Hidden dimensions of fractal dimension

hidden dimensions of fractal dimension

Try to measure the thread length of the coastline of England of the atlas. Then do the same with marine cartography. Interestingly, the second time will go much more. If you then go to England and measure its coastline with carpentry meter, this length will be even greater. Continue this process until in your hands not be drawing yardstick by which you can measure shoreline piece of particle, atom after atom. Mandelbrot is often used this example, arguing that the coastline of England has infinite length.

The meaning of this impractical experiment consists in that the distance must be comparable in scale, location and details. Mandelbrot later determined that the fractal dimension of coastline of England 1.25.

In 300 BC The Book of Euclid began with some definitions, which were:

  1. point is that no parts.
  2. line is length without width.
  3. surface is that which has only length and width.

    In Book XI, he added:
    Bulk

  4. figure is what has length, width and height.

    dimensional concept has been highlighted in these definitions. Everyone knows that the point has 0 size, the line has dimension 1 is a two-dimensional square and cube – a three-dimensional. This dimension is called topological .
    It is used over the millennia, but it appears incorrect in the study of fractals.

    One of the ideas arising from fractal geometry was the idea of ​​non-target values ​​for the number of dimensions in space. Mandelbrot called non-target dimensions such as fractal dimensions 2.76 .

    Peano

    Explore fractals, called Curve paean . Upon its creation is necessary to start with leg and replace it with this figure:

    Then, each of the segments is replaced by the same figure, and repeating this process indefinitely, we get square. Follow the slideshow below to see this process. Did you
    what’s the problem? A fractal consists of lines, which is the topology of the printed dimension 1. But this is inaccurate because the figure – a square with dimension 2.

    F finite number, greater
    0
    I infinite number
    dimension Quantity
    Points Length Area Volume
    D = 0 F 0 0 0
    0 I 0 0 0
    D = 1 I F 0 0
    1 I I 0 0
    D = 2 I I F 0
    2 I I I 0
    D = 3 I I I F

    We can not use the topology of the printed dimensions of fractals, but instead should be used etc. fractal dimension .
    In general, the formal definition of fractal is: this is a figure whose fractal dimension is greater than the topology of the printed. It is amazing that the fractal dimension must be an integer and can be fractional.
    As can be seen from the table to the left, the complexity of figure uvelichiva with dimensionality. How can accurately calculate the fractal dimension can be seen here .

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What is a fractal?

6 Responses to “Hidden dimensions of fractal dimension”

  1. стиано says:

    А за времето нещо то не е ли 4аст от формацията на фракталите имам предвит 4е е нужно време да се “”копира фрактал”" може ли да се из4исли или това са само геометри4ни фигури които нямат нищо общо с времето

    • Vanya says:

      Не са статични геометрични фигури. Както разбрахте, те се образуват с итерации и се развиват във времето. Има теории, че и времето е фрактално ;)

  2. morsono says:

    Здравейте, чудесен сайт! Имам въпрос, свързан с твърдението, че бреговата линия на Англия има безкрайна дължина. Тогава всяка линия, освен ако не е идеална, има безкрайна дължина. Така ли да разбирам?

  3. Rumi says:

    Невероятен сайт. Благодаря!!! :)

  4. В.Диков says:

    По отношение на “безкрайната” дължина на бреговата линия на Англия, аз мисля, че този израз не е коректен. Според мен, промяната на точността на измерване води до все по- добро приближаване на резултата до истинската стойност на дълижината на тази мислена линия. В математиката този проблем е известен като граница, към която клони сойността на даден израз, когато броят на членовете на редицата расте!
    В конкретния случай – точността на измерване в метри ще се коригира, ако измерваме в сантиметри, милиметри и т.н., но никога няма да стане БЕЗКРАЙНО голямо число!

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