koch snowflake zoom

It's possible to continuously zoom into a fractal and experience the same behavior.The Koch curve is named after the Swedish mathematician Here is an animation showing the effect of zooming in to a Koch curve.The typical way to generate fractals is with recursion.

Now that we know G, we can work out 4/9ths of G too.We can subtract the 4/9ths of G from G, and you can see that all the terms, except the first, of the infinite series cancel:This gives an equation for 5/9ths of G which we were able to rearrange to find the total of the infinite series G.Finally, now that we have a value for G, when can substitute this back into the original formula to give an equation for the infinite sum. The Rule: Whenever you see a straight line, like the one on the left, divide it in thirds and build an equilateral triangle (one with all three sides equal) on the middle third, and erase the base of the equilateral triangle, so that it looks like the thing on the right.

The best GIFs are on GIPHY. If you’d like to explore more fractals, there are more listed One of the “classic” fractals is the Koch snowflake, named after Swedish mathematician Helge von Koch (1870–1924). The areas enclosed by the successive stages in the construction of the snowflake converge to The Koch snowflake can be constructed by starting with an The Koch snowflake is the limit approached as the above steps are followed indefinitely. (I’ll drop the red coloring now that you get the idea. Let's keep with the notation that the length of the side of initial triangle is The area of the first iteration is simply the area of the base triangleFor the next iteration the area of the snowflake is increased by the three red triangles shown in the diagram below(Three new triangles with sides of length s/3):For the next iteration we add an additional 12 smaller triangles. In order to find the sum, it helps if we clean this up a little.Then we can pull out any additional 1/4 from the bracket (in order to multiply each term inside by four):Using the following equality (flipping the exponents), we can move the square to the denominator and the increasing power to the numerator.Now all the exponents are the same order and we can combine them and pull the 4 inside each of the terms. It looks like the most intricate snowflake that ever fell out of the sky.Here’s an interesting question: What is the perimeter of the Koch Snowflake? Let's consider what happens with each iteration (and we only need to consider one side; to get the total, we simply multiply by three).Each time we step down, the length of each side is replaced by What about the area? Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (Squares can be used to generate similar fractal curves. The Koch Snowflake fractal is, like the Koch curve one of the first fractals to be described.

The Koch Curve has the seemingly paradoxical property of having an infinitely long perimeter (edge) that bounds a finite (non-infinite) area.

But depending on the thickness of your drawing utensils and how big your first iteration is, you can draw one of the 5 th or even 7 th order.

The Koch snowflake is one of the earliest fractal curves to have been described.

But depending on the thickness of your drawing utensils and how big your first iteration is, you can draw one of the 5 th or even 7 th order. The Koch Snowflake is a fractal based on a very simple rule. This fact is really mind-boggling when you consider that the Koch Snowflake has a finite area.

The progression for the area converges to 2 while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve. Even though we know the length of the all the line segments is increasing with each step, it's looking like the area is not getting that much bigger with successive terms? In the diagram below, I have added a circle around the snowflake. It can be seen by inspection that the snowflake has a smaller area than the circle as it fits completely inside it. (You could try this on other shapes, too, but I’ll leave those variations for you to explore.) This is a little more complicated to calculate.First let's consider what happens to the number of sides.When we first start out, there are 3 sides to the triangle, each of length one unit.On the next iteration, there are 12 sides, each of length 1/3 unit (Each of the three straight sides of triangle is replaced with four new segments). Other great examples are snowflakes and ice crystals: To create our own fractal snowflake, we once again have to find a simple procedure we can apply over and over again.