Snowflake Investigation Report

1. INTRODUCTION

Snowflakes, feathery ice crystals that typically display a delicate six-fold symmetry might be the most distinguished mathematical art in the world, from the center of the snowflake, to the outskirts of their anatomy. When inspected, one notices that the different limbs of the snowflakes are mathematically constructed of fractal triangles. With that said, the aim of my exploration is to generate an equation that will aid me in representing a snow flake of any size, in order to do so I would need to explore specific areas of mathematics such as: symmetry, graphical representations, sequences and series and many more. As a consequence I will be focusing on the mathematical concept of: fractal triangles, in order construct my

. .

And so on to u_n

2. FORMULA GENERATION

(A) Perimeter formula generation

When looking at the general shape of the generated snowflake, I noticed that the perimeter of the snowflake keeps on increasing from u_1 until u_nwhich is essentially u_∞ at a constant rate, consequently having an infinite perimeter.

Recursive Sequence u0 u1 u2 u3 u4 u5 un l 4l3

16l9

64l27

256l81

1024l243

As described earlier, with the original segment l, it is then divided into 3 to give l/3, but after I inscribed the new segments it equally took the length of l/3 because it forms and equilateral triangle resulting in:

But now the new length with the equilateral triangle formed=l/3×4 because, I now have a line with 4 segments thanks to the 2 inscribed lines, consequently ∑▒l=(4l)/3

So, if the original triangle, u_0= l. Then, u_1 with an additional equilateral triangle = 4/3×u_0 because the new shape formed will be 4/3 bigger, consequently the sequence will be generated as

If A is the Area of the resulting Snowflake then

A=A_0+3/9 A_0+12/81 A_0+48/729 A_0…

A=A_0+(3(4)^0)/9 A_0+(3×4)/((9)^2 ) A_0+(3×(4)^2)/((9)^3 ) A_0…

Where (3×4^n) is the number of triangles where n≥0

A=A_0 [1+(3(4)^0)/9+(3×4)/((9)^2 )+(3×(4)^2)/((9)^3 )…]

A=A_0 [1+∑_(k=1)^n▒(3×4^(k-1))/9^k ]

But ∑_(k=1)^n▒(3×4^(k-1))/9^k is a Convergent Geometric series,

U_1=3/a_1 r=4/a_1 ∑_(k=1)^∞▒(3×4^(k-1))/9^k =(3⁄9)/(1-4⁄9)

∑_(k=1)^∞▒(3×4^(k-1))/9^k =8/5

∴A=A_0 [1+3/5]

A=8/5 A_0

But A_0=√3/4 l^2

A=(8/5)(√3/4 l^2)…(2/5)(√3/1 l^2)

Finally A=(2√3)/5 l^2

The equation above is the Area of a Snowflake. I can conclude that a Snowflake has a finite Area.

(C) Relating The Area To That Of A Circle

As a Snowflake grows it gets circular and turns to a circle. So I then thought of comparing the Areas of both the Snowflake and that of a circle in order to verify the similarities between what I observed while working on the Snowflake with a