Koch Tessellation Similarity

Geometry Level 2

If a tan snowflake has area $T$ , and a dark green snowflake has area $G$ , what is $\frac{T}{G}$ ?

4 3 5 2

1 solution

Zandra Vinegar Staff
Oct 12, 2015

Consider enveloping the two snowflakes each in a hexagon that connects the 6 primary snowflake tips. By similarity, the snowflakes fill the same % of each hexagon, therefore we simply need to find the ratio between the side lengths of the hexagons.

The bright green triangle is a 30-60-90 triangle. Therefore, the 1D ratio between the widths of the small snowflake and the large is 1: $\sqrt{3}$ . Therefore the ratio of their areas is $(\sqrt{3})^2 = 3$ .

Interestingly, this problem shows us a characteristic of fractals. If we try just to use the fractal's perimeter, without putting a hexagon around it, we will get strange results. Here's what happens: when a polygon is scaled by a factor k, its area is multiplied by k^2. If we are talking about volumes, a solid's volume would be multiplied by k^3. This works like this because we are working on a 2D (k^2) or 3D (k^3) world. With fractals, however, the relation between their perimeters and areas is not that straight forward. We could argue that since each green snowflake is surrounded by 3 tan snowflakes and each tan snowflake is surrounded by 6 green snowflakes, the perimeter of a tan snowflake is 6*(1/3)=2 times bigger than that of a green snowflake, but that is not the case. In fact, the relation between perimeter and area is defined not on a 2D world or a 3D world, but using a number called the "fractal dimension".

More information about this:

https://en.wikipedia.org/wiki/Coastline paradox https://en.wikipedia.org/wiki/Fractal dimension

A Former Brilliant Member - 5 years ago

Sorry.. i couldn't understand your solution.

Millon Das - 5 years, 1 month ago

Koch Tessellation Similarity

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