Fractals PartI

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<ul><li><p>5/19/2018 Fractals PartI</p><p> 1/24</p><p>2/22/</p><p>IES-01 Fractals and Application</p><p>Attendance: 10 marks</p><p>Assignments: 10 marks</p><p>Class Performance: 5 marks</p><p>(Weightage: Absent : 0, Late : , Present : 1)</p><p>Geometry : developed as a collection of tools for understanding</p><p>the shapes of nature.</p><p>For millenia, Symmetryhas been recognized as a powerful principle in geometry,</p><p>and in art.</p><p>We begin by reviewing the familiar forms of symmetry, then show that fractalsreveal a new kind of symmetry, symmetry under magnification.</p><p>Many shapes that at first appear complicated reveal an underlying simplicity when</p><p>viewed with an awareness of symmetry under magnification.</p><p>We begin by reviewing the familiar symmetries of nature: symmetry undertranslation, reflection, and rotation.</p><p>We are familiar with three forms of symmetry, exhibited approximately in manynatural and manufactured situations. They are translational, reflection, androtational</p><p>Less familiar is symmetry undermagnification :</p><p>zooming in on an object leaves the shape</p><p>approximately unaltered.</p><p>Here we introduce some basic geometry of fractals, with emphasis on the IteratedFunction System (IFS) formalism for generating fractals.</p><p>In addition, we explore the application of IFS to detect patterns, and also severalexamples of architectural fractals.</p><p>First, though, we review familiar symmetries of nature, preparing us for the newkind of symmetry that fractals exhibit.</p><p>The geometric characterization of the simplest f ractals is self-similarity: theshape is made of smaller copies of itself. The copies are similar to the whole:</p><p>same shape but different size.</p><p>The simplest fractals are constructed by iteration. For example, start with a filled-in triangle and iterate this process:</p><p>For every filled-in triangle, connect the midpoints of the sides and remove themiddle triangle. Iterating this process produces, in the limit, the SierpinskiGasket.</p><p>The gasket is self-similar. That is, it is made up of smaller copies ofitself.</p><p>We can describe the gasket as made of three copies, each 1/2as tall and1/2 aswide as the original. But note a consequence of self-similarity:</p><p>each of these copies is made of three still smaller copies, so we can say thegasket is made of nine copies each1/4 by 1/4 of the original, or 27 copieseach 1/8 by 1/8, or ... . Usually, we prefer the simplest description.</p><p>This implies fractals possess a scale invariance.</p></li><li><p>5/19/2018 Fractals PartI</p><p> 2/24</p><p>2/22/</p><p>More Examples of Self-Similarity</p><p>The gasket is made of three copies of itself, each scaled by 1/2, and two copiestranslated. With slightly more complicated rules, we can build fractals that arereasonable, if crude, approximations of natural objects.</p><p>Later we will find the rules to make these fractals.</p><p>For now, to help train your eye to find fractal decompositions of objects, try tofind smaller copies of each shape within the shape.</p><p>The tree is not so hard, except for the trunk.</p><p>The Mandelbrot set: a different nonlinear transformation gives the most famous ofall fractals.</p><p>Fractal landscapes: With more sophistication (and computing power), fractalscan produce convincing forgeries of realistic scenes.</p><p>Making realistic-looking landscapes isdifficult enough, but doing this so they can</p><p>be stored in small files is remarkable.</p><p>Fractals in nature: after looking at so many geometrical and computer-generatedexamples, here is a short gallery of examples from Nature</p><p>Fractals found in nature differ from our first mathematical examples in twoimportant ways:</p><p>the self-similarity of natural fractals is approximateor statisticaland</p><p>this self-similarity extends over only a limited range of scales.</p><p>To understand the first point, note that many forces scuplt and grow naturalfractals, while mathematical fractals are built by a single process.