gn.general topology – Fractal plane continuum with the order $ omega $?

continuum means compact and connected.

The order $ ord (x) $ from one point $ x $ in a continuum $ X $ is defined as the smallest ordinal number $ alpha $ so that $ X $ has a neighborhood base of open sets $ x $ with not more than $ alpha $ shows her limits.

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The Sierpinski triangle has three order points $ 2 $Countable many order points $ 4 $ (the vertices of the other triangles) and all other points are in order $ 3 $,

The Sierpinski carpet has order $ mathfrak c = | mathbb R | $ at each of his points.

I'm looking for something between the Sierpinski triangle and the Sierpinski rug.

Question. Is there a fractal plane continuum that has order $ omega $ at each of his points?

Fractal can be loosely interpreted as "self-similar" or "simple recursive construction".

Is the function $ sinh (x) / x $ fractal at small values ​​of $ x, y $ or do I see rounding errors in the calculation?

I asked Wolfram Alpha to give me a solution for an integral function. Https:// between + x-a + and + x% 2Ba +)% 2F (2onexp (-mx)))
and it gave me an expression that is equivalent
$ z = frac { sinh (xy)} {xy} $,

It also created a contour plot of the value of $ z $ for the domain $[-0.002, The surface z is not smooth, but has a complex (fractal?) Appearance.

If you use the formula $ z = sinh (x) / x $I calculated values ​​in Excel. I found a similar non-soft behavior for small values ​​of $ x: $ ($ 0 <x <$ 0.00002).

My question is: does $ z = sinh (x) / x $ have a fractal behavior for small values ​​of $ x $or is the irregularity (shown in Wolfram Alpha and Excel) due to rounding errors in the calculators?

Fractal – set with changed lower count dimension for the box, which is strictly below the Hausdorff dimension

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Fractal – Multifractal Package – Description of Multifractals

I'm trying to use this multifractal package, which is included in the classic book by Baumann – Mathematica for Theoretical Physics II. The problem is that the code is not working, that is, I have no charts. In my idea, this package could help with fractal modeling. Unfortunately it does not work. The problem is that I have not found a mistake. I reported the code below. Can you help me?

begin package["MultiFractal`"];
clear[Dq, Tau, Alpha, MultiFractal];
MultiFractal :: usage = "MultiFractal[p_List,r_List] calculates that
multifractal spectrum D_q for a model based on the probabilities
p and the scaling factors r. This feature records five functions
Tau (q), D_q (q), Alpha (q), f (q) and f (alpha). ";
(* --- Calculate the multifractal dimensions --- *)
dq[p_List, r_List] : =
block[{l1, l2, listrg = {}},(*---length of the lists---*)
l1 = Length[p]; l2 = length[r];
If[l1 == l2,(*---variation of q and determination of D_q---*)
Do[gl1 = Sum[p[[j]]^ q r[[j]]^ ((q - 1) Dfractal), {j, 1, l1}]- 1;
Result = FindRoot[gl1 == 0, {Dfractal, -3, 3}];
Result = -Defractal /. Result;
(* --- collect the result in a list ---- *)
attach[listrg, {q, result}], {q, -10, 10, .101}], To press[" "];
To press["  Lengths of lists are different!"];
listrg = {}];

(* ---- calculated dew --- *)
dew[result_list] : =
block[{l1, listtau = {}},(*----lengths of the lists---*)
l1 = Length[result];
(* --- calculate Tau --- *)
 listtau, {result[[k, 1]],
Result[[k, 2]](1 - result[[k, 1]])}], {k, 1, l1}];

(* --- Legendre transformation --- *)
alpha[result_List] : =
block[{l1, dq, listalpha = {}, listf = {}, listleg = {}, mlist = {},
 pl1, pl2},(*---lengths of the lists---*)l1 = Length[result];
(* --- determine the differential dq --- *)
dq = (result[[2, 1]]- Result[[1, 1]]2;
(* --- calculate alpha by numerical differentiation --- *)
 listalpha, {result[[k, 
   1]](Result[[k + 1, 2]]- Result[[k - 1, 2]]) / dq}], {k, 2,
l1 - 1}];
l1 = length[listalpha];
(* --- compute f and put the result in a list --- *)
 listf, {result[[k, 
   1]], - (Result[[k, 1]]List Alpha[[k, 2]]- Result[[k, 2]])}];
List Alpha[[k, 2]]= -listalpha[[k, 2]], {k, 1, 12}];
(* --- List of Legendre Transformations --- *)
Do[Append[Listleg{listalpha[AppendTo[Listleg{listalpha[Anhängen[listleg{listalpha[AppendTo[listleg{listalpha[[k, 2]]listf[[k, 2]]}];
attach[mlistlistf[mlistlistf[mlistlistf[mlistlistf[[k, 2]]], {k, 1, l2}];
(* --- pl f and alpha against q --- *)
pl1 = ListLinePlot[listalpha, Joined -> {True, False}, 
 AxesLabel -> {"q", "[Alpha]"}, Prologue -> Thickness[0.001]];
pl2 = ListLinePlot[listf, Joined -> {True, False},
AxesLabel -> {"q", "f"}, Prologue -> Thickness[0.001]];
show[{pl1, pl2}, AxesLabel -> {"q", "[Alpha], f "}];
(* --- plot the Legendre transformation f against Alpha --- *)
ListLinePlot[listleg, AxesLabel -> {"[Alpha]"," f "}];
(* --- print the maximum of f = D_ 0 --- *) maxi = max[mlist];
To press[" "];
To press["   D_0 = ", maxi]];

(* --- Calculate the multifractal properties --- *)
Multifractal[p_List, r_List] : =
block[{listDq, listTau},(*---determine D_q---*)listDq = Dq[p, r];
ListLinePlot[listDq, Joined -> {True, False},
AxesLabel -> {"q", "Dq"}, Prologue -> Thickness[0.001]](* --- calculate Tau --- *) listTau = Tau[listDq];
ListLinePlot[listTau, Joined -> {True, False}, 
 AxesLabel -> {"q", "[Tau]"}, Prologue -> Thickness[0.001]](* --- Determine the hoarding exponent --- *) Alpha[listTau]];

The End[];

final packet[];

Analysis – How can I tell if a system correlates to a fractal?

So I'm studying the one-dimensional Abelian Sandpile for a research project. I mainly investigate if there is a relationship between this model and the Farey sequence.

I've flipped through some papers, mostly well above my level, but I came across a paper titled "Toppling Distributions in One-Dimensional Abelian Sandpiles" by P. Ruelle and S. Sen, stating that the one-dimensional case does not Show criticality. If someone could explain what that means, I would be very grateful.

Finally, I wanted to read the article by Levine, Pegden, and Smart, who relates the two-dimensional case of the Abelian Sandpile to the Apollonian Circles to gain further insights into the problem. However, it requires a background in PDEs. what I currently lack. Does anyone know if Evans's PDE book discusses this?

Which pattern type is generated by mod (x * y, w)? Is this an interference pattern, a simple fractal, or another pattern class?

When working with shaders I stumbled over this pattern several times. The visuals are very interesting – similar to moiré patterns, but more complicated.

I can not quite understand how it works, and I would like to know more about it, but I could not figure out what it is. Any information or tidbits about this pattern would be greatly appreciated.

Sample Example 1

Sample Example 2

Example 3

When applying a scale factor, this pattern has fractal-like properties, with each node appearing to be a smaller version of a similar pattern (but not infinite).

Sample example scaled

Here is a link to the shader where you can test the input parameters: