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Topic: **Systems/Chaos theory**

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# Burning Ship Fractal

The **Burning Ship fractal**, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:

- $z_{n+1}=(|\operatorname {Re} \left(z_{n}\right)|+i|\operatorname {Im} \left(z_{n}\right)|)^{2}+c,\quad z_{0}=0$

in the complex plane $\mathbb {C}$ which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations.

### Discussed on

- "Burning Ship Fractal" | 2016-09-26 | 252 Upvotes 65 Comments

# Digital sundial

A **digital sundial** is a clock that indicates the current time with numerals formed by the sunlight striking it. Like a classical sundial, the device contains no moving parts. It uses no electricity nor other manufactured sources of energy. The digital display changes as the sun advances in its daily course.

### Discussed on

- "Digital sundial" | 2014-07-16 | 232 Upvotes 47 Comments

# Hofstadter's butterfly

In condensed matter physics, **Hofstadter's butterfly** describes the spectral properties of non-interacting two dimensional electrons in a magnetic field. The fractal, self-similar, nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of computer graphics. The name reflects the visual resemblance of the figure on the right to a swarm of butterflies flying to infinity.

The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect, and D.J. Thouless has been awarded the Nobel prize in physics in 2016 for the discovery that the wings of the butterfly are characterized by Chern integers, the quantized Hall conductances discovered in 1980 by Klaus von Klitzing for which he has been awarded the Nobel prize in 1985. The colors in the diagram reflect the different Chern numbers.

### Discussed on

- "Hofstadter's butterfly" | 2014-06-20 | 41 Upvotes 10 Comments

# Lakes of Wada

In mathematics, the **lakes of Wada** (和田の湖, *Wada no mizuumi*) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of *one* of the lakes, the other two lakes' boundaries also contain that point.

More than two sets with the same boundary are said to have the **Wada property**; examples include **Wada basins** in dynamical systems. This property is rare in real-world systems.

The lakes of Wada were introduced by Kunizō Yoneyama (1917, page 60), who credited the discovery to Takeo Wada. His construction is similar to the construction by Brouwer (1910) of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum.

### Discussed on

- "Lakes of Wada" | 2018-10-07 | 66 Upvotes 15 Comments

# Cantor function, a.k.a. devil's staircase: increasing function with 0 derivative

In mathematics, the **Cantor function** is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow, by construction.

It is also referred to as the **Cantor ternary function**, the **Lebesgue function**, **Lebesgue's singular function**, the **Cantor–Vitali function**, the **Devil's staircase**, the **Cantor staircase function**, and the **Cantor–Lebesgue function**. Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).

### Discussed on

- "Cantor function, a.k.a. devil's staircase: increasing function with 0 derivative" | 2020-06-11 | 96 Upvotes 52 Comments

# Menger Sponge

In mathematics, the **Menger sponge** (also known as the **Menger cube**, **Menger universal curve**, **Sierpinski cube**, or **Sierpinski sponge**) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

### Discussed on

- "Menger Sponge" | 2020-10-05 | 27 Upvotes 12 Comments