Complex dimensional shapes

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Offline TGlad

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« on: September 27, 2017, 04:56:58 AM »
I think you can extend my idea for negative dimensional shapes to imaginary and complex dimensions too.

Fractal dimension quantifies how the size of a shape grows as you zoom in. For instance for a line we get a doubling of the size (the number of pixels) each factor 2 zoom:

The dimension is the exponent of this exponential graph (\( e^{Dz} \) where \( z \) is the zoom level), so if the dimension is imaginary we should get a sinusoidal graph. 

So here is an example for a type of dust, it is 2.86i dimensional:


and here's one that is 1.95i dimensional:


A shape that grows in size as well as oscillating has a real and an imaginary dimension, so it has a complex dimension. Since cell counts are only positive and real, when we say a shape is \( c \) dimensional for complex number \( c \), it can be taken to mean that we measure the size of the shape as size = \( a m^c + a^* m^{c^*} + b m^{\Re(c)} \) where \( m \) is metres or units of distance, \( \Re(c) \) means the real part of \( c \), \( a \) and \( b \) are constants such that \( b>=\Re(a) \) and * refers to the complex conjugate. The constant \( a \) can be complex, in which case its angle (or arg) defines the phase of the shape's sinusoidal growth. As shown above, the actual observed size is not a sine wave, and the oscillation shape varies if you change the size or shape of the window, but its period remains fixed, and that's what the dimension \( D \) extracts. 

https://tglad.blogspot.com.au/2017/09/complex-dimensional-geometry.html

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/complex-dimensional-shapes/374/

Offline v

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« Reply #1 on: September 28, 2017, 10:30:34 PM »
Is it possible to take, say, a 2 dimensional structure and modify it just enough so that it's now a 2.1 dimensional structure and so on, 2.2 2.3 etc, until you reach 3 and draw it as an animation being continuously deformed from 2d to 3d.  All those images should be the same or similar structure, like a square turns into a cube or a circle to a sphere.  Would make for an interesting visual

Offline claude

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« Reply #2 on: September 29, 2017, 12:16:34 AM »
@v yes you can construct something like that, here's an example embedded in 2D, turning a 1D line into a 1.75D spiral (going higher resulted in very slow rendering times, theoretically you could go all the way to 2D resulting in a disc)
https://mathr.co.uk/blog/2011-12-31_the_sky_cracked_open.html click the pic for video
A trivial version embedded in 3D space going from 2D to 2.75D (or eventually 3D) would be to extend the curve into the 3rd dimension so it is a topological plane that gets rolled up, but that's not very interesting (just a cartesian product of a fractal curve with a regular line segment).

Offline TGlad

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« Reply #3 on: September 29, 2017, 01:24:50 AM »
Here are some other examples, a 1D line going to a 2D plane (attached below).

and here's a 2D surface morphing into a 2.4D surface:

both described here.

One could make an animation, or even better, an interactive demo: http://glslsandbox.com/e#4851.17  (disclaimer: this is a bit bad quality).

Though, none of this is connected to the OP  :P

Offline Fraktalist

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« Reply #4 on: September 29, 2017, 08:00:45 PM »
Great visualization! The line to plane explains so much on just a single glance..
well done!

Offline TGlad

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« Reply #5 on: October 01, 2017, 01:47:13 PM »
Thanks Frank, I made those a while back.

Anyway, the thing I find interesting about these complex dimensional shapes is that the oscillating density necessarily means any single snapshot has sparse areas. The smaller the imaginary part, the larger the sparse areas.
Mandelbrot popularised fractal dimension but he also wanted some way to distinguish more evenly spaced fractals and ones with sparse areas. He called this the measure of lacunarity. People came up with different ways to measure this, but they all seem quite ad-hoc compared to fractal dimension. Using the imaginary part of the fractal dimension as the lacunarity might tie the two basic metrics of a shape together nicely.

Offline claude

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« Reply #6 on: March 05, 2018, 08:01:23 PM »
I see complex dimensions mentioned here:

https://math.stackexchange.com/a/2677204/209286 Do all fractals have this property?
Quote
[3] Lapidus, Michel L.; van Frankenhuijsen, Machiel, Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics. New York, NY: Springer (ISBN 0-387-33285-5/hbk; 0-387-35208-2/ebook). xxiv, 464 p. (2006).  http://dx.doi.org/10.1007/978-0-387-35208-4 https://zbmath.org/?q=an:1119.28005

But the PDF is paywalled.

Offline lkmitch

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« Reply #7 on: March 06, 2018, 04:53:37 PM »
I see complex dimensions mentioned here:

https://math.stackexchange.com/a/2677204/209286 Do all fractals have this property?
But the PDF is paywalled.

You can google "Lapidus complex dimensions" and find some free articles that may give a sense of what is being considered.

Offline TGlad

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« Reply #8 on: March 07, 2018, 12:53:25 PM »
Thanks for the link Claude (I can access it). I couldn't find anything on this before and now I see this 300 page book on the subject, with about 10 prior papers!

