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Fractals

What is a fractal? A fractal is infinitely detailed and displays self-similarity. It has a similar, although not usually identical, appearance at any magnification.

Sierpinski Triangle

This is perhaps the most well-know fractal accessible to pupils in Key Stages 3 and 4.

Simple instructions for creating a Sierpinski triangle,

Many questions arise while constructing the Sieprinski triangle:

What is the additional shaded area at each stage?

What fraction of the entire shape is shaded at each stage?

Could you develop a formula which would calculate the total fraction which is shaded at the nth stage of the triangle's construction?

 

Koch's Snowflake

Koch's Snowflake after 4 stages

 

First draw at least 4 stages of the Koch snowflake -see how.

What is the perimeter of the snowflake at stage 1? At stage2?

Work out the perimeter of the snowflake at each stage.

What will the perimeter be after n stages?

 

What about the area of the snowflake - what is the area at each stage? Can you develop a formula which will work out the area after n stages?

Hint: work with fractions, not decimals

How does the Koch snowflake illustrate the concept of infinity?

 

Compound Area

With acknowledgements to Richard Goodman, whose article in MT, March 2000 was inspirational.

1.Working on 0.5cm squared paper, draw an 8 x 8 square (olive green) in the middle of the page.

2.On each side of the square draw an isosceles right-angled triangle (pink) so that the hypotenuse is the side of the square.

3.On each exposed side of the triangle draw a square (orange).

4.On each exposed side of the square draw a triangle (purple) as before.

5. Repeat steps 3 and 4 for as long as you can.

How long can you continue?

What is the area of each new shape you add?

What is the total area of the shape at the end of each stage?

What is the total area after the completion of the nth stage?

If you were to continue this process indefinitely, what shape would you end up with?

Could you prove this?

 

Paper Folding

The Stair-Step fractal

The Stair-Step fractal is easy for pupils to create using just a sheet of A4 paper, a pair of scissors and a ruler.

Click here for detailed instructions for the creation of a Stair-Step fractal.

Questions that could arise:

What is the added 'volume' of the Stair-Step at each stage of its creation?

What is the 'volume' after 4 stages?

What is the total volume at the nth stage of the staircase?

 

Koch's Snowflake

 

A 3-D fractal

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