... 2008 Fractal Geometry A fractal is generally “a rough or fragmented geometric shape that can be subdivided into parts.” One of the ways that fractal geometry is used is in the area of medicine. In the past few years fractal analysis techniques have gained increasing attention in signal and image processing, especially in medical sciences such as Pathology. Fractal analysis has found applications in the detection of coding regions in DNA and measurement of the space-filling properties of tumors, blood vessels and neurons. Fractal concepts have also been usefully incorporated into models of biological processes, including cell growth, blood vessel growth, periodontal disease and viral infections. Other very interesting applications are founded in medical imaging Fractal analysis is widely used in image processing, both in characterizing shapes of objects and in assessing texture. Breast masses present shape and texture characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses. The fractal dimension of the contour of a mass may be computed either directly from the two dimensional contour or from one-dimensional signatures derived from the contour. Other ways that fractal geometry is use is in biology with different applications and techniques use to classify and distinguish various types of cells. The use of fractal dimension...
Words: 520 - Pages: 3
...Chaos and Fractals What is a Fractal A fractal is a never ending pattern. Fractals are never ending complex patterns that are self-similar across different scales. They are created by by being repeated over and over in a feedback loop. Fractals are everywhere. They can be found in nature, algebra, and math. Nature Algebra Geometry Mandelbrot Set This was created in 1980 shortly after the first personal computer in order to calculate numbers thousands and sometimes millions of times. Equation (old)Z=(new)Z2+C We start by plugging a value for the variable C into the simple equation below. Each complex number is actually a point in a 2-dimensional plane. The equation gives an answer, Znew . We plug this back into the equation, as Zold and calculate it again. We are interested in what happens for different starting values of C. When you square a number it gets bigger. Then you square the answer and it get even bigger.. Eventually, it goes to infinity. This is the fate of most starting values of C. Some values of C do not get bigger, but get smaller, or alternate between a set of fixed values. These are the points inside the Mandelbrot Set, which we color black. Outside the Set, all the values of C cause the equation to go to infinity, and the colors are proportional to the speed at which they expand. Ju Fractals in Nature Natural fractals can be found anywhere in nature. Even in our body. It can be the blood vessels in our arms, or...
Words: 286 - Pages: 2
...Chaos and Fractals Chaos is a fairly new science. The definition of chaos is complete confusion and disorder, but more generalized to everyday speech it means a random pattern that can occur. (Mirriam - Webster) For scientist this can be a good thing. Henry Adams is quoted as saying "Chaos often breeds life as order breeds habit." (STSCI) Fractals simply put, is a repeat design that as you zoom in is repeated indefinitely. Above is the Sierpenski Triangle, as you zoom in you see three more of the triangle, and in each of those is three more triangle, and as you zoom in you see three more triangles. I imagine that on a big enough scale, or with a strong enough computer there is no end to the number of triangles you can put in the Sierpenski Triangle. (STSCI) If you break a straight line into an equal number N than the equation to describe those parts would be r=1/N(1/d) In the case of the Sierpenski Triangle, a self repeating object, the equation is D = log (N) / log (1/r) (STSCI) The Koch Snowflake is designed by: * divide a line segment into three equal parts * remove the middle segment (= 1/3 of the original line segment) * replace the middle segment with two segments of the same length (= 1/3 the original line segment) such that they all connect (i.e. 3 connecting segments of length 1/3 become 4 connecting segments of length 1/3.) The top row of shapes is the pattern to make the snowflake with a triangle, the bottom shape is a blowup of the edge...
