the example you've given; a lot more can be found here http://reciprocal.systems/research/units.php. The nature of reality itself is actually an expression of powers or dimensionality. Take s/t the basic expression for a speed is just a change in space over time. Your acceleration function s/t^2 is a change in space over time with how that expression itself changes over time. The function s/t^3 by extension is just how acceleration itself changes over time, this is what is known as jerk. Matter is probably the most obvious and direct expression of dimensionality with it being represented by s^3/t^3. This takes us back to Larson's original postulates; motion exists in three dimensions. If you are looking for a direct analogy in the realm of living systems I don't think you will find a one to one correlation since living systems are compounded interactions of multi dimensional motions. Living systems interact through these exponential expressions, but are not themselves a direct expression of them in an arithmetical sense.

[daniel]

Spaceman brings up the relevant points regarding dimensionality, but I would like to clarify...

Mathematics tends to reduce relationships to their simplest form--and in the process, loses the conceptual connection with Nature. To understand what is going on, you have to determine what the original concepts were--and adjust the math to make sense again.

Exponents represent two different concepts:

1. Dimensionality: x², as a concept, is not x*x but x*y, where x=y. The math guys substitute "x=y" in the equation x*y to get x*x, simplified to x². The whole idea that you are dealing with TWO dimensions (that just happen to have the same magnitude) gets lost.
2. Rate of change: in your example, s/t² is acceleration--but the formula is conceptually incorrect. It is actually (s/t)/t, where the latter "t" is clock time--not a dimension OF time--and is how the single dimension of speed is changing with respect to clock time. Force, t/s², has the same problem--it is (t/s)/s, which is how energy (t/s) changes with respect to distance (s).

Physics screws this up all the time, because when they see the same MAGNITUDE for a dimension, they just equate them. Larson points this out in BPOM with electron charge (q), where they sometimes use q as having dimensions of space (the NUMBER of electrons) and other times, units of energy (the CHARGE of electrons). Since the electron is 1 unit of space and the charge is also 1 unit, they don't see a difference--and that substitution gives incorrect equations. (This is discussed in Bruce's video introduction on reciprocalsystem.org).