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proofmathisbeautiful: Because it’s that time of year again!! Also…this is a reblog of one of my very first posts on Proof! :) The Von Koch snowflake! The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows: 1) remove the middle third of each side, 2) build a new equilateral triangle on each of the resulting gaps, 3) repeat steps 1 and 2 for the new object. If you keep going indefinitely, you end up with the fractal shown on the right. The outline of the snowflake is incredibly crinkly. In fact, it does not contain any straight line pieces at all: the middle third of any piece of straight line has been replaced by two sides of a triangle, creating a spike. Mathematicians measure the crinkliness of a fractal by the fractal dimension, a generalisation of our ordinary notion of dimension. The outline of our snowflake is too crinkly to be one-dimensional. On the other hand, it clearly is not two-dimensional either, since it contains no area. In this case, the dimension lies between 1 and 2, in fact it is equal to log(4)/log(3) = 1.2619. (via proofmathisbeautiful) Another cool point to consider: this snowflake has infinite length, even though it is confined to a finite two dimensional space :-)

proofmathisbeautiful:

Because it’s that time of year again!!

Also…this is a reblog of one of my very first posts on Proof! :)

The Von Koch snowflake!

The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows:

1) remove the middle third of each side,
2) build a new equilateral triangle on each of the resulting gaps,
3) repeat steps 1 and 2 for the new object.

If you keep going indefinitely, you end up with the fractal shown on the right.

The outline of the snowflake is incredibly crinkly. In fact, it does not contain any straight line pieces at all: the middle third of any piece of straight line has been replaced by two sides of a triangle, creating a spike. Mathematicians measure the crinkliness of a fractal by the fractal dimension, a generalisation of our ordinary notion of dimension. The outline of our snowflake is too crinkly to be one-dimensional. On the other hand, it clearly is not two-dimensional either, since it contains no area. In this case, the dimension lies between 1 and 2, in fact it is equal to log(4)/log(3) = 1.2619.

(via proofmathisbeautiful)

Another cool point to consider: this snowflake has infinite length, even though it is confined to a finite two dimensional space :-)

Source: proofmathisbeautiful

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    Repeating pattern!!!! Fractal yeah!
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    Have to love a subject where every snowlflake...mathematical wonder.
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