Implicit differentiation example — Helga von Koch’s snowflake curve (1904)

HelgavonKoch snowflakel

Let us continue further our exploration of IITJEE Calculus. Especially, implicit differentiation.

Start with an equilateral triangle, calling it curve 1. On the middle third of each side, build an equilateral triangle pointing outward. Then, erase the interiors of the old middle thirds. Call the expanded curve curve 2. Now, put equilateral triangles, again pointing outward, on the middle thirds of  the sides of curve 2. Erase the interiors of the old middle thirds to  make curve 3. Repeat the process, as shown, to define an infinite sequence of plane curves. The limit curve of the sequence is Koch’s snowflake curve.

The snowflake curve is too rough to  have a tangent at any point. In other words, the equation F(x,y)=0 defining the curve does not define y as a differentiable function of x or x as a differentiable function of y at any point. We will encounter snowflake again when we study length.

Please download the attached figure.

More later…

Nalin Pithwa

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