Fractals inception-style and the father of fractals: Benoit Mandlebrot

I am currently reading Chaos Making A New Science by James Gleick and my mind seems to be lost in a perceptual run-through of the beauty of fractals.  The term fractal was coined by the amazing Benoit Mandelbrot.  Here I refer to his science but it is worth mentioning that he lived a very fascinating life including escaping Nazi persecution upon fleeing his native Poland to France and thus the science he was able to conduct in his life is all that much more precious.


His research and 'knack' for seeing fractals has resulted in him remaining one of the most inspirational scientists I can think of.  I am not alone in this thought; James Gleick in Chaos himself says:

"Benoit Mandelbrot was the one who let us appreciate chaos in all its glory—the noisy, the wayward, and the freakish, from the very small to the very large. He invented a new and slightly nebulous field of study—a kind of geometry, for want of a better description—and he invented that recondite name for it, fractal.^"

Certainly a 'character' I highly recommend his TED talk (below).

 

"The beauty of geometry is that it is a language of extraordinary subtlety 

that serves many purposes".   B. Mandelbrot

So what is a fractal? 

Fractal is a word introduced by Mandelbrot to describe a mathematical concept also witnessed in the natural world of self-similarity across scales.  (Inception style!)   

An example of this can be seen below in a loop continually zooming in on a a Mandelbrot Set image across scales.  You see the shape (large circle on right attached to a small circle on the left) but as you zoom in you see that this shape is made by many smaller versions of the same shape.  Zoom in on any one of those and you will see a smaller self-similar FRACTAL of the original shape and so on.  Repeat as long as your brain can tolerate the fractalness!  Me?  My mind is already blown by about 3 fractals!  ;)

Fractals are mathematically non-differentiable.  Needless to say mathematically-speaking fractals are quite complex and I am not a mathematician so I will instead focus on a few cool examples. 

Fractals in nature:

An example of fractals in nature- broccoli.

Mountains- fractal.

The airways of a mammalian (dog) lung.  From: Dr. Robert Henry DMV of the University of Tennessee College of Veterinary Medicine via:  http://classes.yale.edu/fractals/panorama/biology/physiology/animallungs/animallungs.html

Finally- to me feathers are also fractals- but I could be wrong (what do you think?)

Mandlebrot ran computer simulations of complex numbers in mathematical operations approaching infinity to reveal the patterns of fractals that result.  In addition to the resulting images that are the poster-children of Fractal research (examples above and below)- he published several books with lovely fractal images.

In his famous paper written and published in Science (1967) Mandlebrot clarifies the concept of the fractal further titled: "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"

Here he presents a case where ... 

...say you were to measure the coast of Britain using a measuring instrument of the size indicated below you can imagine your work process and end result would look something like this:

 Now lets say you realize you've missed some little pieces of the coast (see there on the left-side?) so you decide to do this again but now with a smaller measuring implement.  Britain suddenly gets much larger (more coastline) (see below)!

 This process could be repeated again and again with a smaller measuring implement (see you still missed some bits of the coast) and you would find that the coast is still larger...still missed some?  Repeat- and boom!  The coastline is longer (and so on till you find yourself crawling along the coast measuring around pebbles and sand to get each contour of the coast!).

Why is the coast longer and longer - the finer scale you use to measure it?  Well the coastline is fractal and the closer you look the more curves/lines indents and whatnot it has. As you add these imperfections to your measurements your 'estimate' of the coastline with be longer, longer and LONGER!  Inception-style!

Ok- as this post is not really about bats I should stop digressing and get back to the bats!  

This has just been brief introduction to the lovely world of fractals and I hope you are now equally excited as I am about  fractals and the legacy of Mandlebrot.  Fractals seem to be a place where mathematicians, biologists, and artists can have many an inspiring exchange!

 

Note: What is 'inception style'?  

'Inception style' refers to a concept from the movie Inception (term coined by graduate students in the OEB program at UMass Amherst in conversational settings).  Inception was released in 2010, directed by Christopher Nolan and stars Leonardo DiCaprio.  I don't want to ruin it for you so I'll suggest you Netflix it or at least watch this trailer to see how Inception Style is another example of fractals!   

Ie. Fractals in Hollywood?  

 (Perhaps it only works if things get 'smaller' as you repeat (I'm being careful to not say too much and ruin it for you- but think on it if you have seen it))!

The Count Loves Bats: learning to count using bats!

Bats and Sesame Street!  Just another case of bats being depicted in a fun way - although perhaps still rather cliche/mythical manner considering they are chilling with Count Dracula!  At least here they are quite cute so maybe kids who saw this would LIKE the bats in the clip.

