- diagonal of a rectangle is sqrt( a^2 + b^2),
- diagonal of a cuboid is sqrt( a^2 + b^2 + c^2), not say cube_root(x^3+y^3+z^3) etc.
- value of probability density func. of a standard normal distribution is proportional to e^(-x^2/2)
Nature seems to like the Golden Ratio quite a bit. From flowers to ammonite shells.
Nah, the preferred function is the identity. The length of a rectangle is 1 * the length of the rectangle!
Have to study a bit more on it but its value also comes from a quadratic equation (again, the square function) : x^2-x-1=0
The strength of long range forces (gravitation and electromagnetism) decreases as the square of distance.
The distance traveled by Newton’s Apple is proportional to the square of time travelling.
The time a Staggering Drunkard spends walking is proportional to the square of net distance traveled.
Blame Pythagoras.
The reason many physical qualities scale in proportion to a distance measurement to the exponent of 2 is purely an artifact of the geometry of three dimensional space. The standard normal deviation has an exponent of 2 because it is symmetric and unimodal. None of this is in any way mystical or representitive of some kind of underlying natural law.
The golden ratio is, as Quartz notes, a quantity that shows up in many natural features and structures for reasons that are not fully understood, and is also aethetically appealing in art, architecture and music. However, that it is an algebraic number of order 2 is unremakable given that it is a ratio of lengths of a rectangular area. The relationship (a+b)/a = a/b can be rearranged into a normal quadratic equation just by multiplying both sides by the product of the denominators (a*b) and rearranging.
There are many other quantities that have pervasive and subtle relationships to geometry and the natural world but the most interesting are the transcendenal numbers e and π that are intimately involved in periodic and functions and conic sections (ellipses, parabolas, and hyperbolas). The relationship between these two numbers provides the basis for all of trigonometry (see [POST=13014360]this post[/POST] for an explanation) and is the only case I know of where two unique transcendental numbers can be combined in a function which produces a non-zero integer. Nature, it would seem, prefers periodic functions (in the form of what we term as “waves”) right down at the quantum level wheve every particle interaction can be represented by interference of oscillatory functions.
Stranger
In many cases, it’s a square function because it’s an integral of a linear function. Something accelerates linearly, and the distance traveled is the integral of speed (i.e. time squared). Area of a shape is the integral of the length of the sides of the shape. etc.
Understood in context you provide.
Nice.
(You, as well as Nature :))
Many thanks for contributing everyone.
Keep adding your thoughts.
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If Standard normal distribution’s probability density function had a form say e^(-mod(x)), that would also satisfy our initial checks (symmetric, unimodal).
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I know how the quadratic equation for golden ratio came. It is not unremarkable imho because if that quadratic equation is unremarkable, then so is the golden ratio number itself.
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We know the proofs of Pythagoras theorem, std. Normal distribution etc. , so it’s not mysterious in that sense but it’s still interesting for me note the square function in them and in few more things which some members here have noted.
But instead of ‘nature’s preferred’, I should call it something else, I am not sure what.
You include the diagonal of a rectangle and the diagonal of a cuboid as two different examples, but they’re the exact same thing.
Just because a cuboid is 3 dimensional and rectangle is 2 dimensional.
The one is just the degenerate version of the other.
You are looking at the ease of derivation. Please look at its significance, it gives the resultant displacement when you move some distances in 3 orthogonal directions in space.
The diagonal of a cuboid, or indeed, and two points in Euclidian space, is a straight line which lies in a family of two dimensional plans rotated about its length. Any straight line defined in the dimensions can be transformed onto any one of those planes by a suitable selection of Euler angles about some origin colinear with the line to reduce the coordinates to at most two dimesions, and potentially to one dimension, which is exactly what the Pythagorian theorem does implicitly. There is noth8ng mystical about this; it is very ordinary analytic geometry.
Stranger
Squares also tend to appear when there is a natural reason for the output to be positive.
For instance, energy is (mostly) always positive. However, velocity can be negative, and so the kinetic energy equation includes velocity squared. That’s not how you would derive it from first principles, but it’s strangely convenient that it also has the effect of ensuring positive energy.
Or, consider approximating some samples with a line. One of the most useful metrics is least squares: that is, minimize the sum of squared differences between the samples and the approximation. We could not use the simple differences, since the negatives cancel the positives and we would not get a good fit. Squaring the differences makes the error always positive, which makes it easier to minimize. But why a square and not an absolute value function, or a fourth power? Well, the alternatives have worse properties. The square turns out to be the best.
The Pythagorean theorem has been mentioned. But the slightly more general way of looking at it is that it gives us a distance between two points regardless of how we label their coordinates. The point (-3, -4) is 5 units from the origin, and we don’t have to make the coordinates positive to start with; the squaring eliminates the negatives. The Minkowski metric (x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup]) is a bit different, but likewise requires the squares to behave consistently (when considering purely spacelike distances, it works the same as Pythagoras).
Actually, some would argue that least-absolute-value does have better properties for curve-fitting, in that it’s less influenced by outliers.
In general, lower-powered functions show up more often than higher-powered functions. The OP correctly notes that squares show up more often than cubes, and cubes more often than fourth powers, and so on. But the pattern continues at least one step further: Linear functions also show up more often than squares. The OP probably just didn’t notice this because linear functions seem uninteresting compared to square functions.
(I say “at least one step further” because constants, that is, zeroth-power functions, arguably show up more often even than linear functions, but there, it’s tough to even say whether something should be considered a “function”, or how many there are.)