Words: 377 - Pages: 2
...EMM310 Assessment Item 2 Due: 15th October Length: 10 – 12 pages (Assessment marking criteria & Appendix 1 not included in page count) Measurement and Geometry A student: - makes, compares, sketches and names three-dimensional objects, including prisms, pyramids, cylinders, cones and spheres, and describes their features MA2-14MG | Working mathematically A student: - uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM - checks the accuracy of a statement and explains the reasoning used MA2-3WM | | Outcome/s | Lesson activities/ content | Prior knowledge | Relation to other strands | Other KLAs | Diverse learners | 1 | Measurement and GeometryMA2-14MG Working mathematically MA2-1WMMA2-3WM | - Revise 2D shapes- Find out prior knowledge of 3D objects – what do the students already know? Are there any misconceptions?- Using large versions of various 3D shapes, identify each object. Discuss the features of each shape e.g. faces, edges etc. - As a class, place the objects into groups based on similar features. Ensure students use reasoning for placing shapes into a certain group | - Students are already familiar with recognising and describing 3D shapes from stage 1 | Working mathematically MA2-1WM,MA2-3WM | EnglishEN2-1A | Visual Auditory/ linguistic | 2 | Measurement and GeometryMA2-14MG | - Discuss features of 3D shapes describing similarities and differences – focus on language e.g. faces, vertex, base, side...
Words: 3508 - Pages: 15
...patterns of a snowflake and try to make a mathematical sense out of it. In addition, I wanted to explore how to easily calculate the area of a complicated shape as snowflake. As I began to research how exactly I can approach the question, I encountered fractal geometry, which seemed to explain and address my question. Hence, by using a fractal, Von Koch Island, I have tried to make a “mathematical sense” out of snowflakes by exploring its...
Words: 1005 - Pages: 5
...CHAOS THEORY It is a field of study in mathematics, with applications in several disciplines including, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Chaotic behavior can be observed in many natural systems, such as weather and climate. This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincare maps. This latter idea is known as sensitive dependence on initial conditions , a circumstance discovered by Edward Lorenz (who is generally credited as the first experimenter in the area of chaos) in the early 1960s. DEFINITION: It is the study of non linear dynamics, in which seemingly random events are actually predictable from simple deterministic equation. Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: * How much uncertainty we are willing to tolerate in the forecast? * How accurately we are able to measure its current state? * Which time scale is depending on the dynamics of the system? The two main components of chaos theory are the ideas that systems - no matter how complex they may be - rely upon an underlying...
Words: 4143 - Pages: 17
...Greatsword/Longbow Recommended Build * In Zeal you want Fiery Wrath for 7% more damage that is almost always up unless you are soloing with scepter. In that case take Zelous Scepter. As long as you are using Greatsword you definitely want Zealous Blade for a reliable 5% DPS modifier and the cooldown reduction on Greatsword. If you’re not using Greatsword, go ahead and use Kindled Zeal. In the grandmaster slot you will want to take Symbolic Avenger with pretty much any weapon exept Scepter. When you are using Scepter you can take Shattered Aegis since that’s pretty much the only time you don’t have any symbols. * In Virtues we take Unscathed Contender – 20% more damage while under Aegis, or Master of Consecrations – Consecrations last longer and their cooldowns are reduced. It depends on whether we are able to keep up the Aegis buff or need to use Consecrations such as Wall of Reflection. If you need to maintain Projectile Defenses for your party then go with Master of Consecrations. In the second slot we can take Supreme Justice or Absolute Resolution depending whether we need Condition Removal or not. Absolute Resolution removes 3 conditions from each ally and yourself when activating VIrtue of Resolve and Supreme Justice will increase Burning duration and decrease the amount of hits needed to trigger burning from 5 to 3. The final choice is not really hard, it’s just what you need at a specific encounter. Permeating Wrath will change the burning Virtue of Justice applies...
