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Summary on fractals
Topic: Fractals
by Amelia, 2018 Cohort
Note: This entry was created in 2018, when the task was to “summarise a key reading”, and so may not represent a good example to model current primer entries on.
Fractals refers to both fractal art and the mathematical concept of fractals. Fractal concepts have been discussed and published in academic writing since the 17th century, but it wasn’t until the 20th century that Benoit Mandelbrot first published the word fractal, where he applied fractal dimensions to geometric patterns, specifically in nature. The precise definition for the mathematical concept is widely disagreed upon, however a common definition for a geometric fractal, which was posed by Mandelbrot is: “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.
Wikipedia notes that all definitions seem to contain the key characteristics of infinite self-similarity, iteration and a high level of detail. To explain self-similarity and iteration, imagine zooming in on a man with a man inside his head, and no matter how much you zoom it looks the same or pretty much the same- this isn’t a fractal because it isn’t detailed enough but displays self-similarity and iteration. It is also important to note that fractals are not limited to geometric patterns, for example they can also be used to describe space and
Another key feature of a fractal is its dimensions. For example, take a Koch snowflake, as pictured. An example of a naturally occurring fractal is the Koch snowflake which displays all the above characteristics but I will also use it to discuss the complex nature of fractal dimension. As you can imagine by looking at the final picture of the snowflake if this pattern were to repeat infinitely, there would be infinitely many points. This means that you cant describe the fractal by saying where each point lies, and must instead discuss the relation to other pieces of the fractal. In contrast to the infinite perimeter of the shape, the area remains strictly limited. To understand this, picture a circle drawn around the snowflake you know that the fractal will never expand beyond this. It should also be noted that this pattern would replicate in a 3D fractal with the volume being restricted, while the surface area is infinite.
Fractals, such as the Koch snowflake exist throughout nature and fractals pattern can be implemented within scientific research to make new discoveries. For instance, fractal patterns in leaves are being used to determine how much carbon is contained in trees the specifics of how this is done are too complex for the scope of this primer. As well as Fractal patterns being used to assess the authenticity of Jackson Pollock works, as they were produced by pouring paint directly on a canvas and have distinct fractal patterns, unable to be replicated by robots.
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If there are inaccuracies, or opportunities for significant improvement on this topic, feedback is welcome on how to improve the resource.
You can improve articles on this topic as a student in "Unravelling Complexity", or by including the amendments in an email to: Chris.Browne@anu.edu.au
Note: This entry was created in 2018, when the task was to “summarise a key reading”, and so may not represent a good example to model current primer entries on.
Fractals refers to both fractal art and the mathematical concept of fractals. Fractal concepts have been discussed and published in academic writing since the 17th century, but it wasn’t until the 20th century that Benoit Mandelbrot first published the word fractal, where he applied fractal dimensions to geometric patterns, specifically in nature. The precise definition for the mathematical concept is widely disagreed upon, however a common definition for a geometric fractal, which was posed by Mandelbrot is: “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.
Wikipedia notes that all definitions seem to contain the key characteristics of infinite self-similarity, iteration and a high level of detail. To explain self-similarity and iteration, imagine zooming in on a man with a man inside his head, and no matter how much you zoom it looks the same or pretty much the same- this isn’t a fractal because it isn’t detailed enough but displays self-similarity and iteration. It is also important to note that fractals are not limited to geometric patterns, for example they can also be used to describe space and
Another key feature of a fractal is its dimensions. For example, take a Koch snowflake, as pictured. An example of a naturally occurring fractal is the Koch snowflake which displays all the above characteristics but I will also use it to discuss the complex nature of fractal dimension. As you can imagine by looking at the final picture of the snowflake if this pattern were to repeat infinitely, there would be infinitely many points. This means that you cant describe the fractal by saying where each point lies, and must instead discuss the relation to other pieces of the fractal. In contrast to the infinite perimeter of the shape, the area remains strictly limited. To understand this, picture a circle drawn around the snowflake you know that the fractal will never expand beyond this. It should also be noted that this pattern would replicate in a 3D fractal with the volume being restricted, while the surface area is infinite.
Fractals, such as the Koch snowflake exist throughout nature and fractals pattern can be implemented within scientific research to make new discoveries. For instance, fractal patterns in leaves are being used to determine how much carbon is contained in trees the specifics of how this is done are too complex for the scope of this primer. As well as Fractal patterns being used to assess the authenticity of Jackson Pollock works, as they were produced by pouring paint directly on a canvas and have distinct fractal patterns, unable to be replicated by robots.