</p><p>For the second point, the forces responsible for a natural fractal structure areeffective over only a limited range of distances.</p><p>The waves carving a fractal coastline are altogether different from the forcesholding together the atoms of the coastline.</p><p>One way to guarantee self-similarity is to build a shape by applying the sameprocess over smaller and smaller scales. This idea can be realized with a process</p><p>called initiators and generators.</p><p>The initiator is the starting shape.</p><p>The generator is a collection of scaled copies of the initiator.</p><p>The rule is this: in the generator, replace each copy of the initiator with a scaledcopy of the generator (specifying orientations where necessary).</p><p>The initiator is a filled-in triangle, the generator the shape on the right.</p><p>Sierpinski Gasket How can we turn "connect the midpointsand remove the middle triangle" intoinitiators and generators?</p></li><li><p>5/19/2018 Fractals PartI</p><p> 3/24</p><p>2/22/</p><p>Koch curve</p><p>Tents upon tents upon tents ... makes a shape we shall see is very strange, acurve enclosed in a small box and yet that is infinitely long.</p><p>Take as initiator the line segment of length 1, and as generator the shape on</p><p>the right.</p><p>Though its construction is so simple, the Koch curve has some properties thatappear counterintuitive.</p><p>For example, we shall see that it is infinitely long, and that every piece of it, nomatter how small it appears, also is infinitely long.</p><p>Cantor set</p><p>Cut all the tents out of the Koch curve and we are left with something thatappears to be little more than holes. But we can be fooled by appearances.</p><p>Again, take as initiator the line segment of length 1, but now the generator</p><p>is the shape shown below.</p><p>Here is a picture of the Cantor set resolved to the level of single pixels.</p><p>Although so much has been removed that the Cantor set is hardly present atall, we shall find this fractal in many mathematical, and some physical and even</p><p>literary, applications.</p><p>Fractals in the Kitchen</p><p>Cauliflower is a wonderful example of a natural fractal. A small piece of acauliflower looks like a whole cauliflower.</p><p>Pieces of the pieces look like the whole cauliflower, and so on for several moresubdivisions.</p><p>Here is a picture of a cauliflower and a piece broken from it.</p><p>As -------- cook, the boiling batter forms bubbles of many different sizes, giving riseto a fractal distribution of rings.</p><p>Some big rings, more middle-size rings, still more smaller rings, and so on.</p></li><li><p>5/19/2018 Fractals PartI</p><p> 4/24</p><p>2/22/</p><p>Some breads are natural fractals. Bread dough rises because yeast producesbubbles of carbon dioxide.</p><p>Many bubbles are small, some a middle-size, a few are large, typical of thedistribution of gaps in a fractal.</p><p>So bread dough is a foam; bread is that foam baked solid.</p><p>Kneading the dough too much breaks up the larger bubbles and givesbread of much more uniform (non-fractal) texture</p><p>Do fractals have practical applications?How about an invisibility cloak?</p><p>On Tuesday, August 28, 2012, U.S. patentnumber 8,253,639 was issued to Nathan</p><p>Cohen and his group at FracTenna, for awide-band microwave invisibility cloak,based on fractal antenna geometry</p><p>The antenna consists of an innerring, the boundary layer, that</p><p>prevents microwaves from beingtransmitted across the inside of</p><p>this ring. This is the region thatwill be invisible to outsideobservers. Surrounding theboundary layer are sixconcentric rings that guidemicrowaves around theboundary layer, to reconverge atthe point antipodal to where they</p><p>entered the cloak.</p><p>On the left is a magnification of one of the outer rings of the cloak. On the right isthe boundary layer fractal.</p><p>If fabricated at the sub-micron scale, instead of the current mm scale, thistechnology should act as an optical invisibility cloak.</p><p>In late August, 2012, Cohen's group cloaked a person. Interesting times ahead.</p><p></p><p>Now down to work. We learn to grow fractal images, but first must build up themechanics of plane transformations.</p><p>Geometry of plane transformations is the mechanics of transformations thatproduce more general fractals by Iterated Function Systems</p><p>To generate all but the simplest fractals, we need to understand the geometryof plane transformations. Here we describe and illustrate the four features of</p><p>plane transformations</p><p>Affine transformations of the plane are composedof scalings, reflections, rotations, and translations.</p><p>Scalings</p><p>The scaling factor in the x-direction is denoted r.</p><p>The scaling factor in the y-direction is denoted s.</p><p>Assume there are no rotations. Then if r = s, thetransformation is a similarity</p><p>otherwise it is an affinity</p><p>Note the scalingsare always toward</p><p>the origin. That is,the origin is</p><p>the fixed pointofall scalings.</p><p>Reflections</p><p>Negative r reflects across the y-axis.</p><p>Negative s reflects across the x-axis.</p><p>Reflection across both the x- and y-axes is equivalent to rotation by 180about the origin</p><p>Rotations</p><p>The angle measures rotations of horizontal lines</p><p>The angle measures rotations of vertical lines</p><p>The condition = gives a rigidrotation about the origin.Positive angles arecounterclockwise</p></li><li><p>5/19/2018 Fractals PartI</p><p> 5/24</p><p>2/22/</p><p>Translations</p><p>Horizontal translation is measured by e</p><p>Vertical translation is measured by f.</p><p>The matrix formulation of an affine transformation that involves scaling by r in thex-direction, by s in the y-direction, rotations by and , and translations by e andf.</p><p>We adopt this convention:</p><p>scalings first, reflections second, rotations third, and translations last.</p><p>This order is imposed by the matrix formulation.</p><p>Emphasizing this order, the components of a transformation areencoded in tables of this form</p><p>With this encoding of transformations of the plane, we can make fractals usingthe method called Iterated Function Systems (IFS)</p><p>Generating fractals by iterating a collection of transformations is the IteratedFunction System(IFS) method, popularized by Barnsley, based on theoretical workby Hutchinson and Dekking. We use a simple example to see how it works</p><p>Iterated Function Systems</p><p>To illustrate the IFS method, we show how a specific set of IFSrules generates a Sierpinski gasket</p><p>We begin with a right isosceles Sierpinski gasket. Certainly, the gasket can beviewed as made up of three copies of itself, each scaled by a factor of 1/2 in</p><p>both the x-and y-directions</p><p>To determine the translation amount of each piece, takesome point of the whole fractal (the lower left corner, for</p><p>example) and observe where that point goes in eachpiece.</p><p>Here we derive the rules for the right isosceles Sierpinskigasket</p><p>Invariance of the Gasket</p><p>Note that applying all three of these transformations to the gasket gives the gasketagain</p><p>That is, the gasket is invariant under the simultaneous application of these threetransformations.</p><p>What happens if we apply these transformations to some shape other thanthe gasket?</p><p>What happens if we apply these transformations to the resulting shape?</p><p>What happens if we iterate this process?</p><p>Here is an instance of this idea applied to a sketch of a cat</p><p>We observe a sequence of pictures thatconverges to the gasket, independently ofthe starting shape.</p><p>For concreteness we illustrate this converge using the gasket rules. Becauseall the transformations are applied at each iteration, this is called</p><p>the determinisitc algorithm.</p><p>Specifically, suppose T1, ..., Tn are contractions, and P0 is any picture.For example,</p><p>T1(x,y) = (x/2, y/2),</p><p>T2(x,y) = (x/2, y/2) + (1/2, 0),</p><p>T3(x,y) = (x/2, y/2) + (0, 1/2),</p><p>and P0 =</p><p>Generate a sequence of pictures</p><p>P1 = T1(P0) ... Tn(P0)</p><p>P2 = T1(P1) ... Tn(P1)</p><p>...