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.
How Mandelbrot's fractals changed the world 18 October 2010Last updated at 14:15 By Jack Challoner Science writer Fractals have become a common sight, thanks to computer imagery In 1975, a new word came into use, when a maverick mathematician made an important discovery. So what are fractals? And why are they important? During the 1980s, people became familiar with fractals through those weird, colourful patterns made by computers. But few realise how the idea of fractals has revolutionised our understanding of the world, and how many fractal-based systems we depend upon. On 14 October 2010, the genius who coined the word - Polish-born mathematician Benoit Mandelbrot - died, aged 85, from cancer. Unfortunately, there is no definition of fractals that is both simple and accurate. The best way to get a feeling for what fractals are is to consider some examples. They are all complicated and irregular: the sort of shape that mathematicians used to shy away from in favour of regular ones, like spheres, which they could tame with equations.
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
Box-and-Whisker Plots: Interquartile Ranges and Outliers Box-and-Whisker Plots: Interquartile Ranges and Outliers (page 3 of 3) Sections: Quartiles, boxes, and whiskers, Five-number summary, Interquartile ranges and outliers The "interquartile range", abbreviated "IQR", is just the width of the box in the box-and-whisker plot. That is, IQR = Q3 – Q1. The IQR is the length of the box in your box-and-whisker plot. (Why one and a half times the width of the box? Find the outliers, if any, for the following data set: To find out if there are any outliers, I first have to find the IQR. Outliers will be any points below Q1 – 1.5×IQR = 14.4 – 0.75 = 13.65 or above Q3 + 1.5×IQR = 14.9 + 0.75 = 15.65. Then the outliers are at 10.2, 15.9, and 16.4. The values for Q1 – 1.5×IQR and Q3 + 1.5×IQR are the "fences" that mark off the "reasonable" values from the outlier values. By the way, your book may refer to the value of "1.5×IQR" as being a "step". Find the outliers and extreme values, if any, for the following data set, and draw the box-and-whisker plot.
Using Body Awareness For Deeper Meditation & We The Change, Personal Development for Conscious People | Exploring How Self-Improvement Plays a Role in Global Consciousness I would like to share with you a very powerful meditation technique I use called ‘body awarenes s’. Even though this practice is primarily used during meditation, I have found great use in utilizing body awareness in my every day life as well. For those of you who already have a meditation practice, I highly recommend trying the following steps. For those of you who do not currently meditate, the practice brings a great sense of peace, joy, perspective and fulfillment into your life. I highly recommend you giving it a whirl and have written two excellent pieces on ‘meditation for beginners’. How To Meditate For Beginners20 Steps for Quieting The Mind Body awareness is extremely healthy and initiates the flow of fresh oxygenated blood to body areas that usually fall below the radar of your consciousness: Using the Body The big question for beginners is: where do I focus my attention during meditation? First, it is always helpful to make your practice of meditation ‘formal’. nostrils.
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
The Mandelbrot Set : Good Math, Bad Math The most well-known of the fractals is the infamous Mandelbrot set. It’s one of the first things that was really studied *as a fractal*. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s. It’s a beautiful example of what makes fractals so attractive to us: it’s got an extremely simple definition; an incredibly complex structure; and it’s a rich source of amazing, beautiful images. So what is the Mandelbrot set? Take the set of functions fC(x)=x2+C where for each fC, C is a particular complex constant. * m(0,C)=fC(0) * m(i+1,C)=fC(m(i,C)) If m(i,C) doesn’t diverge (escape) towards infinity as i gets larger, then the complex number C is a member of the Mandelbrot set. If we use that definition of the Mandelbrot set, and draw the members of the set in black, we get an image like the one above.
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. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. 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 based on equality sign
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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 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. In reported papers the carbon sponge developed by Gao’s team is the record holder of lightest material, with 0.16 mg/cubic centimeter, lower than the density of helium. The basic principle of developing aerogel is to remove solvent in the gel and retain the integrity. The title of review in Nature is “Solid carbon, springy and light”. The new material is just like a new-born baby. Source: Zhejiang University Related:
[ wu :: fractals ] A Short And Entertaining Introduction to Fractals usually one's first response to fractals is simply this: they are beautiful! indeed, they are visually arresting, and there are many reasons why. perhaps one reason is that they exhibit extreme levels of symmetry, a property we have gravitated toward throughout human history, whether it be in our architectural designs, in our scientific theories, in our religions, or even in the facial structures of the opposite sex. another reason could be that the same self-replicative patterns can be found strewn throughout our natural universe, in vapor trails, snail shells, evergreens, cauliflowers, and snowflakes ... just to name a few. but perhaps most enticing is a reason most people would never guess -- mathematical brevity. many of these stunning patterns are governed by very simple-looking equations consisting of only a few symbols!