Words: 1449 - Pages: 6
...ajuda na percepção de que há um mundo dentro de mundos e que a maioria das formas são representações de si mesmo em escalas maiores. Palavras-Chave: Fractais, Benoît Mandelbrot, koch, escala. LISTA DE FIGURAS Figura 01 – Conjunto de Cantor 11 Figura 02 – Curva de Koch 20 Figura 03 – Dimensão Fractal 37 Figura 04 – Fractais Naturais 37 Figura 05 – Fractais Fisiológicos 37 SUMÁRIO 1. INTRODUÇÃO 13 2. PROPRIEDADES 15 3. CONJUNTO DE CANTOR 17 3.1 QUAL O TAMANHO DO CONJUNTO CANTOR? 17 4. CURVA DE KOCH 17 5. DIMENSÃO FRACTAL 17 6. APLICAÇÕES DOS FRACTAIS NA BIOLOGIA 17 6.1 FRACTAIS NATURAIS 18 6.2 FRACTAIS BIOLÓGICOS 19 7. CONCLUSÃO 20 REFERÊNCIAS 1 INTRODUÇÃO Fractal é "uma forma geométrica áspera ou fragmentada que pode ser subdividida em partes, cada uma das quais é (pelo menos aproximadamente) uma cópia de tamanho reduzido do todo "[1]. Esta propriedade é chamada de auto-similaridade. O termo foi cunhado por Benoît Mandelbrot em 1975 e foi derivada da palavra latina “fractus” que significa "quebrado" ou "fraturado". 2 PROPRIEDADES Um fractal tem as seguintes características: * Estrutura fina em pequenas escalas. * É auto-semelhante (pelo menos aproximadamente). * É muito irregular para ser facilmente descrita em geometria euclidiana tradicional. * Tem uma definição simples e recursiva. Por parecem-se semelhantes em todos...
Words: 1140 - Pages: 5
...irst some questions: What is time? In a physical sense? In psychological terms? What does time do? How does it work? Can it be transcended? Time in many ways is like space. In physics, time and space are woven together, like a fabric upon which all matter lies. At the limits, near the speed of light, movement in time is yoked to movement in space. As spatial speed increases, temporal speed slows. Recent quantum physics, in the area of non-local phenomena, suggests that both time and space are not as they appear on our scale of existence. It appears that particles, separated in both space and time, interact, in a simultaneous manner. Indeed, in one of the strangest experimental effects, the future may causally impact the past (the implications of that one will make your head spin). Distant particles are somehow connected, are somehow not distant. It is as if the space and time between did not really exist, nor the proposed distinction. Rather, these physicists (cf. Yakir Aharonov, Jeff Tollakson, and Menas Kafatos) suggest that perhaps there is an underlying singularity or unity to matter, across both time and space. Many spiritual traditions, philosophies, songs, and so on have suggested similar ideas: "We are one, heartache to heartache....love is a battlefield" Pat Benatar. Beyond funny ‘80's rocker references, such notions are at the heart of spiritual practices, across the various world traditions, even mainstream Christianity which proposes that God exists outside the...
Words: 933 - Pages: 4
...The idea that Africans have contributed little to world civilization is one which many in the West have for a long time assumed and taken for granted. Thanks in part to depictions of Africa which rarely extend past civil wars, famine and the primitive; information about Africa’s past advances and accomplishments have continued to remain obscure and little known. Since first contact between Europe and Africa the history of Africa has been fundamentally dominated by the way Europeans have portrayed themselves in relationship to that continent. So that most of what we read and see about Africa tends to say -- either directly or indirectly -- more about the history of European colonialism and its biases toward Africa than it does about the real Africa and its people (see Ahmad, 1987). The majority of people today of all backgrounds, including those of African ancestry, tend to know little about Africa and its history outside of the transatlantic slave trade and perhaps colonialism. While even in these instances knowledge about these events can be at times, limited. The African continent is too often conceived of as one with no legitimate history before contact with Europeans. Formal anthropological research is now showing that this notion could not be further from the truth. In the bible Ham's sons are believed to have fathered the peoples of Africa. Of Ham's four sons, Canaan, fathered the Canaanites, while Mizraim fathered the Egyptians, Cush the Cushites and Phut the "Libyans"...
Words: 1957 - Pages: 8
...362 Chapter 9. Root Finding and Nonlinear Sets of Equations } a=b; fa=fb; if (fabs(d) > tol1) b += d; else b += SIGN(tol1,xm); fb=(*func)(b); Move last best guess to a. Evaluate new trial root. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). } nrerror("Maximum number of iterations exceeded in zbrent"); return 0.0; Never get here. } CITED REFERENCES AND FURTHER READING: Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood Cliffs, NJ: PrenticeHall), Chapters 3, 4. [1] Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), §7.2. 9.4 Newton-Raphson Method Using Derivative Perhaps the most celebrated of all one-dimensional root-finding routines is Newton’s method, also called the Newton-Raphson method. This method is distinguished from the methods of previous sections by the fact that...
