## Friday, December 25, 2009

## Friday, December 4, 2009

# Hilbert curve and fourier transform

Hilbert curve is a fractal curve of Hausdorf dimension 2 that fills the interior of a square. It provides continuous mapping from real interval [0;1] to unit square. This mapping is not invertible: some different points of [0;1] are mapped to the same points of the unit square.

Let us assign a color to each point of the unit interval, in simple repeating manner, and map the colored points to the square. This procedure will produce a monotonous flat pattern, which is not actually periodic.

Not an exciting sight. However 2D Fourier transform of this pattern reveals amazingly complex fractal structure:

The picture above shows filtered (invert, despeckle and colorization filters were used) and scaled version of the transformed pattern. Apparently, it contains multiple images of some "rounded rectangle" shape. This shape always appears in the Fourier transform of the every repetitive Hilbert curve pattern. What is the equation of this shape? I don't know.

## Monday, October 19, 2009

# Irregular Mandelbrot fractals

A fragment of the irregular Mandelbrot set. |

The famous Mandelbrot set is defined by the recurrent relation:

$$z_{n+1}=z_n^2+c.$$

Despite at the first glance this formula may seem simple, the set of points c, defining stable orbits of z, has incredibly complex structure:

However, when the first impression of infinite variation fades, you'll probably notice that the whole image is based on the same pattern, repeated with several variations (well, that's why it is called fractal, after all). And it is pretty boring.

Equations, other than canonical \(z_{n+1}=z_n^2+c\), can produce some different patterns,

Cubic Mandelbrot variation. |

Non-homomorphic mapping functions produce significantly different patterns, but they are often too "dirty":

A Mandelbrot-like fractal, obtained from the non-homomorphic function. |

### So, where the novelty hides?

Meet the idea: the**irregular Mandelbrot set**.

Instead of repetitive iteration of the same function, iterate two (or more), in an never-repeating pattern:

$$z_{n+1}=f_{Idx(n)}(z_n)+c.$$

Where \(Idx(n)\) is a non-periodic function, returning integers in the range [1..k], and \(f_1(z),...,f_k(z)\) are the different mapping functions. Non-periodicity guarantees that there are no stable cycles in Z.

Results seem interesting, at least for me. Because the mapping functions are homomorphic, there is no effect of "smearing", but non-periodicity destroys all of the repetitive, stable patterns. Here are few sample pictures:

Black-and-white palette is not a requirement, but to my artistic sense, these fractals look better without colors. See more in the album.

## Saturday, January 24, 2009

# 36π/127 conjecture failed

My recent conjecture that the non-differentiable function$$f(x) = \sum_{i=0}^\infty \frac{sin(2^i x)}{2^i}$$

reaches the maximum at the point \(x_0=36\pi/127\) appears to be false (thanks to people from sci.math group for pointing it out).

This is demonstrated by the following graph of the differential quotient \(\frac{f(x_0)-f(x_0-h)}h\), where \(x_0=\frac{36\pi}{127}\).

*h*) graph of the quotient (

*f*(

*x*0)-

*f*(

*x*0-

*h*))/

*h*. Horizontal axis is

*h*.

*h*between \(10^{-29}\) and \(10^{-28}\) the quotient becomes negative, which means that there exists such h, that \(f(x_0+h) > f(x_0)\). Apparently, the extremal value is h≈1.998e-29, giving \(f(x_0+h) - f(x_0) \approx 3.06\dot10^{-32}\).

f(x_{0})= | 1.329833276287310850440418286206506387707650784... |

f(x_{0}-1.998e-29)= | 1.329833276287310850440418286206535446963412182... |

There is still a possibility that the maximum is reached at some rational multiple of π, but it never be as simple, as 36π/127. I feel a strange mix of frustration and enlightenment regarding this failure, the result seems absolutely counter-intuitive for me. Why \(x_0/\pi\) is so incredibly close to the rational value? Is it just a fantastic coincidence, or a consequence of a some deeper facts?

This result strongly resembles the "Almost Integer" values in mathematics. it can be re-formulated in the following way:

$$ N = \frac{127}\pi \mathrm{argmin} \sum_{i=0}^\infty \frac{sin(2^i x)}{2^i} = 36+\epsilon $$ where \( \left| \epsilon \right| < 10^{-27} \). I.e.

*N*is an almost integer value.

## Sunday, January 18, 2009

# Billiard balls and fractals

Consider ideal billiard, consisting of round balls, moving and colliding without friction in the rectangular box.

Then let us measure, how much time needs first ball (green one at the picture) to get to the right wall. Below are the images, showing how time until wall collision depends from initial ball velocity. Horizontal and vertical coordinates at the picture plane correspond horizontal and vertical components of the initial ball velocity, color shows, how much time needed ball to reach the right wall. All pictures are 800x600, about 600K

## Tuesday, January 6, 2009

# The 36π/127 conjecture

There is a problem that bothers me for a long time. Consider function f(x), defined in the following way:

$$f(x) = \sum_{n=0}^\infty \frac{sin(2^n x)}{2^n}.$$

*f*(

*x*) is continuous (due to the uniform convergence of partial sums).

_{max}= 0.8905302... (dash line at the graph above) which effectively equals to \(x_0=\frac{36}{127}\pi\). I am looking for the proof of this fact, but without success.

Following plot may give a key to the solution. It is a logarithmic graph, showing how the function behaves in the close vicinity of the point \(x_0\).

Graph of the f(x) in the vicinity of the point x0. |

Periodical structure appears only if x is a rational fraction of \(\pi\). For other values, graph shows chaotic structure: