# Recent Explorables

# Albert & Carl Friedrich

December 11, 2018

This explorable illustrates the geometric and dynamic properties of the physical process of diffusion and its intimate relation to a mathematical object known as a random walk. It also illustrates graphically the implications of the central limit theorem that explains why we so often (normally) observe Gaussian distributions in nature. In the context of random walks this means that in the long run and from a great distance the paths of different types of walks become statistically indistiguishable.

read more
# Critically Inflammatory

November 14, 2018

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 spontaneously seeded (lightning strikes) forest fires that spread across areas of dense vegetation.
For a broad range of parameters the model generates a dynamic equilibrium in which the size of single fires has no characteristic scale.

read more
# Dr. Fibryll & Mr. Glyde

November 12, 2018

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.

read more
# Facebooked Flu Shots

October 9, 2018

This explorable illustrates and compares three vaccination strategies in complex networks. In this type of model nodes are people and network links potential transmission paths. "Vaccination" then means that nodes are disconnected from the network because they can no longer acquire or transmit a disease. Vaccination thus effectively dilutes the network. Two strategies, A and B, are straightforward to understand. A third one, C, is a bit odd and counterintuitive at first glance:

read more
# The Blob

September 29, 2018

This explorable illustrates an important feature of complex networks: the emergence of the giant component. Networks often have multiple components. A component is a part of the network where we can find a path between any two nodes by traversing links. Two different components, however, are disconnected. When the number of links in the network is too small, the network will consist of many little components. Only when the number of links is sufficiently large, the network will have a single component.

read more
# Stranger Things

September 12, 2018

This explorable illustrates the structure and beauty of strange attractors of two-dimensional discrete maps. These maps generate sequences of pairs of number \((x_n,y_n)\) where the index \(n=0,1,2,...\) denotes the step of the iteration process that starts at the point \((x_0,y_0)\). The map is defined by two functions \(f(x,y)\) and \(g(x,y)\) that determine the point \((x_{n+1},y_{n+1})\) given \((x_{n},y_{n})\):
\[x_{n+1}=f(x_{n},y_{n})\]
\[y_{n+1}=g(x_{n},y_{n})\]
The orbit of such a map is the sequence of points \((x_n,y_n)\) in the plane.

read more
# I herd you!

August 6, 2018

This explorable illustrates the mechanism of herd immunity. When an infectious disease spreads in a population, an individual can be protected by a vaccine that delivers immunity. But there's a greater good. Immunization not only projects the individual directly. The immunized person will also never transmit the disease to others, effectively reducing the likelihood that the disease can proliferate in the population. Because of this, a disease can be eradicated even if not the entire population is immunized.

read more
# Scott's World*

August 2, 2018

This explorable illustrates a dynamic model for pattern formation in a growing community of microbes. Many microbial organisms exhibit collective behavior when a community of them expands say on a surface with nutrients. These patterns are often very beautiful and rich in structure.
If you are interested in exploring the beauty of microbial pattern formation have a look at Life at the Edge of Sight - A Photographic Exploration of the Microbial World a recent book by Scott Chimileski and Roberto Kolter that contains amazing photography of these patterns (and much more).

read more
# Yo, Kohonen!

July 30, 2018

This explorable illustrates the dynamics of a self-organizing map (SOM), specifically a neural network known as the Kohonen Map. The Kohonen map is a model for self-organization of biological neural networks and how the brain can learn to map signals from an input space, e.g. visual stimuli in the visual field, to a two dimensional layer of neurons, e.g. the visual cortex in the brain, in such a way that neighborhood properties in the stimulus space are conserved as best as possible: Neurons that are neighbors in the neural network should respond to stimuli that are also close in stimulus space.

read more
# Echo Chambers

July 27, 2018

This explorable illustrates a network model that captures how opinion dynamics might work in a population and how clusters of uniform opinion might naturally emerge from an initially random system. The model is a variant of a model introduced by P. Holme and M. Newman in a 2006 paper: Nonequilibrium phase transition in the coevolution of networks and opinions.
The model describes a network with a fixed number of nodes and links.

read more
# Kelp!!!

July 9, 2018

This explorable illustrates how fractal growth patterns can be generated by stochastic cellular automata. Cellular automata are spatially and temporally discrete dynamical systems that are conceptually very straightforward but can generate unexpected complex behavior, often fractal-like structures reminiscent of patterns we see in natural systems.
Here we are dealing with a one-dimensional cellular automaton in which a horizontal line in the display represents the state of the system and time evolves vertically from bottom to top.

read more
# Surfing a Gene Pool

July 3, 2018

This explorable is about pattern formation in a model system for the growth of a bacterial population in a petri dish in which the bacterial population is made up of a mix of two or three mutant strains that have identical fitness and reproduction properties. In the model system, an initially well mixed drop of bacteria with an equal amounts of every mutant strain is positioned in the center of the petri dish.

read more
# Knitworks

June 26, 2018

This explorable illustrates network growth based on preferential attachment, a variant of the Barabasi-Albert model that was introduced to capture strong heterogeneities observed in many natural and technological networks. It has become a popular model for scale-free networks in nature.
Preferential attachment means that nodes that enter the network during a growth process preferentially connect to nodes with specific properties. In the original system, they preferentially connect to existing nodes that are already well connected, increasing their connectivity even further.

read more
# If you ask your XY

June 20, 2018

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.
The model is defined by a two dimensional square lattice. Each lattice site hosts a magnetic dipole, illustrated by the little needles below. They can freely rotate around their central pivot and align with the magnetic field of their surrounding.

read more
# Ride my Kuramotocycle!

April 14, 2018

This explorable illustrates the Kuramoto model for phase coupled oscillators. This model is used to describe synchronization phenomena in natural systems, e.g. the flash synchronization of fire flies or wall-mounted clocks.
The model is defined as a system of \(N\) oscillators. Each oscillator has a phase variable \(\theta_n(t)\) (illustrated by the angular position on a circle below), and an angular frequency \(\omega_n\) that captures how fast the oscillator moves around the circle.

read more
# Maggots in the Wiggle Room

April 8, 2018

This explorable illustrates an evolutionary process in an "ecosystem" of interacting species (cartoon maggots, in this case). Individuals move around in their enviroment, replicate and eat each other. Optionally, mutations can generate new species. The system is similar to the Explorable A Patchwork Darwinge, only a bit more animalistc and dynamically slightly different. However, for this one here, you need a bit more patience in order to observe interesting effects.

read more
# A Patchwork Darwinge

February 27, 2018

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.

read more
# Into the Dark

February 12, 2018

This explorable illustrates how a school of fish can collectively find an optimal location, e.g. a dark, unexposed region in their environment simply by light-dependent speed control. The explorable is based on the model discussed in Flock'n Roll, which you may want to explore first.
This is how it works: The swarm here consists of 100 individuals. Each individual moves around at a constant speed and changes direction according to three rules:

read more
# Barista's Secret

February 6, 2018

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. Percolation theory is the mathematical tool designed for understanding these processes.
Percolation is best understood in terms of the following physical situation: Let's say you have a porous medium, e.g. ground coffee in your percolator or porous soil of some depth. You pour a liquid on the medium and would like to know whether the liquid can make it through the medium and if so how this depends on the porosity of the medium (i.

read more
# Lotka Martini

January 28, 2018

This explorable illustrates the dynamics of a predator-prey model on a hexagonal lattice. In the model a prey species reproduces spontaneously but is also food to the predator species. The predator requires the prey for reproduction. The system is an example of an activator-inhibitor system, in which two dynamical entities interact in such a way that the activator (in this case the prey) activates the inhibitor (the predator) that in turn down-regulates the activator in a feedback loop.

read more
# Double Trouble

January 12, 2018

This explorable illustrates the beautiful dynamical features of the double pendulum, a famous idealized nonlinear mechanical system that exhibits deterministic chaos. The double pendulum is essentially two simple pendula joined by a bearing. It's a classic complex system in which a simple setup generates rich and seemingly unpredictable behavior. The only force that is acting on it is gravity. There's no friction.
Initially, the pendulum is raised to some position and has therefore some potential energy with respect to the central pivot.

read more
# Critical HexSIRSize

January 5, 2018

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.
First press the Play button. While the system is doing its thing, keep on reading.

read more
# Orli's Flock'n Roll

December 30, 2017

This explorable can be sluggish when viewed in Safari. It works much better in Chrome/Firefox
Important Note This explorable is an updated version of the previous Flock'n Roll Explorable. This new version is much better. It has Orli's Magic Switch. The implementation of the magic switch was suggested by Orli (*), a 6yr old girl and future collective behavior scientist who --rightfully-- pointed out that the original Flock'n Roll Explorable lacked creativity in the chosen color scheme.

read more
# Hokus Fractus!

December 29, 2017

This explorable illustrates one of the simplest ways to generate fractals by an iteration process in which elements of a structure are replaced by a smaller version of the whole structure. Similar to the Weeds & Trees Explorable, these structures can be viewed as Lindenmayer systems. A great variety of examples of such fractals exist.
Here you can explore some of the most famous ones (e.g. the Koch Snowflake and the Sierpinski Triangle) and a few that aren't so well known (e.

read more
# Particularly Stuck

October 9, 2017

This explorable illustrates a process known as diffusion-limited aggregation (DLA). It's a kinetic process driven by randomly diffusing particles that gives rise to fractal structures, reminiscent of things we see in natural systems. The process has been investigated in a number of scientific studies, e.g. the seminal paper by Witten & Sander.
The Model: The system is initialized by say 300 diffusing particles that move about randomly in the plane.

read more
# Cycledelic

October 3, 2017

This explorable of a pattern forming system is derived from a model that was designed to understand co-existance of cyclicly interacting species in a spatially extended model ecosystem. Despite its simplicity, it can generate a rich set of complex spatio-temporal patterns depending on the choice of parameters and initial conditions.
The Model: The foundation of the model is a set of 3 species A, B and C that are distributed in space and locally interact in a cyclic way: When species A (red) encounters species B (green), A "eats" B and replicates.

read more
# Kick it like Chirikov

October 3, 2017

In this explorable you can investigate the dynamics of a famous two-dimensional, time discrete map, known as the standard or Chirikovâ€“Taylor map, one of the most famous simple systems that exhibits determinstic chaos.
The Model: The system is defined by the iterative equations
[\begin{align} p _{n+1} &= p _n+K\sin(x_n) \mod 2\pi\newline x _{n+1} &= x _n +p _{n+1} \mod 2\pi \end{align}]
for the two variables $x$ and $p$.

read more
# Epidemonic

October 3, 2017

This explorable illustrates the dynamics of the SIRS epidemic model, a generic model that captures disease dynamics in a populations or related contagion phenomena.
The Model: Susceptible individuals (S) can be infected by coming in contact with other infected (I) individuals. Once infected they can transmit the disease until they recover (R) and become immune. After some time immunity wanes and individuals become susceptible again.

read more
# Keith Haring's Mexican Hat

October 3, 2017

This explorable illustrates one of the most basic pattern forming mechanism: Local excitation and long range inhibition. This mechanism or similar ones are responsible for patterns observed in neural tissue, animal fur and spatial heterogeneity in social systems.
This is how it works:
The state of the two dimensional system is defined by a state variable $u(x,t)$ at each position $x$ at time $t$. The state variable can have values in the range $[-1,1]$.

read more
# Spin Wheels

September 10, 2017

This explorable illustrates some interesting and beautiful properties of oscillators that are spatially arranged on a lattice and interact with their neighbors.
This is how it works: Each pixel in the panel on the right represents an oscillator. The state of each oscillator $n$ is defined by its phase $\theta_n(t)$, the angle of a circle. Because of this, we represent the state of each oscillator by a color in the cyclic rainbow spectrum.

read more
# Weeds & Trees

August 7, 2017

This explorable illustrates how fractal patterns observed in natural systems, particularly structural properties of some plants, can approximately be modeled by simple iterative models. Sometimes these models are refered to as Lindenmayer systems.
The pattern that you see is generated by a basic structure to start with and applying a simple replacement rule.
Initially we have a root branch (stem) of base length $L_0$. To this root we attach 3 leaf branches of lengths $l_1$, $l_2$, $l_3$ and at angles $\theta_1$, $\theta_2$, and $\theta_3$, respectively.

read more
# Flock'n Roll

August 2, 2017

This explorable can be sluggish when viewed in Safari. It works much better in Chrome/Firefox
This explorable illustrates of an intuitive dynamic model for collective motion (swarming) in animal groups. The model can be used to describe collective behavior observed in schooling fish or flocking birds, for example. The details of the model are described in a 2002 paper by Iain Couzin and colleagues.
Here's a short summary of how it works:

read more