{"id":178,"date":"2023-02-01T00:03:29","date_gmt":"2023-02-01T00:03:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/putting-it-together-geometry\/"},"modified":"2023-02-01T00:03:29","modified_gmt":"2023-02-01T00:03:29","slug":"putting-it-together-geometry","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/putting-it-together-geometry\/","title":{"raw":"Putting It Together: Fractals","rendered":"Putting It Together: Fractals"},"content":{"raw":"\n[caption id=\"attachment_2141\" align=\"aligncenter\" width=\"550\"]\"Bubble<\/a> Foam consists of bubbles packed together in a fractal pattern.[\/caption]\n\n \n\nLet\u2019s use what we have learned about fractals to study a real-world phenomenon:  foam.  By definition a foam<\/strong> is any material made up of bubbles packed closely together.  If the bubbles tend to be very large, you might call it a froth<\/strong>.\n\n \n\nWe know that bubbles like to form spheres, because the sphere is the most efficient shape for minimizing surface area around a fixed volume.  When you blow a soap bubble on a warm spring day, for example, that bubble will be approximately spherical until it pops.\n\n \n\nHowever, spheres do not pack together very nicely.  There\u2019s always gaps between the adjacent spheres.  So when a foam forms, there may be some number of large bubbles, interspersed with smaller bubbles in the gaps, which in turn have even smaller bubbles in their gaps and so on.  The foam is approximately self-similar on smaller and smaller scales; in other words, foam is fractal.\n\n \n\nLet\u2019s take a look at a two-dimensional idealized version of foam called the Apollonian gasket<\/strong>.  This figure is created by the following procedure.  It helps to have a compass handy.\n
    \n \t
  1. Draw a large circle.<\/li>\n \t
  2. Within the circle, draw three smaller circles that all touch one another.  In technical terms, we say that the circles are mutually tangent<\/strong> to one another.<\/li>\n \t
  3. In the gaps between these circles, draw smaller circles that are as large as possible without overlapping any existing circles.  If you do this correctly, the new circle will be tangent to two of the circles from step 2 as well as the original big circle.<\/li>\n \t
  4. Continue in this way, filling each new gap with as large a circle that will fit without overlap.  Notice that with each new circle, there will be multiple new gaps to fill.<\/li>\n<\/ol>\nThe process should continue indefinitely, however you will eventually reach a stage in which the gaps are smaller than the width of your pencil or pen.  At that point, you can step back and admire your work.  A computer-generated Apollonian gasket is shown in the figure below.\n\n[caption id=\"attachment_2143\" align=\"aligncenter\" width=\"324\"]\"Apollonian<\/a> The Apollonian gasket is a fractal that can be used to model soap bubble foam.[\/caption]\n\n \n\nBecause the Apollonian gasket is only approximately self-similar, there is not a well-defined scaling-dimension.  However, if you look at any \u201ctriangular\u201d section within three circles, it looks like a curved version of the Sierpinski gasket.  Recall, it requires 3 copies of the Sierpinski gasket in order to scale it by a factor of 2.  So we would expect the fractal dimension of the Apollonian gasket to be close to:\n

    [latex]D={\\large\\frac{\\log(3)}{\\log(2)}}\\approx1.585[\/latex]<\/p>\nIn fact, using a more general definition of fractal dimension, it can be shown that the dimension of the Apollonian gasket is about 1.3057.  This implies that the gasket is somehow closer to being one-dimensional than two-dimensional.  In turn, a foam made of large bubbles, like the froth on top of your latte is more two-dimensional than three-dimensional.  Remember this next time you get an extremely frothy drink; there\u2019s very little substance to it!\n\n \n\nIf you would like a more detailed instructions on how to make your own Apollonian gasket, they are provided at the following website: http:\/\/www.wikihow.com\/Create-an-Apollonian-Gasket<\/a>\n","rendered":"

    \"Bubble<\/a><\/p>\n

    Foam consists of bubbles packed together in a fractal pattern.<\/p>\n<\/div>\n

     <\/p>\n

    Let\u2019s use what we have learned about fractals to study a real-world phenomenon:  foam.  By definition a foam<\/strong> is any material made up of bubbles packed closely together.  If the bubbles tend to be very large, you might call it a froth<\/strong>.<\/p>\n

     <\/p>\n

    We know that bubbles like to form spheres, because the sphere is the most efficient shape for minimizing surface area around a fixed volume.  When you blow a soap bubble on a warm spring day, for example, that bubble will be approximately spherical until it pops.<\/p>\n

     <\/p>\n

    However, spheres do not pack together very nicely.  There\u2019s always gaps between the adjacent spheres.  So when a foam forms, there may be some number of large bubbles, interspersed with smaller bubbles in the gaps, which in turn have even smaller bubbles in their gaps and so on.  The foam is approximately self-similar on smaller and smaller scales; in other words, foam is fractal.<\/p>\n

     <\/p>\n

    Let\u2019s take a look at a two-dimensional idealized version of foam called the Apollonian gasket<\/strong>.  This figure is created by the following procedure.  It helps to have a compass handy.<\/p>\n

      \n
    1. Draw a large circle.<\/li>\n
    2. Within the circle, draw three smaller circles that all touch one another.  In technical terms, we say that the circles are mutually tangent<\/strong> to one another.<\/li>\n
    3. In the gaps between these circles, draw smaller circles that are as large as possible without overlapping any existing circles.  If you do this correctly, the new circle will be tangent to two of the circles from step 2 as well as the original big circle.<\/li>\n
    4. Continue in this way, filling each new gap with as large a circle that will fit without overlap.  Notice that with each new circle, there will be multiple new gaps to fill.<\/li>\n<\/ol>\n

      The process should continue indefinitely, however you will eventually reach a stage in which the gaps are smaller than the width of your pencil or pen.  At that point, you can step back and admire your work.  A computer-generated Apollonian gasket is shown in the figure below.<\/p>\n

      \"Apollonian<\/a><\/p>\n

      The Apollonian gasket is a fractal that can be used to model soap bubble foam.<\/p>\n<\/div>\n

       <\/p>\n

      Because the Apollonian gasket is only approximately self-similar, there is not a well-defined scaling-dimension.  However, if you look at any \u201ctriangular\u201d section within three circles, it looks like a curved version of the Sierpinski gasket.  Recall, it requires 3 copies of the Sierpinski gasket in order to scale it by a factor of 2.  So we would expect the fractal dimension of the Apollonian gasket to be close to:<\/p>\n

      [latex]D={\\large\\frac{\\log(3)}{\\log(2)}}\\approx1.585[\/latex]<\/p>\n

      In fact, using a more general definition of fractal dimension, it can be shown that the dimension of the Apollonian gasket is about 1.3057.  This implies that the gasket is somehow closer to being one-dimensional than two-dimensional.  In turn, a foam made of large bubbles, like the froth on top of your latte is more two-dimensional than three-dimensional.  Remember this next time you get an extremely frothy drink; there\u2019s very little substance to it!<\/p>\n

       <\/p>\n

      If you would like a more detailed instructions on how to make your own Apollonian gasket, they are provided at the following website: http:\/\/www.wikihow.com\/Create-an-Apollonian-Gasket<\/a><\/p>\n\n\t\t\t

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      Candela Citations<\/h3>\n\t\t\t\t\t
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      CC licensed content, Original<\/div>