Claude
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4 answers
6 votes
2k views
Is there a koch circle?
9 votes

Parts of the Mandelbrot set look like circles with smaller circles attached recursively: They aren't quite exact circles (except for the one centered at $-1+0i$), but the property that the radius of ...

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2 answers
12 votes
469 views
Grid spacing, iterations used in the 1978 first published rendering of the Mandelbrot set?
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8 votes

I wrote a small Haskell program using the Diagrams library: import Diagrams.Prelude hiding (aspect) import Diagrams.Backend.SVG.CmdLine (B, defaultMain) import Data.Complex main :: IO () main = ...

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1 answers
7 votes
403 views
What is the shortest path to a "little Mandelbrot" from $i$?
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8 votes

$i$ is indeed the tip of a filament. However, mini-Mandelbrot copies are dense in the boundary of the Mandelbrot set, which means that for any radius $r > 0$ you can find a little Mandelbrot copy ...

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8 answers
80 votes
10k views
Koch snowflake paradox: finite area, but infinite perimeter
7 votes

You can fill the finite extruded Koch snowflake 3-D volume with a finite amount of 3-D paint. But the Koch snowflake, while topologically a 1-D line, has Hausdorff dimension $\log_3 4 \approx 1.2618\...

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3 answers
1 votes
337 views
Is there an iterative graphing program that lets you graph custom fractals like the Mandelbrot set?
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7 votes

fractint is one of the oldest programs, it has a formula interpreter. xaos also supports user-settable formulae. shadertoy probably has many escape-time fractal experiments on it that you can edit ...

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2 answers
4 votes
20k views
90 degree counter-clockwise rotation around a point
7 votes

Subtract the point, rotate around origin, add the point back: $$(-(y-b)+a, (x-a) + b)$$

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3 answers
6 votes
5k views
Number of normals to a parabola from a given point
7 votes

If $(p_x,p_y)$ are the coordinates of a point off the parabola and we'd like the line from $(p_x,p_y)$ to the point $(x,x^2)$ on the parabola to be normal to the parabola, then we need: $$ \begin{...

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1 answers
7 votes
1k views
How is this fractal produced?
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7 votes

The start value $x_0$ is determined by the coordinates of each pixel. Pixels are coloured acccording to how quickly the orbit for that pixel diverges (escape time colouring). Black pixels remained ...

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6 answers
22 votes
9k views
A way to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal
6 votes

As this graph shows, the behaviour of normalized iteration counts of points near the Mandelbrot set varies widely, indicating that attempts at a formula based on scale factors is doomed to fail. In ...

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1 answers
5 votes
2k views
Why does the mandelbrot set seem to end at a copy of itself?
6 votes

Pretty sure it's an aesthetic choice, as Mark says in his comment. It arguably makes a more satisfying end to end inside a mini copy (or just before). Alternatives are to end outside the set (which ...

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2 answers
12 votes
820 views
Is The *Mona Lisa* in the complement of the Mandelbrot set.
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6 votes

I think yes Consider the sequence of "westernmost" islands increasing in period: Here are some examples, you can see them increasing in hairyness / spinyness. Period 20: Period 30: Period 40: ...

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2 answers
6 votes
3k views
Determine coordinates for Mandelbrot set zoom.
6 votes

One strategy to find some interesting (at least to my own subjective taste) places (which might not make interesting videos) semi-automatically is to spot patterns in the binary expansions of external ...

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2 answers
8 votes
3k views
How to compute a negative "Multibrot" set?
6 votes

https://commons.wikimedia.org/wiki/File:Multibrot_Lupanov_power-2_Q1.png suggests using the Lyapunov exponent as a stability criterion, before performing periodicity checking for colouring. The ...

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1 answers
7 votes
200 views
Art hobbyist seeking maximally dispersed Latin Square of order 7
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5 votes

This is not a method of construction, but a brute force exhaustive search. Not mathematically elegant, but may eventually give you a maximally dispersed Latin square. Wikipedia says there are $...

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1 answers
6 votes
720 views
How to draw a Mandelbrot Set with the connecting filaments visible?
5 votes

You want to colour the complement of the Mandelbrot set using the exterior distance estimate. For each pixel you calculate the running derivative (w.r.t. $c$) as well as the $z$ iterate, then at the ...

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3 answers
4 votes
5k views
Value to use as center of Mandelbrot Set zoom?
4 votes

You want a point on the boundary of the Mandelbrot set. Even so, "simple" ways to choose boundary points are not so interesting, because they eventually loop. Points at rational internal angles (...