Romanian flag

Interestingly enough Count Dracula makes me think of Romania (Transylvania is where the original 'Count' was supposed to be from) and a quick search revealed that indeed Romania has its own Romanian Bat Protection Association (here)!

Furthermore, they COUNT bats!  (Sorry I couldn't resist- but really... look there they are counting bats!)


Oddly there are no vampire bats in Europe so where the mythical connection between bats and vampires came from in this region remains something of a mystery to me but I am sure someone somewhere has a hypothesis regarding where this mythical relationship originated.  Of further interest is what made Sesame Street producers decide that they needed a 'count to count' is interesting and clever.  I am not sure how many kids loved him growing up but I would like to be counted in that number!

Transylvanian coat of arms (Transylvania is a historical region of Romania).

True Facts About Bats

This video is too amazing (i.e. hilarious) to not share.  The video was made by this guy.  It is fairly accurate and includes top-notch humor.  Enjoy!  (P.s.  I think 'Eric' is an Artibeus jamaicensis... meaning he is even more awesome than depicted just here!)

Belize bats

Over the next few days I will be finalizing my preparations to head off for a quick (5 day) trip to the field.  Specifically, I will be going to the bat species-rich country of Belize with a group of other bat biologists for a collaborative batting effort!  I hope to sample a few bats others are working with as well as get some great photos of less common species.  I'll admit that I am excited to catch my old friend from my dissertation: the Jamaican fruit bat (Artibeus jamaicensis).  Strange how a species you work with extensively can become like a dear-friend you miss when you don't see it for awhile and that you look forward to catching up with.

My mad dash to pick up essentials included:
Bug spray (DEET included unfortunately a must for the mosquito-prone in the tropics),
batteries for headlamps (*a important item for hours of bat work- I also have 3 headlamps packed 'just in case'),

black velvet for photography of tissues/bats etc.,
duct tape (just case you don't recall why see this post).

I also had to grab a suite of lab supplies as well.  Of course these you cannot just pick up at Target!  A favorite that will surely last the duration of my Postdoc tenure?  Parafilm!  Parafilm is a fun tape-like wax that you can use to seal tubes and other containers and is extremely useful and fun to use!  Really I would suggest that Parafilm is the scientific version of duct tape.  It is incredibly useful.

I leave Monday and look forward to sharing information about the bats we catch!   With any luck I will have many an update during my trip (we should have internet access making this possible)!  Meanwhile tomorrow will consist of packing the rest of my lab supplies: calipers, pesolas (a spring-based scale used to weigh bats), leather gloves (for handling bats) among other things.

Meanwhile- as I sit and write this- a song (Tropical Iceland by the Fiery Furnaces to be exact) came to mind.  Really this song is a little silly but I've always loved how catchy it is and well as you might suspect leaving the North East for the tropics made it come to mind again.  Give it a listen!  If nothing else the cartoons are both strange and cute.  Yes and 'strange and cute' makes me think of some of the bats I hope to catch (see? full circle)!   Now feel free to weigh in on the image below is it strange and cute or just strange?

'Don't hate me because I am beautiful!l' an image of Centurio senex the Wrinkle-Faced bat (image by Laval from the American Society of Mammalogists image library). 

Crowd funding and hemipenes

I know its been some time since I've posted a blog- but- it is time!  Why you might ask?  Well primarily because I have a few minutes to catch my breath and more importantly tell you about an exciting project that has been listed on Rockethub for crowd funding.

(If you need an photo to grab your curiosity look no further!)

Photo: C. Gilman (from http://www.rockethub.com/projects/21959-hemipenes-evolutionary-novelty-or-a-new-twist-on-an-old-trick)

This lovely project titled Hemipenes: evolutionary novelty, or a new twist on an old trick will investigate the development of hemi...wait did I say pens? that sounds sort of like 'penis'...  yes hemipens!  That's right two penises!  Lizards happen to have two (as do other squamates) but how do these two functional intermittent organs develop?  Do they develop the same way that a usual 'ol penis does?  Well- my dear friend and labmate Casey Gilman - has a project ready to go to find out!  Stop by her Rockethub page to see her COOL video to learn more!  (For example- do you find yourself curious if they use both at once?  Go to find out by watching Casey's video)!

Why should you care about hemipenes? 

Well according to Casey a PhD candidate at UMass Amherst "Because of differences in general morphology (two versus one), and some observable differences during development, hemipenes have been considered by some to be an evolutionary novelty. However, there is still no agreement about whether hemipenes and the single penis found in other amniotes are homologous (that is, they come from a common ancestor), or are indeed novel."