Find Area of Snowflake

  • Thread starter Sczisnad
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In summary: Therefore, the area is equal to 1/2 + 2/3 + 3/4 + ... which is a divergent series. This means that the area is infinite and cannot be calculated.
  • #1
Sczisnad
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http://imageshack.us/photo/my-images/835/mathproblem.jpg/

I was thinking of an interesting shape (I drew it in paint for help), and if it would be possible to find the area of it. I started to write a series to represent the area but ran into trouble because I think that parts of the snowflake might overlap.

the formula that I started to work on is incomplete: 1 + 3/4 + [(Σ,∞,n=2) 12(3^(n-2))+...(incomplete)]

The "12(3^(n-2))" is the number of boxes in that layer, I was going to multiply that by the area of the individual boxes in that layer and then subtract the overlap. But that is where the problems might be.
 
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  • #2
With what rate does the size of the boxes decrease? Your first box is 1x1, what size do the four boxes attached to that have? And what size do the boxes attached to that have?
 
  • #3
opse sorry, when i was calculating it, I used a factor of 1/2 for shrinking the squares. However if that causes the squares to overlap I would like to find out what the minimum rate of size decrease is.
 
  • #4
Sczisnad said:
opse sorry, when i was calculating it, I used a factor of 1/2 for shrinking the squares. However if that causes the squares to overlap I would like to find out what the minimum rate of size decrease is.

No, the squares won't overlap. However, I do expect that the entire plane get's filled this way and that the area is equal to 4.

Let's calculate this. First, let's see how many squares we have to add at each step:

At step 1, we have 1 square
At step 2, we add 4 squares
At step 3, we add [itex]3\cdot 4=12[/itex] squares
At step 4, we add 324 squares
At step n, we add 3n-24

Now let's see what area we add every step:

At step 1, we add an area of 1.
At step 2, for each square we add, we will add an area of 1/4. However 1/4 of that area will overlap with our first square. So we only add an area of 1/4-1/16=3/16. We have 4 squares to add so we get 3/4.
At step 3, for each square we add, we will add an area of 1/16. However, 1/4 of that area will overlap with our original squares. So we only add an area of 1/16-1/64=3/64. We have 12 squares to add, so we get 9/16.
At step n, we add a square of dimensions 21-n. So we add an area of 22-2n. However, 1/4 of that area will overlap with the original squares. So we only add an area of

[tex]\frac{1}{2^{2n-2}}-\frac{1}{2^{2n}}=\frac{3}{2^{2n}}[/tex]

We have 3n-24 squares to add, so we add

[tex]\frac{3^{n-1}}{2^{2n-2}}[/tex] area.

So eventually we get the following sum:

[tex]\sum_{n=1}^{+\infty}{\frac{3^{n-1}}{2^{2n-2}}}=\frac{4}{3}\sum_{n=1}^{+\infty}{\left(\frac{3}{4}\right)^n}[/tex]

Evaluating this gives an area of 2. Like expected.

Try to modify this argument if you add a square with a smaller area each time.
 
Last edited:

What is the typical method for finding the area of a snowflake?

The most common method for finding the area of a snowflake is to use a microscope or magnifying glass to measure the length of each side of the snowflake. Then, using the formula for the area of a regular hexagon, which closely approximates the shape of a snowflake, the area can be calculated.

Can the area of a snowflake be accurately measured?

Due to their intricate and unique shapes, it is difficult to accurately measure the area of a snowflake. However, using advanced imaging techniques and mathematical algorithms, scientists have been able to estimate the area of snowflakes with a high degree of accuracy.

How does the area of a snowflake affect its melting rate?

The area of a snowflake does not directly affect its melting rate. However, larger snowflakes may take longer to melt due to their larger surface area, which allows for more heat to be absorbed before the snowflake completely melts.

Can the area of a snowflake be used to determine its age?

No, the area of a snowflake cannot be used to determine its age. While snowflakes form in a predictable pattern, the rate at which they grow and accumulate layers is heavily influenced by environmental factors such as temperature and humidity.

Is there a relationship between the area of a snowflake and its shape?

Yes, there is a relationship between the area of a snowflake and its shape. Generally, larger snowflakes have more complex and intricate shapes, resulting in a larger area. However, this relationship is not always consistent and can vary depending on the environmental conditions during snowflake formation.

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