</p><p>Pk+1 = T1(Pk) ... Tn(Pk)</p><p>This sequence converges to a unique shape, P, the only (compact)shape invariant under the simultaneous application of T1, ..., Tn:</p><p>P = T1(P) ... Tn(P) That is,</p><p>Because of this convergence property, P is</p><p>called the attractor of the IFS {T1, ... , Tn}.</p></li><li><p>5/19/2018 Fractals PartI</p><p> 6/24</p><p>2/22/</p><p>Inverse problems</p><p>finding the transformations to produce a given fractal</p><p>Given a fractal F, the Inverse problem is to find affinetransformations T1, ..., Tn for which</p><p>F = T1(F) ... Tn(F)</p><p>Here we present a method to solve this problem</p><p>Solving the inverse problem takes just two steps.</p><p>1. Using the self-similarity (or self-affinity) of F, decompose F as F = F1 ... Fn, whereeach Fi is a scaled c opy of F.</p><p>2. For each piece F i, find an affine transformation Ti for which Ti(F) = Fi. By "find an affine</p><p>transformation" we mean find the r, s, , , e, and f values.</p><p>Remarkably, solving the inverse problem has only two steps:</p><p>Because the transformations can involve rotations, reflections, and scalingsby different factors in different directions, decomposition is not always assimple a task as it may seem at first. Here are some examples of morecomplicated decompositions.</p><p>Decomposition</p><p>This fractal is an instructive example for people who haveseen the gasket and a f ew of its relatives.</p><p>The primacy of the gasket in early examples of fractalsmakes this shape one of the easiest to recognize.</p><p>The most common response to first seeing this picture is,"It's half a gasket."</p><p>But we don't have rules for making half of a fractal.</p><p>The main lesson here is that we're looking for scaled copies of thewhole shape, and the whole shape is not a gasket.</p><p>Tracing small copies of the outline of the whole shape, perhapscutting them out of paper, is a good way to build up intuition for this</p><p>process. Here's a decomposition.</p><p>Note the bottom left pieceis a reflected copy of the</p><p>whole shape</p><p>In the x-direction we see the familiarCantor middle thirds set; in the y-direction</p><p>just a line segment. Again, look for scaledcopies of the whole shape. Here's</p><p>a decomposition.</p><p>An additional problem is that decompositions never are unique. Here aresome examples of different decompositions of the same f ractal.</p><p>Find a decomposition of this fractal intosmaller copies of itself.</p><p>Here's one decomposition, andhere's another.</p><p>Usually, we try to find a decomposition intothe smallest number of pieces, keeping inmind that each piece must be a contractedcopy of the whole shape.</p><p>We have already seen one decompositionof this fractal.</p><p>When we give up the requirement that thepieces be similar to the whole, new</p><p>possibilities appear.</p></li><li><p>5/19/2018 Fractals PartI</p><p> 7/24</p><p>2/22/</p><p>1. Using the self-similarity (or self-affinity) of F, decompose F as F = F1 ... Fn, whereeach Fi is a scaled c opy of F.</p><p>2. For each piece F i, find an affine transformation Ti for which Ti(F) = Fi. By "find anaffine transformation" we mean find the r, s, , , e, and f values.</p><p>Solving the inverse problem has only two steps:</p><p>(a) Trace the main features of the fractal and cut outsmaller copies of the tracing.</p><p>(b) To allow for reflections, flip the small copies and on theback trace over the lines on the front. Label the frontimage with a small F, to distinguish it from its reflection,and to indicate the original orientation.</p><p>(c) Place the small copies, perhaps rotating or reflectingthem, to make a copy of the original fractal.</p><p>Examples</p><p>This fractal can be decomposed into threepieces:</p><p>Note the top and bottom left pieces havethe same orientation as the entire fractal,</p><p>while the bottom right piece is reflectedacross a vertical line</p><p>Keeping in mind that our transformation rules allow only reflections across thex- and y-axes, some care must be taken with the...</p></li></ul>