Transition Year Maths Koch Snowflake


The Koch Snowflake is a fractal shape which has an infinitely long outline but encloses a normal finite space, first described by Helge von Koch in 1904. Draw a simple equilateral triangle. On each of the three sides place another equilateral triangle exactly one third the size and in the middle of the side. The length of each side is now four thirds of the original length. Now continue the same process with each of the new sides. Every time, the overall length of the shape is four thirds times the previous. This can be done forever making the shape infinitely long but the area of the shape will never grow beyond the bounds of the hexagon (no. 2 below).


1. 2. 3. 4. 5. NOTE Coastline of Ireland


How long is the coastline of Ireland? If we used a 10km long measuring stick to measure it, we might get a length of 2,500km. However if we were to ‘zoom in’ closer to the coastline and measure it with a 1km long measuring stick we might get a length of over 15,000km. As we measure it in more and more detail and go in and around every cove it will grow to 25,000km. If we look close enough we can make it 100,000km or any length we wish.


The reason for this is that a coastline is also fractal in nature and therefore the length of the outline will depend on the measuring stick (or unit of length) that we use to measure it. Like the Koch Snowflake the coastline of Ireland has a potentially infinite length but it encloses a finite area.


Infinity


Infinity is an idea that can cause problems in mathemathics due to its paradoxical nature. If we try to imagine an enormous number that comes close to infinity, there are still more than this number of numbers between 0 and 1.


The number infinity is a paradox. Call the number NI. This number has some great properties: NI + 1 = NI


NI × 2 = NI Hilbert’s Hotel


David Hilbert, a German mathematician (1862 – 1943), came up with the concept of Hilbert’s Hotel. Hilbert’s Hotel has infinitely many rooms, and it is entirely full. If one more guest arrives, where can he be put? The management simply moves each guest to the next room, the guest in room 1 moves to room 2 etc. No one has to leave, so the latecomer gets room 1. If an infinitely large coachload of guests arrive, can they


158 NI × NI = NI With these properties in mind let’s look at Hilbert’s Hotel. NINI = NI.


A fractal is a never-ending pattern that repeats itself at different scales. Although they are very complex they are created by repeating a simple process. Fractals appear everywhere in nature from sea shells to snowflakes to swirling galaxies. Fractals differ from the usual Geometric patterns you will encounter. With normal geometric patterns (like the ones in Chapter 5) the scale of the changes in the pattern is constant. With Fractals the patterns are much more complex and often the pattern replicates at every scale.


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