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 ). ).
Complexity The complexity of a physical system or a dynamical process expresses the degree to which components engage in organized structured interactions. High complexity is achieved in systems that exhibit a mixture of order and disorder (randomness and regularity) and that have a high capacity to generate emergent phenomena. Complexity across Scientific Disciplines Despite the importance and ubiquity of the concept of complexity in modern science and society, no general and widely accepted means of measuring the complexity of a physical object, system, or process currently exists. The lack of any general measure may reflect the nascent stage of our understanding of complex systems, which still lacks a general unified framework that cuts across all natural and social sciences. While a general measure has remained elusive until now, there is a broad spectrum of measures of complexity that apply to specific types of systems or problem domains. General Features of Complexity Components. Emergence.
Computational complexity theory Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. Closely related fields in theoretical computer science are analysis of algorithms and computability theory. Computational problems A traveling salesman tour through Germany’s 15 largest cities. Problem instances A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. Representing problem instances
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 .
Thinking Enterprise Home 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
Simple animation 1, aircraft radial engine 2, oval Regulation 3, sewing machines 4, Malta Cross movement - second hand movement used to control the clock 5, auto change file mechanism 6, auto constant velocity universal joint 6.gif 7, gun ammunition loading system 8 rotary engine - an internal combustion engine, the heat rather than the piston movement into rotary movement # Via World Of Technology. 1, inline engine - it's cylinders lined up side by side 2, V-type engine - cylinder arranged at an angle of two plane 3, boxer engine - cylinder engine arranged in two planes relative
Game of Life News: Oblique Life spaceship created Andrew J. Wade has recently built a self-replicating configuration in Life. It consists of two stable configurations equipped with Chapman-Greene construction arms, and a volley of gliders circulating between them. The announcement was made on this forum thread . The spaceship propagates at the impressively slow speed of (5120,1024)c/33699586. Undoubtedly, this creation will lead to an avalanche of discoveries in Life. It differs from the Standard Architecture in a variety of ways. The configuration uses three Chapman-Greene construction arms at each end of the tape: two perpendicular arms for construction, and a third arm for destruction. This is the thirteenth explicitly constructed spaceship velocity in Life, although it facilitates an infinite number of related velocities.
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
Living Code: The importance of visual programming Python has a well-earned reputation for being easy to use and to learn, at least for people who have learned programming in other languages first. Lately my kids have been very interested in programming, and I've found that Python doesn't come as easily to 6-11 year olds as it does to adult programmers. So I see two approaches to this problem, if it is a problem. One, let them use other languages than Python. Two, find (or make) ways for Python to be more approachable. Let's look at both of these. Scratch For languages other than Python, there are some really good choices. Learn more about this project Scratch is great for learning and for sharing. One option that is often suggested as a step up from Scratch is GameMaker, which apparently is a very nice commercial system that lets kids build games. Quartz Composer Another interesting system we've been playing around with lately is Quartz Composer. eToys One more tool we've begun to explore is Squeak/eToys. . Turtles Turning now to Python.
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
Instances of Fractal Evolution « Mindsoul’s Weblog When couple of years ago (Nov 2006) I experienced author and biologist Dr. Bruce Lipton‘s 2-day presentation in his seminar produced by Spirit 2000 in San Francisco, I was totally blown away by the depth of insight I got into how my biology, my health, and how my health at the cellular level, is in my complete control through my beliefs (both conscious and subconscious). But one new thing Bruce added at the end of his presentation was his talk about how patterns of life and evolution are similar to patterns of fractal geometry. Dr. Fractal Geometry Fractal geometry is derived by a recursive mathematical formulas, and incredible shapes were produced when computers were strong and fast enough to be able to execute these very computing-intensive formulas. Concept of Fractal Evolution is that evolution follows a similar pattern, meaning “pattern of the whole is seen in the parts of the whole” quoting from Dr. Digital Technology and Unified Field Theory Dr. Evolution of the Web Like this:
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.