I have now noticed that the perimeter is increasing in a geometric sequence, by ⅓ of its value each time. It is also increasing by a larger amount each time, and so is a divergent series.
For example: 27 × 1⅓ = 36
36 × 1⅓ = 48
48 × 1⅓ = 64 and so on…
After considering this, I have discovered that this is connected to how I first calculated the perimeter. It is evident on the table above that I multiplied the previous perimeter by 4, then divided by 3 to find the next perimeter. So, the perimeter can be calculated by multiplying the previous one by .
27 × = 36
36 × = 48
48 × = 64
let n = shape number let p = perimeter
so, in other words,
27 × = 36, which is shape no. 2
27 × = 48, which is shape no. 3
27 × = 64, which is shape no. 4
the general rule for the perimeter is:
Area
I will now investigate how the area increases with each shape. Instead of working out the area of each shape in cm³, I will count the number of triangles in each shape, as this is more efficient.
In the first snowflake (see back of project) there were 81 triangles. The sides of the whole triangular shape are each 9 centimetres long, the area of one of the triangles can be found by squaring the length of one side ( 9 ² = 81).
I will calculate the area of each snowflake by breaking it down into individual triangle shapes. For example, shape 2 consists of one large triangle, from shape 1, and three new, smaller triangles:
The area of shape 2 can therefore be calculated by:
area of large triangle + ( 3 × area of small triangle).
In order to work out the area in this manner, I will have to first work out the area of each triangle added on and the amount of triangles added on with each shape.
This table shows how the area of each small triangle decreases with each shape:
So, I now know that the area of the smallest triangle is divided by 9 with each new shape.
This table shows how the number of new triangles in each shape increases:
From shape 2 onwards, the number of new triangles is multiplied by 4 each time. However, the first shape is an exception. I must remember this odd result later when forming equations.
Now, by combining these two tables, I am able to work out the total area added on to each shape. The area of each small triangle × the number of small triangles will give me the area added on each time.
Using my experience from working out the perimeter, I know that the area added on is increasing by with each shape. I may need to use this common ratio in equations later on.
By adding up the areas added on, I can now work out the total area of each snowflake.
There are no obvious patterns in this sequence, but the area added on is getting smaller with each shape, so it may be a convergent series. I think it may be a geometric progression, an area of maths I have not yet studied. To find out how to construct an equation, I refer to the S.M.P. 16-19 Foundation A-Level book.
I have found out that to find the rule for a geometric progression this equation can be used:
n = number of term
r = common ratio
a = first term
I must now adapt this equation to suit my sequence. The first shape’s area of 81 is irregular because, earlier in my project, I used an odd result to calculate this number. So, I must use 27, the area of shape 2’s smallest triangles, as my first term. My common ratio is , as I worked out earlier. I substituted these figures into the equation:
I then tested this equation:
This equation is not correct. I will now try adding on 81 to my equation, as previously I had totally discarded it for being an exception to the pattern. As 27 is really the second term and not the first, I will also change ‘n’ in my equation to ‘n-1’:
This equation works. It can be used to find the area of any number of snowflake.
As the area series is convergent, I can also find the sum to infinity.
Using the Foundation A-Level book again, I have found the following equation:
I can now adapt this equation to suit my sequence:
This answer of 129.6 is consistent with my previous results as the area gets closer and closer to this figure, without ever reaching it. This is evident on the graph I have drawn (see back of project).
129.6 is 1.6 times the original area of the first snowflake.
