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# Pattern formation

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organisation and the common principles behind similar patterns.

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. The role of genes in pattern formation is best understood in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (fruit fly).

## Biology

Animal markings, segmentation of animals, phyllotaxis, neuronal activation patterns like tonotopy, Predator-prey equations' trajectories.

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue assume complex forms and functions by coordinated cell fate control [1][2]. Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues.

### Anterior-posterior axis patterning in Drosophila

One of the best understood examples of pattern formation is the patterning along the future head to tail (antero-posterior) axis of the fruit fly Drosophila melanogaster. The development of Drosophila is particularly well studied, and it is representative of a major class of animals, the insects or insecta. Other multicellular organisms sometimes use similar mechanisms for axis formation, although the relative importance of signal transfer between the earliest cells of many developing organisms is greater than in the example described here.

## Chemistry

see reaction-diffusion systems and Turing patterns

• Belousov-Zhabotinsky reaction
• Liesegang rings

## Physics

Bénard cells, Laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads.

## Computer graphics

Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects [3] [4].

A popular photoshop plugin, KPT 6 included a filter called 'KPT reaction' that produced reaction-diffusion style patterns based on a seed image. A similar effect can also be achieved, with a little patience, by repeatedly sharpening and then blurring an image in many graphics applications. If other filters are used, such as emboss or edge detection different types of effects can be achieved.

## Analysis

The analysis of pattern-forming systems often consists of finding a PDE model of the system (the Swift-Hohenberg equation is one such model) of the form

$\frac{\partial u}{\partial t} = F(u,t)$

where F is generically a nonlinear differential operator, and postulating solutions of the form

$u(\mathbf{x},t) = \sum_j z_j(t) e^{i\mathbf{k}_j\cdot\mathbf{x}} + z_j(t)^* e^{-i\mathbf{k}_j\cdot\mathbf{x}}$

where the zj are complex amplitudes associated to different modes in the solution and the $\mathbf{k}_j$ are the wave-vectors associated to a lattice, e.g. a square or hexagonal lattice in two dimensions. There is in general no rigorous justification for this restriction to a lattice.

Symmetry considerations can now be taken into account, and evolution equations obtained for the complex amplitudes governing the solution. This reduction puts the problem into the form of a system of first-order ODEs, which can be analysed using standard methods (see dynamical systems). The same formalism can also be used to analyse bifurcations in pattern-forming systems, for example to analyse the formation of convection rolls in a Rayleigh-Bénard experiment as the temperature is increased.

Such analysis predicts many of the quantitative features of such experiments - for example, the ODE reduction predicts hysteresis in convection experiments as patterns of rolls and hexagons compete for stability. The same hysteresis has been observed experimentally.