Reddit reviews The Fractal Geometry of Nature

Pretty sure Xfocus was being sarcastic.



See "The Fractal Geometry of Nature" by Benoit B. Mandelbrot (the Mandelbrot - the man that coined the word).

I'm in a similar boat with you. I went through calculus in high school, graduated university with a B.A. in music, but have recently taken a keen interest in developing an actual understanding of math.

Aside from music, I have a strong background in philosophy, and from philosophy, so do the natural sciences extend and I've taken advantage of that. Math was discovered through raw observation of the world and through the concourse of logic, and so I have designed for myself the study of math through the source works of where the math originated, for practical and ontological purposes. Here's a few books that I've picked up and began reading:


A History of Greek Mathematics, Vol. 1: From Thales to Euclid https://www.amazon.com/dp/0486240738/ref=cm_sw_r_cp_apa_RljGybYRSB723

The Mathematical Principles of Natural Philosophy: The Principia https://www.amazon.com/dp/1512245844/ref=cm_sw_r_cp_apa_AmjGyb14R4B2V

Euclid's Elements https://www.amazon.com/dp/1888009187/ref=cm_sw_r_cp_apa_7mjGybZ97DBR7


Introduction to Mathematical Philosophy https://www.amazon.com/dp/1420938401/ref=cm_sw_r_cp_apa_OnjGybQ0078ZX

The Fractal Geometry of Nature https://www.amazon.com/dp/0716711869/ref=cm_sw_r_cp_apa_lojGybPPY25P4

The study of equations and formulas had been unfulfilling and unengaging until I framed it with the historical context of the natural sciences. I'm still a novice to this approach, but I believe it to be of merit- Ive also see some indication (when researching my own self-study method) that this is more similar to the method which Waldorf schools teach math and science as opposed to the traditional American Public school classroom, which as I grow older and reflect upon the majority of my experiences in classrooms, were uninspired, with the exception of very few memorable educators.

You could even base your study on other, less abstract interests than the interest of learning mathematics, such as an interest in modern physics or economy (or Comp sci, anything that utilizes math). Using that interest as a guide, you would be more clear minded to reverse-engineer your own individually purposed self-study. Such a direction of interest would certainly help for you to be able to design your course and keep you engaged. I hate how I've worded most of this Frankenstein of a comment; it's unnecessarily verbose and unorganized, but it's late and I'm tired to I'm not gonna edit it, nevertheless, hopefully you'll get the point(s).

Anyway, I'm curious what other people have to say about this approach, and especially I am open for people to suggest in response here to additional and essential sourcebooks!

I'm really glad you posted this, I'm buying Mandelbrot's book now.

The best description of Fractal Dimension that I am presently aware of is the one presented in Mandelbrot's book: The Fractal Geometry of Nature.

You start off some time in the 19th or early 20th century, when cartographers were trying to work out the length of the coastline of Britain. Despite cartography being so mature of a discipline that we can launch rockets and photograph the Earth from space for the first time and find basically zero surprises compared to what we've already mapped by crawling across the surface like microbes on a watermelon, here we are with a dozen survey teams all reporting lengths for the same portions of British coastline off by factors of 2-5. I mean, it's simply preposterous!

Hell, cartographers from Portugal are reporting coastal lengths for their country — with impeccable methodology, mind you — greater than Spanish cartographers find for the entire Iberian peninsula.

Well, somebody did a meta-analysis and found that reported coastal lengths not only correlate directly with what atomic measurement scale the surveyors used (EG: over how short of a distance do you stop trying to count the winding details), but the correlation was exponential and it followed different exponential constants for different coastlines. For example, shrinking how short a measuring stick you use to measure the coastline of West Britain by N will give you a total length that is longer by about N^1.25, regardless the starting value of your yardstick or the value you choose for N.

Mathematically, this means that if you keep shrinking your yardstick and count every bay, every outcropping of rock, every pebble, every molecule dividing a time-perfect snapshot of sea from land, the total length that you measure will not converge onto any attractor representing the "real" length of the coastline.. it will instead predictably diverge to infinity.

But we get the same effect if we try to measure the "length" of a square area, say 1 foot square. You can try splitting it into square inches, by lining them up in a row and seeing that they measure 144 inches long. Or you can divide smaller into square half-inches.. but now they get to be 288 inches long. And splitting more finely by N always nets you a "length" that is N^2 yardsticks "longer".

So, any mathematician would just patiently explain to somebody trying to find such a length that there isn't one because they're trying to measure magnitude in the wrong number of dimensions, and that the exponential constant they are running against is the number of dimensions they should measure with to get a reliable and finite result.

