Which mathematical objects are more common in nature?

[PainterGuy]

Mod note: Moved from Precalc Homework, as this seems to be a more general question.
Hi,

Which mathematical objects (numbers, functions, figures, etc) are more common in nature? I mean the mathematical objects which could more easily be identified with nature. For example, circles and triangles are most common mathematical figures (or, objects) which could be found in nature. Likewise, sine and cosine functions are really common in nature because many natural phenomena are periodic. Golden ratio is another number which is quite common in nature (but not as common as presented in so many fabricated examples) for its own reasons just like Euler's number, e. Thank you for your help.

 

[Nidum]

Fractals

 

[Ssnow]

Even logarithms are very common in nature: for example the sound level is measured by logarithms (bel and decibel), sometimes the variation of the concentration in a chemical reaction is also logarithm :$\ln{\frac{[A]}{[A_{0}]}}=-kt$ where $[A_{0}]$ is the initial concentration, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth, because you use the logarithmic scale, in thermodynamics they are used to determine the work of particular transformations ... , exponential and logarithms are very common in nature ...
Ssnow

 

[PainterGuy]

Thank you for your help.

But in my humble opinion I don't think that fractals and logarithms are as common as circle, triangle, sine, and cosine because these are some of those mathematical objects which really define the connection of mathematics with physical world. There is no doubt that almost all of mathematics is a way to encode and quantify physical world in a defined manner but mathematical objects like circle, triangle, sine, cosine are really a part of alphabet for that encoding. Thank you.

 

[FactChecker]

You are really asking about how humans measure objects in nature. That depends on what we are interested in. But some mathematical facts offer profound insight into nature.
For instance, a function in time (or space) can be represented by a combination of frequencies. Something like a step function actually can be considered as a combination of frequencies. That should give you additional insight into why cyclic behavior is so common in nature.
Also, the Golden Ratio is much more common than you imply, but it does not occur with perfect accuracy that often in nature. You may be interested in this series of 3 videos on Fibonacci numbers and the Golden Ratio in nature:

 

[Ssnow]

But in my humble opinion I don't think that fractals and logarithms are as common as circle

Note that a circle or a line segment are simply fractals with Hausdorff dimension $=1$.
Ssnow

 

[Delta2]

Well, basically it is already have been said (sine and cosine functions), I just say all the functions that satisfy the homogeneous or inhomogeneous wave equation, because many quantities of nature like temperature or pressure or electromagnetic fields and even gravity field propagate as waves in space time with finite speed. Many of the wave phenomena can be identified with our senses, like water waves or sound waves or the colours in nature (EM/optical waves).

Wave functions are used in Quantum Mechanics as well but ok it is still an open debate whether the quantum mechanical wave function is something real.

 

[scottdave]

Planetary orbits follow an elliptical path. Radioactive decay is exponential. Projectile motion pretty much follows a parabola, depending on air effects.
There are hexagons and cubes (and some other shapes) in crystal lattices. How do we determine which of these are "more common" than others?

 

[jbriggs444]

Superposition and proportionality (addition and multiplication) are ubiquitous in nature. As one would expect from a 1st order expansion using a Taylor series.

 

[scottdave]

Superposition and proportionality (addition and multiplication) are ubiquitous in nature. As one would expect from a 1st order expansion using a Taylor series.

I guess everything can be described with enough Sines and Cosines, if you consider Fourier series.

 

[PainterGuy]

Thank you, everyone!

Also, the Golden Ratio is much more common than you imply, but it does not occur with perfect accuracy that often in nature.

I do agree that the Golden Ratio is very much common in nature but still not as much common as it is shown to be.

 

[Asymptotic]

Which mathematical objects (numbers, functions, figures, etc) are more common in nature? I mean the mathematical objects which could more easily be identified with nature. For example, circles and triangles are most common mathematical figures (or, objects) which could be found in nature. Likewise, sine and cosine functions are really common in nature because many natural phenomena are periodic.

Thank you for your help.
But in my humble opinion I don't think that fractals and logarithms are as common as circle, triangle, sine, and cosine because these are some of those mathematical objects which really define the connection of mathematics with physical world. There is no doubt that almost all of mathematics is a way to encode and quantify physical world in a defined manner but mathematical objects like circle, triangle, sine, cosine are really a part of alphabet for that encoding. Thank you.

In my contrarian opinion, geometry in the natural world I'm familiar with doesn't deal in ideal triangles, circles and spheres, but is a rather more convoluted affair. While walking down a creek in summer, the rocks under my feet have shapes as diverse as the snowflakes that fell upon these waters in winters past, with not one exactly the same as another. Straight lines aren't for stream beds; if one runs much further than a cobble's throw perfectly plumb and square it's a ditch or a canal that you've got there. It is as though nature abhors pure platonic shapes, and tweaks them to suit her fancy.