# “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.

# “Critically Inflammatory”

## A forrest fire model

This explorable illustrates, as a representative of a broad range of dynamic phenomena,
a simple model for the spatial spread of forest fires and the dynamic patterns they generate. In the model
two antagonistic processes interplay, the * reproductive growth of vegetation* that continously covers the landscape with trees susceptible to fires and

*that spread across areas of dense vegetation.*

**spontaneously seeded (lightning strikes) forest fires**# “Dr. Fibryll & Mr. Glyde”

## Pulse-coupled oscillators

This explorable illustrates pattern formation in excitable media. The example explored here
in a system of **pulse-coupled oscillators** that are arranged on a two-dimensional lattice and interact with their neighbors by delivering excitatory pulses to them and receiving them in return. This model is sometimes used to study synchronization and can capture the dynamics of activation in layers of neurons as well as the spatial patterns of signaling molecules that play a role in microbial aggregation processes.

# “If you ask your XY”

## The XY model of statistical mechanics

This explorable illustrates pattern formation and dynamics in the **\(XY\)-model**, an important model in statistical mechanics for studying phase-transitions and other phenomena. It's a generalization of the famous **Ising-Model**. The \(XY\)-model is actually quite simple.

# “A Patchwork Darwinge”

## Evolution: Variation and Selection

This explorable illustrates how the combination of **variation** and **selection** in a model biological system can increase the average **fitness** of a population of mutants of a species over time. Fitness of each mutant quantifies how well it can reproduce compared to other mutants. Variation introduces new mutants. Sometimes a mutant's fitness is lower than its parent's, sometimes higher. When lower, the mutant typically goes extinct, if higher the mutant can outperform others and proliferate in the population. This way mutants with higher fitness are naturally selected.

# “Barista's Secret”

## Percolation on a lattice

This explorable illustrates a process known as percolation. ** Percolation** is a topic very important for understanding processes in physics, biology, geology, hydrology, horstology, epidemiology, and other fields.

**is the mathematical tool designed for understanding these processes.**

*Percolation theory*# “Critical HexSIRSize”

## The stochastic, spatial SIRS model

This explorable illustrates the behavior of a contagion process near its *critical point*. Contagion processes, for example transmissable infectious diseases, typically exhibit a critical point, a threshold below which the disease will die out, and above which the disease is sustained in a population. Interesting dynamical things happen when the system is near its critical point.