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The Koch Snowflake
Transcript of The Koch Snowflake
...but what's a fractal? Basically, a fractal is a curve or geometric figure, each part of which has the same statistical character as the whole: self-similarity. base: a regular shape composed of line segments motif: a recurring design that "interrupts" the base The snowflake is based on a Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" by the Swedish mathematician Helge von Koch. The snowflake can be constructed from the infinite repetition of a three-step process: Starting with an equilateral triangle:
1. Divide each line segment into three equal parts
2. Draw an equilateral triange (pointing out) using the middle segment as base.
3. Remove the middle segment. Properties The Koch snowflake has some curious properties... like the fact that it has an infinite perimeter that encloses a finite area. Each "part" has three sides. Every segment is divided into four new segments, so this is how the number of sides increases: Length of each side of the snowflake (s) after n iterations: So . . . the perimeter of the snowflake after n iterations: With each iteration, a new triangle is added on each side in the previous iteration, so the number of new triangles added in iteration n is: Let a (sub zero, formatting doesn't permit) equal the area of the original triangle. Area of each new triangle added in an iteration is 1/9 of the area of each triangle of the previous iteration, so the area of each triangle added in iteration n is: So . . . the total new area added in iteration n: The total area of the snowflake after n iterations is: but what happens if, instead of an equilateral triangle, we start with a... Square? Pentagon? THE BIG QUESTION can a miniscule animal walk on it and never fall off? Actually, we answered this question in the very beginning. Let's go back, shall we? So yes, yes it can. This presentation is brought to you by: Rice CHAOS & Life Program 2013
"The Koch Snowflake" Group Ananna Anu
Christina Wang Much like the Koch snowflake, the Koch pentagon can be constructed in the same way. The three-step (infinite) process is (almost) exactly the same:
1. Divide each line segment into three equal parts
2. Draw a pengtaon (pointing out) using the middle segment as base.
3. Remove the middle segment. Properties Observations Infinite Perimeter Again, we find that the Koch pentagon has an infinite perimeter.
Let n = number of sides; l = length of each side
n = 5 · 4
l = x · 3
Perimeter = number of sides (n) x length of side (l)
The resulting equation (5x · (4/3) ), when a limit is applied, has an unbounded, infinite perimeter. a -a a a Finite Area Yet again, the Koch pentagon, much like the snowflake an the square, also possesses a finite area.
After calculations, the ratio of the area of an added triangle to its previous triangle is 4/9, because the ratio |4/9| < 1, the area converges to one value. The fractals we have observed have all been structures that are added onto the original base. However, in our weekly lectures, we observed a different kind of fractals where we remove a portion of the base in each iteration. Explanation: the "outside" construction type of fractals are based on the Koch curve and principle. However, the ones we observed in class were... "Sierpinski" fractals, which play on the principle of removing a certain portion of a base's area. So if we wanted to make the "opposite" of a Koch pentagon, it would be a Sierpinski pentagon, and it would look something like this... Sierpinski fractals are actually quite interesting to look at: For even more information about fractals and their implications, PBS Nova has an episode entitled, "Fractals - Hunting the Hidden Dimension." Great. A finite area bounded by an infinite perimeter. Because this is obviously has relevence in real life . . . Actually, it does. Practical applications in cartography: How big must a variation be to be considered? Compare Norwegian fjords to the relatively smooth beaches of Nha Trang. Coastline Paradox Measure a coastline in kilometers. Then meters. Then centimeters. Then millimeters. Smaller and smaller units - how small can you get? The Koch Snowflake- Zooming in on the fractal Well, first... what would it look like? Does it have an infinite perimeter? A finite area? Yep. Koch had a cool beard. Infinite Perimeter Finite Area First you have a side of length a.
When you attach a square, the new length is a + (2/3)a.
Next you attach 3 little squares of side a/9 to the 3 sides of length a/3.
The perimeter increases by an amount equal to the lengths that are perpendicular to the old sides, so this time you add 6 times a/9. This is (2/3)a.
It’s the same every time: you always increase the perimeter by 2/3 of a.
The perimeter increases by (2/3)a every time you add smaller squares. This could be written in a series form. However, this series diverges 2/3 + 2/3 + 2/3 + 2/3 +... = infinity
In other words the lines marked in blue always add up to (2/3)a.
On top, both the blue lines are a/3 and there are two of them. In the next figure the short lines are a/9 and there are 6 of them, so it’s 2a/3 again. A finite area of a Koch square is actually quite easy to prove. This has a finite area. So what keeps this making sense?
The concept of a limit! Let's put a Koch square in it. Area of Koch Square < Finite Area of Rectangle
Therefore... According to logic: the area of a Koch Square is finite.