Mumbai university > Comp > SEM 4 > Computer Graphics

Marks: 10M

Year: May 2015

1. The Koch snowflake (also known as the Koch curve, star, or island) is a mathematical curve and one of the earliest fractal curves to have been described.

2. A Koch curve is a fractal generated by a replacement rule. This rule is, at each step, to replace the middle 131/3 of each line segment with two sides of a right triangle having sides of length equal to the replaced segment.

3. This quantity increases without bound; hen

4. ce the Koch curve has infinite length.

5. However, the curve still bounds a finite area.

6. We can prove this by noting that in each step, we add an amount of area equal to the area of all the equilateral triangles created.

Construction:

1. The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

   • Divide the line segment into three segments of equal length.

   • Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.

• Remove the line segment that is the base of the triangle from step 2.

• After one iteration of this process, the resulting shape is the outline of a hexagram.

1. The Koch snowflake is the limit approached as the above steps are followed over and over again.

2. The Koch curve originally described by Koch is constructed with only one of the three sides of the original triangle.

3. In other words, three Koch curves make a Koch snowflake.