Fractals, Mathematical Constants, and Binary Code

[epepke]

The Mandelbrot set is one of the more famous fractals, but there were others before it, such as the Cantor set (1884?) and the Koch snowflake (1904).

Yes. However, the notion of a fractal dimensionality was pioneered by Mandelbrot and his contemporaries.

 

[Art Vandelay]

It means that when you look at it at different magnifications, it's all prety much the same.

Take a Nautilus shell, which is one of those golden spiral thingies. If you magnify a small chamber, it looks pretty much the same as a large chamber.

That's statistical self-similarity. In the strongest sense of the term, self-similar means that it's possible to take part of the object, magnify it, and end up with an exact copy of the original object.

 

[Dr Adequate]

A quick glance into this area on the 'net covers everything from archiatecture, breeding rabbits, how trees grow, to the da Vinci code. So mathematics can be made to cover natural phenomena and woo-woo constructs, giving woo-woo people some sort of scientific footing.

In the Da Vinci Code, so far as I can make out, the author invents a code which takes knowledge of the Fibonacci sequence to decode. This hardly means that it provides support for woo-woo concepts.

 

[DevilsAdvocate]

In the Da Vinci Code, so far as I can make out, the author invents a code which takes knowledge of the Fibonacci sequence to decode. This hardly means that it provides support for woo-woo concepts.

As I recall, it wasn't nearly that complex. The "code" was just a scrambling of the first few numbers of the Fibonacci sequence.

 

[1984]

Thanks alot. Due to this thread I started my fractal generator this morning and it's now been 6 hours and I'm still playing with fractals. They seem to calm the mind which may lead to a little understanding in how the mind works.

That’s what got me about fractals years ago when I saw a TV documentary on them. I believe it was this one which comes as part of a book DVD package. The book is introduced by Arthur C. Clarke.

In the Da Vinci Code, so far as I can make out, the author invents a code which takes knowledge of the Fibonacci sequence to decode. This hardly means that it provides support for woo-woo concepts.

I’ve been moving towards the same conclusion since starting this thread. Thanks for that, Dr Adequate. I still think I have some mathematical mastery to do if I'm to reason against numerologists and the like.

The Mandelbrot set is one of the more famous fractals, but there were others before it, such as the Cantor set (1884?) and the Koch snowflake (1904).

So the math came before the visuals? I was under the impression that fractals could only exist because of computers. This from the book description of the link above: “The power and the beauty of fractals were only capable of being seen with the advent of computers, which become psychedelic windows on the infinite when using simple fractal equations.” (My underlining.)




Looks like no-one is going to give me an easy answer. Drat! I’ll have to chase up the references above, align them with my science and philosophy books, and post the most well known mathematical equations into their historical context here. Don’t hold your breath. Could be a while, but I have to do it for myself. More input most welcome.

 

[Cabbage]

So the math came before the visuals? I was under the impression that fractals could only exist because of computers. This from the book description of the link above: “The power and the beauty of fractals were only capable of being seen with the advent of computers, which become psychedelic windows on the infinite when using simple fractal equations.” (My underlining.)

Some fractals (like the Mandelbrot set) probably do require computers for their discovery, because a lot of computation is required just to get an idea of what they look like. However, other fractals, such as the Cantor set and Koch snowflake, can be reasonably approximated with a quick sketch once you know how they're defined.

 

[1984]

In a last ditch attempt to avoid hours of personal research into my least favourite topic, I visited a big local bookstore, and came away with the following Math through the Ages by Berlinghoff and Gouvea. Follow the links to the table of contents. Note the subtitle: A Gentle History for Teachers and Others. I'd be one of the "others" . I like the word "gentle". And there's even pictures .

This is the expanded edition. What that means is you get an additional 50 pages or so of sketch points (which I'll read), with questions and essays for each sketch point (which I won't be doing) . For instance, "zero" is mentioned twice in the body of the book. It is then expanded upon as a sketch point over five pages. The book is endorsed by the Mathematical Association of America.

I also came across book endorsed on the back by Arthur C. Clark (amongst others) called 200% of Nothing: An Eye-Opening Tour through the Twists and Turns of Math Abuse and Innumerancy by A.K. Dewdney. Amazon review: "If you know the difference between lies, damned lies, and statistics, give a copy of A.K. Dewdney's 200% of Nothing to your friends to get them up to speed. If you don't know the difference, consider this funny, engaging little book a crash course in numeracy, the mathematical equivalent of literacy." I got the hardback on order which should arrive by the time I've worked my way through the first book mentioned. This way, I won't have to keep referencing my Skeptics Dictionary to identify and reason against woo-woos. I'll be able to do it for myself. I'll be ready to rumble .

I'm chuffed for the responses to this thread which I'd started not knowing why. Anyone want to post math based books that have helped them, I'd still be interested. Not good to rely on one source alone for an answer. But these two texts hit the spot for me.

Now for fractals .

ETA: Author's name.

 

[1984]

This is the expanded edition. What that means is you get an additional 50 pages or so of sketch points *(snip)*

Correction. Both the regular and expanded editions have the sketch points. The expanded edition contains questions and essays to do. As far as I can tell, only the expanded edition is currently available in hardback.

 

[SezMe]

To add some enjoyment just for enjoyment's sake, read A History of PI by Petr Beckmann (no, that first name is NOT a typo). It has very little mathematics in it. Who would have thought a whole book could be written about one little irrational number?

 

[1984]

To add some enjoyment just for enjoyment's sake, read A History of PI by Petr Beckmann (no, that first name is NOT a typo). It has very little mathematics in it. Who would have thought a whole book could be written about one little irrational number?

Thanks for that. Apparently (don't you just love that word?) Pi cannot be definitely attributed to Pythagoras. I got that from my book .

There's a film that has the mathematical symbol for Pi (1998) as its title. This mathematical guy studies universal patterns in mathematics. He's chased by Jewish zealots and finance world heavyweights for his insights. The zealots want him to usher in a new messanic age. The financiers want him for obvious reasons. Very believable numerical stuff that I'd like to be able to identify as woo-woo or not, by my own reasoning.

 

[DevilsAdvocate]

I'm chuffed for the responses to this thread which I'd started not knowing why.

I've heard of "chuff" but never "chuffed". Glad I looked it up before going off.