Butterfly Skeleton

butterfly skeleton
apophysis 

Dancing Dragon

dancing dragon

 apophysis

 

Reverse Hourglass (and other fractals)

The last image has a name: Reverse Hourglass. The others are just images!


They were all made by me, a complete fractal novice, using the cutest (and free) little program called Apophysis. It's super adictive, so I'm sure more fractal posts will follow...you don't mind do you?


Yours in fractality,
JJR

Limits of the Imagination

This evening I'm doing an "academic lecture" as part of a series at school. My lecture is going to be on "The Limit". Here's a preview:
 

Are we getting somewhere?

Maybe not...

When it comes to limits, expect the unexpected!

So let's create an accurate definition:

Any Questions?

To Infinity and (not) Beyond: Part II

photo from the nytimes archives

 
I'm sorry that this post is so belated, but the creative side of my brain has been dead, and understanding Maths  takes a lot of creativity.

everything maths

I promised you a discussion of Mr Cantor and his contribution to infinity... Cantor realised that there is more to understand about infinity than previously thought.

He started by thinking about different sets that we use everyday and think about as "infinite", such as the natural numbers (0,1, 2, 3, 4...). And the real numbers (0, 0.1, 0.0000000002, -0.0547829 etc...). In fact he thought about a lot of sets, and these are just two simple ones. But they will suffice for my highly simplified explanation.

Most of us would agree that there are infinitely many natural numbers. You could keep counting forever (though why you'd want to I'm not sure) , and you would never "run out" of natural numbers - even if what you were counting was the number of stars in the sky!

To briefly introduce some terminology, Cantor called the number of elements in a set the set's cardinality. He called the cardinality of the natural numbers "aleph null" (HINT: the squiggly N is the Hebrew letter aleph).

Well so far so good. All we've really done is make another label for infinity. Who cares?

The really exciting part comes now! Cantor went on to observe that the real numbers behave...differently...to the natural numbers. In fact, he demonstrated that there are infinitely many real numbers between every two natural numbers.

This is the core of the Cantor's insight, and it is demonstrated visually by Cantor sets, which you can learn how to draw here.

The idea is that if you keep dividing the interval between 0 and 1 into thirds, deleting the middle third each time, you could keep dividing forever. Since the real numbers between 0 and 1 essentially represent every possible fractional value between 0 and 1, they can in some sense be represented by the infinitely many "segments" you would have at the end of this process. If there was an end, which there obviously wouldn't be!

And yes, the Cantor set is a kind of fractal - go here for another (probably more rigorous) discussion!

Cantor went on to say (very reasonably in my view) that you can't say that the real numbers have the same cardinality (number of elements) as the natural numbers if there are infinitely many real numbers between 0 and 1 and further between any two natural numbers. It just doesn't make sense! So he labelled the cardinality of the real numbers aleph, with the rider that aleph is bigger than aleph null.

Um, aren't we forgetting something here??? We agreed that both the natural numbers and the real numbers had an infinite number of members. And do you remember Galileo: ...“it is wrong to speak of infinite quantities as being the one greater or less than or equal to the other”?

Cantor disagrees. According to him, there are at least two types of infinity. The first has the same cardinality as the natural numbers. This he called a "countable" infinity. The second has the same cardinality as the real numbers. This he called an "uncountable" infinity.

As you can imagine, this has some interesting consequences...but that's a story for another time.

from here

To Infinity and (not) Beyond: Part I

 

...awesome disney blog where I found this...

This mini-series of posts is dedicated to those of you who never got the full impact of this wicked Disney joke (much like me, until Maths III).

Newsflash: the whole point of infinity is that there is no such thing as "beyond" infinity. Infinity is it. The biggest. The most. The furthest.

To be honest, I'd be lying if I said I understood infinity. I don't even understand Mobius strips, and those are about the best visual representation of infinity we have... Escher drew the best one (of course). Follow the marching ants and you'll see what I mean by the Mobius strip being a visual representation of infinity. Sort of, anyway.

Much like me, the ancient Greeks didn't really like infinity much. In fact some of them refused to believe in infinity at all (you've got to admire these guys for their stubbornness if nothing else). They were forced to acknowledge the idea of unboundedness (time appearing to have no beginning and end, for example) but all applications or references to infinity were... tricky. Irrational numbers like

π

(which has an *infinite* number of digits) were regarded with deep suspicion and generally shunned.

The Arabs used the notion of infinity, because they started solving equations like

x² = 2

which (as you will know) only has irrational roots. They still weren't too keen on infinity though, and didn't examine the alarming endlessness of irrational numbers too closely.

Meanwhile in Europe quite a lot of people used the notion of unboundedness or infinity as a handy notion to explain divinity. Theologians like St Augustine and St Thomas Aquinas used the idea fairly liberally to refer to God, the unlimited being.

In fact most great minds have flirted with the idea of infinity here and there. The famous Galileo said that “It is wrong to speak of infinite quantities as being the one greater or less than or equal to the other.” 

...super-awesome collection of fractal art...

But still no-one had really studied infinity for its own sake. Until one day along came...Georg Cantor.

I think he deserves a post of his own...watch this space!

(oh, and go here to download an article about the history of infinity which goes into a lot more detail. 27 pages of detail, but fairly accessible. strongly recommended if you found this interesting)