That said, one can theoretically measure the coastline of Britain and converge to a finite result so long as they are constantly considering inch^1.25 's, but of probably more use is the understanding that the 1.25 gives us a reliable measure of how "rough" the coastline is: how much extra length one gets from studying another successive factor of detail. :)

All surfaces that remain "rough" or bumpy no matter how far you zoom in can be said to have fractal dimension. From "dusts" of points like the cantor set (log(2)/log(3) ≈ 0.631) between dimensions 0 and 1 .. infinitely complicated collections of elements each dimension 0 to coastlines like the Koch Snowflake (log(3)/log(4) ≈ 1.2619) or foams like the Seirpinski Triangle (log(3)/log(2) ≈ 1.585) between 1 and 2.. infinitely complicated collections of (or kinks in) elements each dimension 1, to surfaces like any given land area on Earth, or foams like the Menger Sponge (log(20)/log(3) ≈ 2.727) with dimensions between 2 and 3 represented by infinitely varied kinks and folds in 2d elements or continued aspiration of 3d elements until all 3d volume is lost. (obviously cantor set and sierpinski triangle can equally be described as aspiration of larger-dimensional solids as well! ;D)

Fractional dimensionality can obviously be extended farther, and even measurably in our own universe one can posit that the gravitational warping of spacetime around infinitely varied mass distribution gives us slightly greater than 4 space+time dimensions prior to even leaving the bounds of mundane general relativity: EG, any attempted measurement of volume * duration of any portion of the universe is doomed to diverge to infinite values by some constant as your measuring stick to account for smaller and smaller curvatures around smaller and smaller gravity wells keeps shrinking.

But in addition to cylindrical and spherical coordinate systems (themselves just elliptical dimensions combined with euclidean ones) it is fun to consider more exotic additions like hyperbolic dimensions (Yeah, you can cross hyperbolic dimensions with Euclidian ones in the same space) or fractional dimensionality or add more Minkowski dimensions because you did remember that we already have one of those, right? Well heck, we can even take that one away and make it Euclidian instead. xD

But yeah, it's true that "adding more Euclidean spatial dimensions to our 3E+1M reality" is a fun thought exercise, and that the result of adding more E is the same as adding more elements to a vector for our linear algebra formulas to nom upon. And there are a ton of fun alternative to consider as well. :)

Patterns are everywhere in nature.

Once I was eating grapes while watching a video about the inter-filamental structure of the universe when my mind suddenly exploded.

When we see the endless variety in nature it's amazing to think of how all life is encoded in DNA. One of the most efficient ways to express a potentially endless variety is using fractals. It's no wonder then that much of the behavior and form of life exhibits fractal patterns.

Jason Silva did a short on the awe of patterns.

Our understanding of much of this came from Mandelbrot's 'The Fractal Geometry of Nature' book (who passed away just a few years ago).

Books on Fractal Geometry tend to have pretty pictures:

Indra's Pearls: The Vision of Felix Klein by David Mumford et al.

Beauty of Fractals by Heinz-Otto Peitgen et al

Fractal Geometry of Nature by Benoit Mandelbrot

For what it's worth New Kind of Science by Stepeh Wolfram has tons of pretty pictures, even if the content is dubious.



you might also want to checkout the Non-Euclidean Geometry for babies and other similar titles.

Fractals are most useful for understanding the character of nature! http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869

It's time we weaned you from the Gaussian. Here's something to help you with your teething.

I found it interesting because of the history behind the fractal and how it entered into maths. If you still feel like not watching it you can look into Mandlebrot book

http://www.amazon.co.uk/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869

the concept you're looking for is fractal.

an important book on the mathematical description of nature called "The Fractal Geometry of Nature" was written about 40 years ago by a guy named Benoit Mandelbrot. in it, he described how iterative natural processes could be described mathematically to model natural phenomena. it's an amazing book, a work of true genius, but heavy reading.

the Fibonacci sequence is not fractal -- that is, self-similar over a broad domain of scales. but some sequence sets are.

in any case, the self-similarity you are observing in this -- how the small branches look just like the big branches but in miniature -- is definitely fractal and just one of the many ways in which human systems represent our nature.

Absolutely!

Math is everywhere and it's just about seeing the patterns emerge from simplicity. My knowledge on this topic has mainly been from my own work in Artificial Life and encoding AI genetic knowledge combined with my general interest in biological patterns (which are everywhere in nature) but the first thing that got many things to click for me was playing around with Turtle Logo in high school that is all about using simple constructs to create amazingly complex structures (i.e. one, two - look familiar?).

Sadly I don't work on my AI research anymore due to ethical concerns so I'm a bit out of date but I'd highly recommend the following that weren't mentioned in the original post though:

• "Godel, Escher, Bach: An Eternal Golden Braid" Book
  
• "The Fractal Geometry Of Nature" Book
  
• Procedural Generation
  
• Gene Expression Programming

I would strongly encourage you to pick up Mandelbrot's book on fractals as it shows the intersection of real-world problems with fractal theory. There are now better introductions now but this is THE CLASSIC reference (and a good read).

Here is the amazon link, but you can often grab it in used bookstores.