Welcome to Fractal Forums

Today I had this idea in the other forum: Escher's pictures "Circle Limit" I to IV... they look kind of fractal, but I'm not entirely sure.

What do the other Escher fans think?

M.C.Escher Circle Limit III in a rectangle (http://vladimir-bulatov.deviantart.com/art/M-C-Escher-Circle-Limit-III-in-a-rectangle-281848653) by Vladimir-Bulatov (http://vladimir-bulatov.deviantart.com/)

Well, I guess the shapes he uses aren't selfsimilar but due to the projection it becomes a fractal.

yes, definitely fractal.
I think they very much are.. same forms, some just slightly warped and decreasing in size. that for me is self-similarity

 

I meant the fishes themselves. :)

Escher's circle limits are tilings of the hyperbolic disc. Nothing fractal about it. The individual tiles just get smaller and smaller towards the edge of the circle.

If one calls this 'fractal', than one should call the square tiles in your bathroom 'fractal' also.

wohoo, I've got fractal tiles in my bathroom!  ;D

seriously:
(https://upload.wikimedia.org/wikipedia/en/5/55/Escher_Circle_Limit_III.jpg)
can't you see it?

the patterns repeat and they get smaller and smaller, so they are scale invariant.
to quote the very first sentence for "fractals" from wikipedia:

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.

what does the koch curve do much differently? or the cantor set?

I would say that Escher's circle limits *can be thought of* as tilings of the hyperbolic disk.  That the projection makes the shapes get smaller and smaller toward the edge is what makes this a fractal image, in my mind.  The bathroom floor is not a fractal because the tiles are the same size.

Well, if you had eyes adapted to hyperbolic geometry, you would see that all the tiles are the same size, when measured with the hyperbolic metric that is valid in the disc.

So it's a question of perspective? If you watch it from a 'hyperbolic geometry'-perspective, all tiles are the same size.
The thing is, probably no eyes in reality are adapted to hyperbolic geometry - (being a purely theoretical concept?).
So if you watch it from a human 'real life' perspective, with my eyes looking on the 2d computerscreen, the visible patterns clearly show fractal characteristics.
I don't see the point in saying 'this must (or even can only) be watched from a certain perspective."
(even if the mathematics behind this are solid)

Could you explain to a non-mathematician in simple words, what the difference between hyperbolic and fractal geometry is?

couldn't you generate eschers picture by using IFS?

I think the difference is the same one I always go on about - if a fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale, then the bathroom tiles fail to meet either of the qualifying criteria.  They certainly are not a natural phenomenon, and it would take quite a lot to convince me that they are a mathematical set.  Both the Koch Curve and The Cantor Set are quite definitely a mathematical set, and thus meet the criteria.

That's just my several hundred pennies worth   :tease:

Yeah, after thinking about it, it's like this: Like the Koch curve, these pictures have infinitely many figures on limited space; to create it, you need an iterative process, also like with many fractals; but you can't cut it in pieces which look exactly like a smaller version of the whole picture, so it's different. Am I forgetting something?

I think you do.
That there is not just strict-self similarity like in the koch curve but also quasi-selfsimilarity like in the mandelbrot-set, where shapes can look very different from each other but still are similar to their 'close relatives'.
A common example for this would be the distorted mini-mandelbrot sets deeper down in seahorse valley.

Well, the definition of fractal includes: They have a number of dimensions that isn't a plain number.

But how would you measure the dimension of Circle Limit X?

I have no idea.

But I'm wondering about what kind of dimensions are we talking about? if it's the fractal dimension as in Hausdorff Dimension, lots of fractals have an integer dimension, like the mandelbrot set with 2.

https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

That's also true. Well, even the experts haven't agreed on how to define fractals.

Just checked the German WP article again and found something we missed:

Mandelbrot himself defined fractals like this: A fractal is a set which Hausdorff dimension is bigger than its Lebesgue covering dimension.
(German: "Ein Fraktal ist eine Menge, deren Hausdorff-Dimension größer ist als ihre Lebesgue’sche Überdeckungsdimension.")

Cantor dust has L(ebesque) 0, but H(ausdorff) 0.6309...

A Hilbert curve has L=1, but H=2.


This allows fractals with a dimension that's exactly 2.