I used the formula N*r^d=1, assumed that N=8 and from there got r=1/16. But I can't imagine how that fractal looks like. Maybe you know an instrument to visualize or just give me a hint.
Thanks in advance!
Bit of a long shot here, but I'm following Falconer's Fractal Geometry (3rd ed) and I'm stubbed by the solution to exercise 5.4. For reference, the solutions manual is available openly here http://higheredbcs.wiley.com/legacy/college/falconer/111994239X/solutions/solutions4-8.pdf
In his estimate for the upper density of the Cantor dust set, Falconer gives the relation
[; H^1 (F \cap B(x, 4^{-k} \sqrt{2})) >= H^1 (E_k) = 4^{-k} ;]
Justified by the fact that [; B(x, 4^{-k} \sqrt{2}) ;] contains a whole square of E_k However we also know that
[; F \cap B(x, 4^{-k} \sqrt{2}) \subset E_k \cap B(x, 4^{-k} \sqrt{2}) ;]
Which would imply the reverse inequality from subadditivity? I'm not really sure how he obtained the greater-than inequality?
Fractals would probably come closest, no? So if the Romanesco broccoli is an example of an approximate fractal in nature, does that mean that the broccoli is a 3D object or a very close approximation of a Hausdorff dimension between two natural numbers? (it cannot be between 2D and 3D since that implies no volume and infinite area... but I can't see it between 1D and 2D either...)
http://imgs.xkcd.com/comics/chaos.png
Greetings! There are many definitions of "measuring" the dimension of a fractal. I am just wondering what insights can we get from computing those. I do not see this much in lit. Examples:
- Is there any relationship between two fractals when their dimensions are equal, "close" to each other, or having a integer difference?
- What does the integer part of the dimension mean? (because "normally" dimensions are natural numbers)
- (Late addition) Is the dimension used in a formula for something?
I also appreciate references for the theorems, or applied math papers demonstrating those insights. Thanks for replies!
In this video, Adam Neely describes a musical fractal to be a piece of music where if slowed down sufficiently, would sound self-similar and this be a fractal. I'm wondering how finding the dimension of this fractal would work and what intuitively this would mean about the piece.
I'm trying to measure the level of fractality in a network: that is, how self-affine the structure of the graph really is. When analyzing time-series I used DFA (detrended fluctuation analysis), which splits the time series into chunks (from the smallest scale to the biggest scale) and then looks at the sum of the fluctuations in each.
Is there a similar approach in network analysis somebody knows of?
Shame! You got those groups that do 2/3 fractals and someone drops, put 't4 dailies' in the lfg, do the 3rd fractal and leave that new person to lfg for the other fractals that they need done, or redo what they've done. Either have to ask if its an actual fresh group everytime and hope they don't lie to you, or host my own group from now on. Just had to rant somewhere ty gg
If so, how much does it vary when taking a narrower perspective and looking at various particular types of local neural circuitry? Or when considering its larger-scale topological arrangements separately?
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Iβm learning about fractals, and struggling to understand the concept of fractal dimensions. I understand itβs not really the same as topological dimension, and is not usually an integer number, but I have encountered a situation where I have a fractal with a capacity dimension of 2, and I just have no idea what this means. Is this fractal somehow now a 2 dimensional object, like a square? I can calculate it, I just have no idea what to do with this βdimensionβ number. Itβs entirely meaningless to me so far.
