## Welcome to Complexity Explorables

This site is designed for people interested in complex systems and complex dynamical processes. *Complexity Explorables* hosts different collections of interactive illustrations of models for complex systems in physics, mathematics, biology, chemistry, social sciences, neuroscience, epidemiology, network science and ecology.

Topics include pattern formation, synchronization, critical phenomena, chaotic dynamics, evolutionary dynamics, fractals, collective behavior, reaction-diffusion systems and more.

The main collection is **Explorables**. Each explorable contains one interactive component and describes a single system. The models are chosen in such a way that the key elements of a system’s behavior can be explored and explained without too much math (there are a few exceptions) and with as few words as possible.

The site also features **Flongs** (short for *“foot longs”*). These are mini tutorials on specific and paradigmatic complex systems that go a bit deeper, feature more interactive elements but usually require a bit more math.

If you want to use Explorables in teaching or presentations, we have a **Slide section**. A slide only contains an Explorables’ interactive element, without the text, and can easily be used as part of a presentation or lecture.

## Most Recent:

# “Prime Time”

## The distribution of primes along number spirals

This explorable is about **prime numbers**. It illustrates interesting patterns that emerge if you arrange the positive natural numbers \(1,2,3,....\) and so forth in a regular pattern in the plane and look at the * distribution of the primes* in the arrangement. Although prime numbers, as one might expect, don't follow some regular repetitive pattern, they are also not distributed completely at random. Instead, in all arrangements streaks and fragments of neighboring primes emerge. These structures are related to

**prime generating polynomials**like the famous polynomial discovered by Leonard Euler.

# “I sing well-tempered”

## The Ising Model

This explorable illustrates one of the most famous models in statistical mechanics: * The Ising Model*. The model is structurally very simple and captures the properties and dynamics of magnetic materials, such as

*ferromagnets*. It is so general that it is also used to describe opinion dynamics in populations and collective behavior in animal groups.

# “Anomalous Itinerary”

## Lévy flights

This explorable illustrates the properties of a class of random walks known as Lévy flights.
To get the most out of this explorable, you may want to check out the explorable **Albert & Carl Friedrich** on ordinary random walks and diffusion first.