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2 answers
7 votes
497 views
Special reptiles - repeating shapes and fractals
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4 votes

The Viper substitution tiling divides a particular triangle into 9 similar sub-triangles without reflection, in a non-trivial way. The same triangle can also be divided into 4 similar sub-triangles ...

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2 answers
3 votes
429 views
Point on Mandelbrot set with copy of the original at a different scale
4 votes

Windows of Periodicity Scaling gives a formula $\beta \Lambda_p^2$ for the size (and orientation, taking it as a complex number) of a minibrot island with nucleus $c$ of period $p$ relative to the top ...

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5 answers
4 votes
1k views
How to plot the Julia Set of $z-z^2$
4 votes

Your Julia set is similar to the one for $z^2+\frac{1}{4}$, as mentioned by other answerers. This Julia set is parabolic, and parabolic Julia sets are hard to compute accurately efficiently. One ...

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1 answers
8 votes
477 views
Continuous path inside the Mandelbrot set connecting i to zero?
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4 votes

Here's a zoom into the Mandelbrot set centered on $i$, with a factor of 10 magnification per frame. As you can extrapolate from the first frame, your path will have to go through an infinite number ...

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1 answers
5 votes
130 views
Why is there a Mandelbrot hidden in this sequence?
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3 votes

Consider iteration of $z \mapsto c^{\sin(z)}$ near $c$: $$z \mapsto c^{\sin(c + z)} - c$$ The Taylor series near $z = 0$ is (via Wolfram Alpha): $$ z \mapsto \left(c^{\sin(c)} - c\right) \\ + \left(\...

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2 answers
5 votes
300 views
How can I reconstruct a Julia set by a given image?
3 votes

The first image may be a zoom into a quadratic Julia set for $f_c(z) = z^2 + c$ near $c = 0.270723273 + 0.575139611 i$, centered on $0$ with zoom factor $5.3$ (zoom factor $1$ would have $\pm i$ at ...

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2 answers
0 votes
184 views
Finding Periodic/Fixed Points in the Julia Sets close to the Period-3 Cardioid
3 votes

For the promenade around the main cardioid, the point in the Julia set you want is a fixed point $z = f_c(z)$. By tuning (renormalization), the point you want in the "little Julia set" at ...

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1 answers
0 votes
118 views
Matching an image with a fractal, advice needed
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3 votes

Shapes can be approximated by Julia sets of rational functions with a large number of roots [1]: we are [...] interested in factored rationals of the form $$ R(q) = e^C \frac{ \prod (q - t_i)^{T_i} }{...

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1 answers
2 votes
130 views
Naming bulbs on the Mandelbrot set
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3 votes

The first thing to note is that each of the hyperbolic components (either cardioid-like or disk-like) is associated to a period, which is a positive integer. The biggest cardioid has period $1$, the ...

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3 answers
3 votes
362 views
Is there a technique to exactly calculate the Hausdorff dimension of the border of this fractal?
3 votes

You may be able to calculate a graph-directed IFS for the boundary of the tile, though the rotated tiles may make this tricky - I'm not sure whether this is classed as a self-affine tile, or whether ...

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1 answers
1 votes
691 views
Mandelbrot set perturbation theory: When do I use it?
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3 votes

First answer: "the $\delta^3$ term remains significantly smaller than the $\delta^2$ term" means that $\left|C_n \delta^3\right| << \left|B_n \delta^2\right|$, usually a few orders of magnitude ...

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3 answers
2 votes
499 views
Are the intriguing and lovely Mandelbrot Set hoops and curls the result of floating point computation inaccuracy?
3 votes

I'm not absolutely sure, but I think the shadowing lemma applies to the Mandelbrot set, meaning that near any rounded-off floating point trajectory there is a true trajectory that started nearby, i.e. ...

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3 answers
6 votes
1k views
Color $27$ unit cube so that by rearranging, they could form a blue $3\times3$ cube, a green one, and a red one?
3 votes

For the $3 \times 3 \times 3$ case it is possible: Haskell code: {-# LANGUAGE FlexibleContexts #-} import Diagrams.Prelude import Diagrams.Backend.Cairo.CmdLine (defaultMain) v x = [x,x,x] e x = [...

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3 answers
2 votes
65 views
Possibility to construct a perpendicular to a line interval
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3 votes

Let $A$, $B$, $P$ be the endpoints of your line segment and your given point respectively. Noting from don-joe's answer that you need the angles $\angle PAB$ and $\angle PBA$ to both be less than $90^...

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