Fractals PartI

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    IES-01 Fractals and Application

    Attendance: 10 marks

    Assignments: 10 marks

    Class Performance: 5 marks

    (Weightage: Absent : 0, Late : , Present : 1)

    Geometry : developed as a collection of tools for understanding

    the shapes of nature.

    For millenia, Symmetryhas been recognized as a powerful principle in geometry,

    and in art.

    We begin by reviewing the familiar forms of symmetry, then show that fractalsreveal a new kind of symmetry, symmetry under magnification.

    Many shapes that at first appear complicated reveal an underlying simplicity when

    viewed with an awareness of symmetry under magnification.

    We begin by reviewing the familiar symmetries of nature: symmetry undertranslation, reflection, and rotation.

    We are familiar with three forms of symmetry, exhibited approximately in manynatural and manufactured situations. They are translational, reflection, androtational

    Less familiar is symmetry undermagnification :

    zooming in on an object leaves the shape

    approximately unaltered.

    Here we introduce some basic geometry of fractals, with emphasis on the IteratedFunction System (IFS) formalism for generating fractals.

    In addition, we explore the application of IFS to detect patterns, and also severalexamples of architectural fractals.

    First, though, we review familiar symmetries of nature, preparing us for the newkind of symmetry that fractals exhibit.

    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:

    same shape but different size.

    The simplest fractals are constructed by iteration. For example, start with a filled-in triangle and iterate this process:

    For every filled-in triangle, connect the midpoints of the sides and remove themiddle triangle. Iterating this process produces, in the limit, the SierpinskiGasket.

    The gasket is self-similar. That is, it is made up of smaller copies ofitself.

    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:

    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.

    This implies fractals possess a scale invariance.

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    More Examples of Self-Similarity

    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.

    Later we will find the rules to make these fractals.

    For now, to help train your eye to find fractal decompositions of objects, try tofind smaller copies of each shape within the shape.

    The tree is not so hard, except for the trunk.

    The Mandelbrot set: a different nonlinear transformation gives the most famous ofall fractals.

    Fractal landscapes: With more sophistication (and computing power), fractalscan produce convincing forgeries of realistic scenes.

    Making realistic-looking landscapes isdifficult enough, but doing this so they can

    be stored in small files is remarkable.

    Fractals in nature: after looking at so many geometrical and computer-generatedexamples, here is a short gallery of examples from Nature

    Fractals found in nature differ from our first mathematical examples in twoimportant ways:

    the self-similarity of natural fractals is approximateor statisticaland

    this self-similarity extends over only a limited range of scales.

    To understand the first point, note that many forces scuplt and grow naturalfractals, while mathematical fractals are built by a single process.

    For the second point, the forces responsible for a natural fractal structure areeffective over only a limited range of distances.

    The waves carving a fractal coastline are altogether different from the forcesholding together the atoms of the coastline.

    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

    called initiators and generators.

    The initiator is the starting shape.

    The generator is a collection of scaled copies of the initiator.

    The rule is this: in the generator, replace each copy of the initiator with a scaledcopy of the generator (specifying orientations where necessary).

    The initiator is a filled-in triangle, the generator the shape on the right.

    Sierpinski Gasket How can we turn "connect the midpointsand remove the middle triangle" intoinitiators and generators?

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    Koch curve

    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.

    Take as initiator the line segment of length 1, and as generator the shape on

    the right.

    Though its construction is so simple, the Koch curve has some properties thatappear counterintuitive.

    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.

    Cantor set

    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.

    Again, take as initiator the line segment of length 1, but now the generator

    is the shape shown below.

    Here is a picture of the Cantor set resolved to the level of single pixels.

    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

    literary, applications.

    Fractals in the Kitchen

    Cauliflower is a wonderful example of a natural fractal. A small piece of acauliflower looks like a whole cauliflower.

    Pieces of the pieces look like the whole cauliflower, and so on for several moresubdivisions.

    Here is a picture of a cauliflower and a piece broken from it.

    As -------- cook, the boiling batter forms bubbles of many different sizes, giving riseto a fractal distribution of rings.

    Some big rings, more middle-size rings, still more smaller rings, and so on.

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    Some breads are natural fractals. Bread dough rises because yeast producesbubbles of carbon dioxide.

    Many bubbles are small, some a middle-size, a few are large, typical of thedistribution of gaps in a fractal.

    So bread dough is a foam; bread is that foam baked solid.

    Kneading the dough too much breaks up the larger bubbles and givesbread of much more uniform (non-fractal) texture

    Do fractals have practical applications?How about an invisibility cloak?

    On Tuesday, August 28, 2012, U.S. patentnumber 8,253,639 was issued to Nathan

    Cohen and his group at FracTenna, for awide-band microwave invisibility cloak,based on fractal antenna geometry

    The antenna consists of an innerring, the boundary layer, that

    prevents microwaves from beingtransmitted across the inside of

    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

    entered the cloak.

    On the left is a magnification of one of the outer rings of the cloak. On the right isthe boundary layer fractal.

    If fabricated at the sub-micron scale, instead of the current mm scale, thistechnology should act as an optical invisibility cloak.

    In late August, 2012, Cohen's group cloaked a person. Interesting times ahead.

    www.fractenna.com

    Now down to work. We learn to grow fractal images, but first must build up themechanics of plane transformations.

    Geometry of plane transformations is the mechanics of transformations thatproduce more general fractals by Iterated Function Systems

    To generate all but the simplest fractals, we need to understand the geometryof plane transformations. Here we describe and illustrate the four features of

    plane transformations

    Affine transformations of the plane are composedof scalings, reflections, rotations, and translations.

    Scalings

    The scaling factor in the x-direction is denoted r.

    The scaling factor in the y-direction is denoted s.

    Assume there are no rotations. Then if r = s, thetransformation is a similarity

    otherwise it is an affinity

    Note the scalingsare always toward

    the origin. That is,the origin is

    the fixed pointofall scalings.

    Reflections

    Negative r reflects across the y-axis.

    Negative s reflects across the x-axis.

    Reflection across both the x- and y-axes is equivalent to rotation by 180about the origin

    Rotations

    The angle measures rotations of horizontal lines

    The angle measures rotations of vertical lines

    The condition = gives a rigidrotation about the origin.Positive angles arecounterclockwise

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    Translations

    Horizontal translation is measured by e

    Vertical translation is measured by f.

    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.

    We adopt this convention:

    scalings first, reflections second, rotations third, and translatio