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Teachers Notes This is a brief but very interesting look, at the Von Koch Snowflake Curve. After introducing the curve and discussing its generation, the students are simply asked to derive the perimeter formula for nth iteration. We then move on to discuss the curve’s finite area and reveal, (by reference to the formula) its infinite perimeter. Students are encouraged to generate a spreadsheet from the formula for the first 50 terms in the sequence to convince themselves of the infinity of the perimeter. (Spreadsheet is at slide 16). There is a printable worksheet if needed at slide 18 (some students may wish to make jottings/notes on it). If you want to extend still further for the very able, then you might wish to see if they can work out the area A n for the nth iteration.

The Von Koch Snowflake

1 3 L 1 3 L 1 3 L The curve is generated from an equilateral triangle by trisecting the sides and constructing this smaller equilateral triangle on each of the sides. This is then repeated ad infinitum. P 0 = L The Von Koch Snowflake

Thinking about the increased length of this side, what will the first new perimeter, P 1 be? 1 3 L 1 3 L 1 3 L P 0 = L P 1 = 4 3 L The Von Koch Snowflake

1 3 L 1 3 L 1 3 L Derive a general formula for the perimeter of the n th curve in this sequence, P n. P 1 = 4 3 L P 0 = L P 2 =( ) L The Von Koch Snowflake

Derive a general formula for the perimeter of the n th curve in this sequence, P n. P 1 = 4 3 L P 0 = L P 2 =( ) L P 3 =( ) L P n =( ) n 4 3 L The Von Koch Snowflake

The area A n of the nth curve is finite. This can be seen by constructing the circumscribed circle about the original triangle as shown. P 1 = 4 3 L P 0 = L P 2 =( ) L P 3 =( ) L A0A0 A1A1 A2A2 A3A3

It is a surprising fact therefore that the perimeter of the curve is infinite. A0A0 A1A1 A2A2 A3A3 P 1 = 4 3 L P 0 = L P 2 =( ) L P 3 =( ) L P n =( ) n 4 3 L

Whatever fixed value you care to make the perimeter of any curve in the sequence it can always be exceeded by choosing a large enough value for n. A0A0 A1A1 A2A2 A3A3 P 1 = 4 3 L P 0 = L P 2 =( ) L P 3 =( ) L P n =( ) n 4 3 L

Use a spreadsheet to compute the first 50 values for the perimeter. Set P 0 = 1. A0A0 A1A1 A2A2 A3A3 P 1 = 4 3 L P 0 = L P 2 =( ) L P 3 =( ) L P n =( ) n 4 3 L

P01 P11.333P P21.778P P32.370P P43.160P P54.214P P65.619P P77.492P P89.989P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P

The Von Koch Snowflake The perimeter of the Von Koch Snowflake Curve is infinite. Just as the coast line of the UK is infinite. The smaller the ruler that you use to measure the coast line, the longer it becomes. Coast line  miles