It could be very useful material but I've only done the quickest skim and it is pretty inpenetrable. We need a good elucidator like John Baez to translate it into English and clarify what their findings actually mean.

Offline Fraktalist

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« Reply #9 on: March 07, 2018, 12:58:26 PM »
sci-hub.la
not sure if legal in all countries, but found the pdf there.

Offline TGlad

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« Reply #10 on: March 08, 2018, 01:05:58 PM »
The complex dimension is defined on 'fractal strings', 'fractal drums/sprays' and 'relative fractal drums'.

- Fractal strings are a set of separated intervals on the number line, e.g. ___ ___ __ _ _    the spacing, location and order of the pieces is unimportant.
The dimension of this as a set is 1, but they define the strings by their boundary (two end points), so all possible fractal strings have a dimension between 0 and 1.

- Fractal drums/sprays seem to be just the same thing but in higher dimensions, e.g. in 2D if could be a set of separate disks or filled squares. Again, they define their dimension just on the boundary, so in 2D I imagine the fractal dimension is between 1 and 2D.

- Relative fractal drums are something like a fractal drum where the boundary (usually a perimeter line in the 2D case) is itself a fractal string.

Anyway, the complex dimension is based on extending the Dirilicht series used to define the string length reduction rate, to the continuous Riemann zeta function. The poles of this function are the dimensions of the object, which can be complex.

It looks like (as in my OP) complex dimensions indicate an oscillation in the growth rate:
Quote
In this book, we develop a theory of complex dimensions of fractal strings (i.e., one-dimensional drums with fractal boundary). These complex dimensions are defined as the poles of the corresponding zeta function. They describe the oscillations in the geometry or the frequency spectrum of a fractal string by means of an explicit formula. Such oscillations are not observed in smooth geometries.

What I don't understand is how they apply this 'complex dimension' formula to normal fractal sets. But it appears that they can and have done because the author talks about applying it to basic fractals like the Koch snowflake and Menger carpet.

What surprises me is that complex dimensions implies dimensions can be negative. But they haven't shown any examples of such a shape, and the only mention of supporting negative dimensions applies to the quite odd 'relative fractal drum', in which the negativity represents smoothness in some sense.

Anyway, I'm still struggling with what is motivating these hundreds of pages of work. The author's descriptions are crazy:

Quote
the theory of complex dimensions aims at unifying many aspects of fractal and arithmetic geometries. e.g. the notion of fractal membrane (i.e., quantized fractal string), a still conjectural (noncommutative) flow on the moduli space of fractal membranes and correspondingly, a flow of zeta functions (or partition functions) and a flow of zeros are used in an essential manner in order to provide a possible new approach to the Riemann hypothesis. The flow of fractal membranes can be viewed as transforming generalized quasicrystals into (self-dual) pure crystals. Conjecturally, this would also provide a proof of the generalized Riemann hypothesis (GRH), whether it be viewed as a (noncommutative) Ricci-type flow or, physically, as a renormalization flow (not unlike the one which presumably describes the time evolution of our universe).
FYI the Riemann hypothesis is a famous unsolved problem, a Millenium prize problem and one of very few still unsolved problems from Hilbert's list in the 1920s.

This quote might get to his motivation better:
Quote
our definition of fractality cannot just be applied to standard geometric objects but is also applicable, in principle, to ‘virtual’ geometries, spectral geometries, dynamical systems, algebraic and noncommutative geometries as well as arithmetic geometries. In fact, along with complex dimensions, it should be used as a unifying tool between these apparently vastly different domains of mathematics. This long-term goal has been one of the central motivations of the author (and his collaborators)

His definition of 'fractality' is basically that you can call something a fractal if it has at least one non-real complex dimension (with positive real part). He claims it encompasses the set of fractals better than any other definition.

(If this is true, it implies that ALL standard fractals have an imaginary component to their dimension)
« Last Edit: March 10, 2018, 12:12:42 PM by TGlad »

Offline TGlad

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« Reply #11 on: March 10, 2018, 12:12:12 PM »
It looks like Lapidus has more recently defined complex dimensions on normal fractals (rather than the quirky strings/drums he started with).
The key paper seems to be here:

"Lapidus zeta functions of arbitrary fractals and compact sets in Euclidean spaces"
Which is based on a new function called the Lapidus (or distance) Zeta Function

The resulting work on complex dimension is perhaps best summed up in this review paper:
"A Survey of Complex Dimensions, Measurability, and the Lattice/Nonlattice Dichotomy"... though it is still pretty heavy.


Anyway, the conclusion is that shapes can and do have a complex dimension, or in fact many. This represents an oscillation in the growth rate as you zoom in. See my attached image, taken from "Fractal Geometry, Complex Dimensions and Zeta Functions" and compare it to the graph under my animation in my OP. Both graphs can be built from sine waves, each being a complex exponent. 

Lapidus's definitions are different from mine, and his don't seem to allow for negative dimensions, but I think the complex dimension concept comes out about the same, I think he'd get the values as I put in my OP.



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