The Mandelbrot Set Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah The Mandelbrot Set. Note: All of the Mandelbrot pictures on this page were generated with the applet on this page! You can click on any of them to see a large version, and you can use the applet to generate those very same pictures, or similar pictures all your own! The first picture ( No1 ) shows a small part of the Mandelbrot set (which is rendered in red). List of Contents What's so special about the Mandelbrot set? What is the Mandelbrot set? z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, remains bounded. You may ask, what's so special about the particular iteration (1), and why do we use complex numbers instead of real ones. Much of the fascination of the Mandelbrot set stems from the fact that an extremely simple formula like (1) gives rise to an object of such great complexity. Consider this picture ( Title ). Now, I know you already clicked on that applet! This is what you should see. Max.
Mandelbrot Set The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable. "The" Mandelbrot set is the set obtained from the quadratic recurrence equation with , where points in the complex plane for which the orbit of does not tend to infinity are in the set. equal to any point in the set that is not a periodic point gives the same result. molecule by Mandelbrot. A plot of the Mandelbrot set is shown above in which values of in the complex plane are colored according to the number of steps required to reach . The adjoining portion is a circle with center at and radius The region of the Mandelbrot set centered around is sometimes known as the sea horse valley because the spiral shapes appearing in it resemble sea horse tails (Giffin, Munafo). Similarly, the portion of the Mandelbrot set centered around and
psykopaint Rare earth element As defined by IUPAC, a rare earth element (REE) or rare earth metal is one of a set of seventeen chemical elements in the periodic table, specifically the fifteen lanthanides, as well as scandium and yttrium. Scandium and yttrium are considered rare earth elements because they tend to occur in the same ore deposits as the lanthanides and exhibit similar chemical properties. List A table listing the seventeen rare earth elements, their atomic number and symbol, the etymology of their names, and their main usages (see also Applications of lanthanides) is provided here. Some of the rare earth elements are named after the scientists who discovered or elucidated their elemental properties, and some after their geographical discovery. A mnemonic for the names of the sixth-row elements in order is "Lately college parties never produce sexy European girls that drink heavily even though you look". Abbreviations The following abbreviations are often used: Spectroscopy
Mu-Ency -- The Encyclopedia of the Mandelbrot Set at MROB A second-order embedded Julia set This is a picture from the Mandelbrot Set, one of the most well-known fractal images in the world. (Click it for a larger version). The Mandelbrot Set is one of my hobbies, and I have collected a large amount of information about it. To organize that information I have created Mu-Ency, a large collection of text files linked to each other. Here are some entries from Mu-Ency: Mandelbrot Set: The mathematical definition. More Pictures: Some entries with pictures of parts of the Mandelbrot Set are: R2, Cusp, Embedded Julia set, 2-fold Embedded Julia set, 4-fold Embedded Julia set, Paramecia, R2.C(0), R2.C(1/3), R2.1/2.C(1/2), R2t series, Seahorse Valley, Delta Hausdorff Dimension, Exponential Map, Reverse Bifurcation. You can also look up specific terms in the index. Coordinates of the image above: Center: -1.769 110 375 463 767 385 + 0.009 020 388 228 023 440 i Width (and height): 0.000 000 000 000 000 160 Algorithm: distance estimator Iterations: 10000
wonders of the world TOP 10 IMPOSSIBLE INVENTIONS THAT WORK « Revolutionizing Awareness Searl Effects Generator by Jeane Manning When Leonardo da Vinci sketched out an impossible invention, fifteenth-century scholars probably put him down. Forget it, Leon. If machines could fly, we’d know about it. Throughout history, experts tell innovators that their inventions are impossible. Perhaps in the 21st century the following inventions will be standard science, and a history student may wonder why 20th-century pundits disregarded them. This class of inventions could wipe out oil crises and help solve environmental problems. Forget the Rube Goldberg mechanical perpetual motion contraptions; they had to stop eventually. Inventors give various names to their space-energy converters. A spiritual commune in Switzerland had a tabletop free energy device running in greenhouses for years, but members feared that outsiders would turn the technology into weaponry. It may have been done before Tesla’s time. The garage inventors come from many backgrounds. 8.
List of mathematical symbols When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image. Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. Basic symbols Symbols based on equality sign Symbols that point left or right Brackets Other non-letter symbols
Space-Time Loops May Explain Black Holes Physics cannot describe what happens inside a black hole. There, current theories break down, and general relativity collides with quantum mechanics, creating what's called a singularity, or a point at whichthe equations spit out infinities. But some advanced physics theories are trying to bridge the gap between general relativity and quantum mechanics, tounderstand what's truly going on inside the densest objects in the universe. This was the first time scientists have applied the full loop quantum gravity theory to black holes, and the results were encouraging, researchers said. "What they have done is a major step, because they have been able to provide a much more complete description of what really happens near the black hole singularity using loop quantum gravity," said Abhay Ashtekar, a physicist who studies loop quantum gravity at Pennsylvania State University, who was not involved in the new research." "This model we've done is extremely simple," Pullin said.
Outline of combinatorics The following outline is presented as an overview of and topical guide to combinatorics: Combinatorics – branch of mathematics concerning the study of finite or countable discrete structures. Essence of combinatorics Main article: Combinatorics Branches of combinatorics Multi-disciplinary fields that include combinatorics History of combinatorics Main article: History of combinatorics General combinatorial principles and methods Data structure concepts Problem solving as an art Living with large numbers Persons influential in the field of combinatorics Combinatorics scholars Journals Prizes See also References External links Combinatorics, a MathWorld article with many references.Combinatorics, from a MathPages.com portal.The Hyperbook of Combinatorics, a collection of math articles links.The Two Cultures of Mathematics by W.
Graphene Aerogel – Lightest Material in The World A research team headed by Professor Gao Chao have developed ultra-light aerogel – it breaks the record of the world’s lightest material with surprising flexibility and oil-absorption. This progress is published in the “Research Highlights” column in Nature. Aerogel is the lightest substance recorded by Guinness Book of World Records. It gets its name due to its internal pores filled with air. In 1931, American scientist Kistler first produced aerogel with silicon dioxide, and nicknamed it “frozen smoke”. In 2011, HRL Laboratory, University of California Irvine, and California Institute of Technology collaborated in developing nickel aerogel with a density of 0.9 mg/cubic centimeter, the record lightest material at that time. It couldn’t even cause deformation on dandelion flower fluffs. Gao Chao’s team has long been developing macroscopic graphene materials, such as one-dimensional graphene fibers and two-dimensional graphene films. The new material is just like a new-born baby. Related:
Engineering ToolBox Meet 115, the Newest Element on the Periodic Table If you've learned all the elements from actinium to zirconium, it's time to head back to the periodic table, where there's a new, extremely heavy element in town. The new element doesn't have an official name yet, so scientists are calling it ununpentium, based on the Latin and Greek words for its atomic number, 115. (Related: Read a feature on element hunters in National Geographic magazine.) In case you forgot your high school chemistry, here's a quick refresher: An element's atomic number is the number of protons it contains in its nucleus. The heaviest element in nature is uranium, which has 92 protons. The man-made 115 was first created by Russian scientists in Dubna about ten years ago. Element 115 will join its neighbors 114 and 116-flerovium and livermorium, respectively-on the periodic table just as soon as a committee from the International Union of Pure and Applied Chemistry (IUPAC) decides on an official name for 115. When you find a new element, it has to be confirmed. No.