Help- what self-similar shapes do you know of?

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Offline TGlad

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« on: November 10, 2018, 11:35:04 AM »
Hi everyone, I am going to be *writing a book* about self-similarity and I need your help !  :thumbs: :D

I would like to get a survey of what 2D and 3D self-similar objects you can think of, by name. But the twist is that it can't have a fractional fractal dimension. So not 'thin' fractals but instead self-similar objects with dense regions, for example the Pythagoras tree.

I'll kick off with four to give you the idea, but I'd really love to get as many well-known types as you can think of. Thank you indeed, the more contributers the better!

- Pythagoras tree
- Mandelbrot set
- Rauzy fractal
- Gosper Island
« Last Edit: November 10, 2018, 11:47:47 AM by TGlad »

Offline Sabine62

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« Reply #1 on: November 10, 2018, 01:20:01 PM »
To thine own self be true

Offline TGlad

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« Reply #2 on: November 11, 2018, 05:15:35 AM »
Thanks, yes I do. They however almost all have non-integer dimensions.
I am interested in what structures people know about, as a sort of survey of which are well-known.

Offline gerrit

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« Reply #3 on: November 11, 2018, 05:43:28 AM »
Certain totally disconnected Julia sets of quadratic polynomial have Hausdorff dimension 1. For example:
https://fractalforums.org/image-threads/25/gerrit-images/565/msg5153#msg5153

Offline Sabine62

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« Reply #4 on: November 11, 2018, 02:29:15 PM »
Already thought so;)

Quote
I am interested in what structures people know about
I have no idea at whom/what kind of audience your book is directed, if at 'about everyone who likes fractals but does not necessarily know a lot about them in mathematical detail'  ;D then next to the pythagoras tree and the mandelbrot set: koch curve, sierpinski triangle, apollonian gasket.

Offline FractalDave

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« Reply #5 on: November 11, 2018, 09:00:23 PM »
Heighway Dragon, Twin Dragon, Tame Twin Dragon, Cesaro Sweep, Levy Dragon.
The meaning and purpose of life is to give life purpose and meaning.

Offline FractalDave

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« Reply #6 on: November 11, 2018, 09:03:06 PM »
Also (any?) Peano Curve.

Offline claude

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Offline TGlad

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« Reply #8 on: November 13, 2018, 01:15:10 AM »
These are great suggestions, thanks everyone so far.

I know it seems lazy of me rather than doing my own search, but it has more weight if these are a from a survey of many people, rather than my own picks of what I think the well known shapes are.

Keep them coming  :thumbs: :thumbs: :thumbs:

Offline rsidwell

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« Reply #9 on: November 13, 2018, 03:12:05 PM »
Any space filling curve will have an integer Hausdorff dimension. The Peano curve (already mentioned) was the first example, but there are plenty of others. The Hilbert curve comes to mind; there are both 2D and 3D versions.

I don't normally focus on Hausdorff dimension, but I remember a few because they were surprising. The Sierpinski tetrahedron is a 3D object with dimension 2. Brownian motion/random walk has dimension 2 in any dimension, including 2D and 3D (assuming you accept statistical self-similarity).

Offline TGlad

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« Reply #10 on: November 14, 2018, 12:10:26 AM »
Thanks! great suggestions, I didn't know that about the Sierpinski tetrahedron.

The simplest examples are just recursive 'filled' shapes, like how the Pythagoras tree is made of filled squares. But there are surprisingly few of these that have names.


Is there a name for one which is a tree of disks? or spheres?


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