Words: 3328 - Pages: 14
...CHAPTER 1 INTRODUCTION TO INFORMATION TECHNOLOGY 1.1 Introduction Modern civilization has become so complicated and sophisticated that to survive one has to be competitive. This compels people to keep themselves informed of all types of happening in the society. And this in turn entails the need for an infrastructure of information. This is the point where information technology or IT becomes most important, as it is the infrastructure that allows us to get information accurately and in time. Before we define information technology, it is important to understand the notion of Data, Information, Technology, and Knowledge. In everyday conversation, people use the terms data and information interchangeably. However, some computer professional make a distinction between the two terms. Data It is the words, numbers, letters, symbol, sound, video and graphics that describe people, events, things and ideas. It is raw facts about people, objects, and events that have little or no meaning. It is the raw material used to create useful information. It becomes information when you use it as the basis for initiating some action or for making a decision. Information It is defined as the words, numbers, letters, symbol, sound, video and graphics used as the basis for human action or decisions. It is data that have been processed and presented in a form suitable for human interpretation, often with the purpose of revealing trends or patterns that can be used in decision-making. It is data...
Words: 12773 - Pages: 52
...SEPERATION PROCESSES II DESIGN PROJECT COAGULATION & FLOCCULATION PROCESSES IN THE PRODUCTION OF POTABLE WATER SUBMISSION DATE: 14 August 2012 COURSE COORDINATOR : Dr. Netatollah Rahmanian GROUP MEMBERS’ NAMES: Derek Lai Chai Zern 14233 Derek Lai Chai Zern 14233 Sean Suraj Jeremiah 14286 Nabila Syahira Bt Azizuddin 14295 Hazwan Farid B Muhammad Puzi 14382 Karrthik S/O Subramaniam 15450 Kiveeyashini D/O Govindasamy 17252 INTRODUCTION Human settlements have always been centred around sources of clean drinking water. As the population increases and the quality of fresh water declines, it has become an engineering challenge to supply sufficient potable water to the meet demands. Of the many unit processes and operations used in water treatment, coagulation and flocculation required a unique combination of chemical and physical phenomena for producing water acceptable for human consumption. Aggregation of fine particulate matter into larger particulates by the use of coagulation and flocculation facilities permits cost-effective removal in subsequent solid separation processes. Particulates inorganic origin such as clay, silt, and mineral oxides generally enter surface water by natural erosion processes and can decrease the clarity of the water to an unacceptable level. Organic particulates, such as colloidal humic and fulvic acids are a product of decay and leaching of organic debris and litter which have fallen in the water...
Words: 14355 - Pages: 58
...Ironically I wasn’t scared, worried about where I would fall. I found myself I in this room. The room and objects were so colorful and golden I cannot describe it! The effect as literally being in a space craft and lifting off into fractal hyperspace dimension via vibrations, phasing and movement I just can't put into words. I had to breathe deeply many times, as I seemed to be flapping folding into myself. Honestly just can't describe it. I closed my eyes and was in some hyper ultra-colourful fractal landscape and in total joy! Everything was just so colorful and I just felt...
Words: 520 - Pages: 3
...------------------------------------------------- Dimension From Wikipedia, the free encyclopedia "0d" redirects here. For 0D, see 0d (disambiguation). For other uses, see Dimension (disambiguation). From left to right, the square, the cube, and the tesseract. The square is bounded by 1-dimensional lines, the cube by 2-dimensional areas, and the tesseract by 3-dimensional volumes. A projection of the cube is given since it is viewed on a two-dimensional screen. The same applies to the tesseract, which additionally can only be shown as a projection even in three-dimensional space. A diagram showing the first four spatial dimensions. In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A surface such as aplane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and itslongitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces. In physical terms, dimension refers to the constituent structure of all space (cf. volume) and its position in time (perceived as a scalar dimension...
Words: 2729 - Pages: 11
