Turbulence Flow visualization of a turbulent jet, made by laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows. Laminar and turbulent water flow over the hull of a submarine In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic property changes. Flow in which the kinetic energy dies out due to the action of fluid molecular viscosity is called laminar flow. Features Turbulence is characterized by the following features: Irregularity: Turbulent flows are always highly irregular. Turbulent diffusion is usually described by a turbulent diffusion coefficient. Via this energy cascade, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow. Integral length scales: Largest scales in the energy spectrum. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. and pressure where ). ).
Random walk Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis). A random walk is a mathematical formalization of a path that consists of a succession of random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks, although they may not be truly random in reality. The term random walk was first introduced by Karl Pearson in 1905. Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology. Random walks explain the observed behaviors of processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. Various different types of random walks are of interest. . . Lattice random walk .
Sacred geometry As worldview and cosmology The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said God geometrizes continually" (Convivialium disputationum, liber 8,2). In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying "God arithmetizes". At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists. Closeup of inner section of the Kepler's Platonic solid model of planetary spacing in the Solar system from Mysterium Cosmographicum (1596) which ultimately proved to be inaccurate Natural forms Art and architecture Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. In Hinduism Unanchored geometry Music See also Notes Further reading External links Sacred geometry at DMOZ
Golden ratio Line segments in the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, The golden ratio is also called the golden section (Latin: sectio aurea) or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number. Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Calculation Therefore, Multiplying by φ gives and History
Patterns in nature Natural patterns form as wind blows sand in the dunes of the Namib Desert. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. Mathematics, physics and chemistry can explain patterns in nature at different levels. History In 1202, Leonardo Fibonacci (c 1170 – c 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci. Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population. In 1917, D'Arcy Wentworth Thompson (1860–1948) published his book On Growth and Form. Causes Types of pattern Symmetry Trees, fractals
Fractal flame Fractal flames differ from ordinary iterated function systems in three ways: Nonlinear functions are iterated instead of affine transforms.Log-density display instead of linear or binary (a form of tone mapping)Color by structure (i.e. by the recursive path taken) instead of monochrome or by density. The tone mapping and coloring are designed to display as much of the detail of the fractal as possible, which generally results in a more aesthetically pleasing image. Algorithm The algorithm consists of two steps: creating a histogram and then rendering the histogram. Creating the histogram First one iterates a set of functions, starting from a randomly chosen point P = (P.x,P.y,P.c), where the third coordinate indicated the current color of the point. Set of flame functions: In each iteration, choose one of the functions above where the probability that Fj is chosen is pj. Each individual function has the following form: The functions Vk are a set of predefined functions.
Snowflake Snowflake viewed in an optical microscope A snowflake is either a single ice crystal or an aggregation of ice crystals which falls through the Earth's atmosphere. They begin as snow crystals which develop when microscopic supercooled cloud droplets freeze. Snowflakes come in a variety of sizes and shapes. Complex shapes emerge as the flake moves through differing temperature and humidity regimes, such that individual snowflakes are nearly unique in structure. Formation Snow crystals form when tiny supercooled cloud droplets (about 10 μm in diameter) freeze. The exact details of the sticking mechanism remain controversial. Symmetry A non-aggregated snowflake often exhibits six-fold radial symmetry. Uniqueness Almost all snowflakes are unique Snowflakes form in a wide variety of intricate shapes, leading to the popular expression that "no two are alike". Use as a symbol Snow flake symbol Gallery A selection of photographs taken by Wilson Bentley (1865–1931):
Fractal art Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The Julia set and Mandelbrot sets can be considered as icons of fractal art. Fractal art (especially in the western world) is not drawn or painted by hand. It was assumed that Fractal art could not have developed without computers because of the calculative capabilities they provide. Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations. Types A 3D Mandelbulb fractal generated using Visions of Chaos There are many different kinds of fractal images and can be subdivided into several groups. Fractal expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example.
Ultra Fractal: Advanced Fractal Software for Windows and Mac OS X Fractint Development Team Homepage Download Fractint 18.21 Free - Create your fractals with the help of this tool. The beauty and complexity of fractals has ensnared many users and not necessarily those with inclinations towards the exact sciences. Nonetheless, looking deep inside one or customizing its parameters would not be possible without the help of a utility like Fractint. Dull interface for splendid shapes With the package unwrapped, you can start the application immediately as it doesn't have to be installed and comes with all the necessary dependencies. Loading files is possible only if they are in GIF format, but opening a fractal is done by selecting the desired formula from the dedicated menu. Customizable fractals and views Each time you choose a certain formula, Fractint offers you the possibility to adjust the parameters prior to opening that fractal. 3D parameters can also be adjusted, so you will be able to modify the default values for rotation, and perspective. Decent performance and feature pack
List of fractals by Hausdorff dimension Deterministic fractals Random and natural fractals See also Notes and references Further reading Benoît Mandelbrot, The Fractal Geometry of Nature, W. External links Mandelbulb The Mandelbulb is a three-dimensional analogue of the Mandelbrot set, constructed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does. White and Nylander's formula for the "nth power" of the 3D vector is where , and They use the iteration is defined as above and is a vector addition. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" controlled by the parameter n. Quadratic formula Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as: which we can think of as a way to square a triplet of numbers so that the modulus is squared. Cubic fractal
Sterling2 download page Sterling2 is based on Sterling, a fractal-generating program written in 1999 by the redoubtable Stephen C. Ferguson. In mid-2007 I contacted Stephen, as I thought that Sterling was an excellent program that lacked one key feature - a formula editor. He told me that adding a formula editor would be a huge job and that in any case the development environment to compile all the parts of Sterling was no longer available, as it is obsolete. However, he encouraged me to do the next best thing, which was to change the formulae in the program. With his help I set up the development environment on my PC and was able to recompile Sterling and to make changes to just one part of the program, ie the formulae. Between June 2007 and August 2008, I spent some 100 to 200 hours changing formulae (that's the quick part) and then testing them to see which ones produced interesting images. Download Sterling2 for Windows by clicking below (437 kbytes) The zip file contains brief instructions.