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Fractal card

By

David Shaw
,

A piece of paper, folded in half, and with a cut to the middle. One side of the cut is folded up.

Fold one of the flaps.

Written by Azmina Hossain

Make a pop-up greeting card with a beautiful fractal pattern.

You will need

  • 2 different coloured A4 pieces of cardboard
  • Ruler
  • Pencil
  • Scissors
  • Glue stick

What to do

The card is partially unfolded, hald the paper sticks out and the other half goes in.

Push the flap inside the card.

  1. Fold one piece of card in half.
  2. Mark a point half way along the fold. Starting from that point, rule a line perpendicular to the fold 75 mm long, or half the width of the folded card.
  3. Cut along the line. You will have two flaps – fold one of them so the original fold meets the far side of the card.
  4. Unfold the card and push the flap inside the card as in the diagram. This is the key step in creating the fractal design.
    Paper folded tol look like a four step staricase.

    Cut and fold a second time.

  5. Fold the card flat. This time it should look like a staircase with two steps.
  6. Mark a point halfway along each fold. Rule lines coming in from these points 37 mm long, or half the height of each step.
  7. Cut along each line. Fold the lower flaps over so the card takes the shape of a staircase with four steps.
  8. Unfold the card and push these flaps inside the card as in the diagram.
  9. Repeat steps 6-8, but this time you will need to make four cuts, each 19 mm long. The card should take the shape of a staircase with eight steps.
    A pop-up card with triangles made of triangles.

    The completed card.

  10. Repeat this process, but this time make eight cuts, each 10 mm long.
  11. Fold the flaps inside the card as in the diagram below. A good way to check if you have folded the flaps correctly is to fold the card flat. It should look like a sixteen step staircase, and when unfolded it will look like a triangle made of triangles!
  12. To turn your shape into a greeting card, fold it flat and put glue on the outside. Fold the second piece of card and slip it around the outside of the first sheet. When you open it, there will be a beautiful pop-up pattern inside your card.

What’s happening?

A fractal is a shape that has self-similarity. That means that one part of the shape looks very similar to the whole shape. In this activity, the shape you made looks like a triangle made of triangles. Each of those triangles looks very similar to the whole triangle.

This shape is known as Seirpinski’s triangle. Sierpinski’s triangle turns up in lots of surprising places. It can appear when calculating probabilities, expanding binomial expressions, running simple programs known as cellular automata, or in the solution to the Towers of Hanoi puzzle. The fact that it appears in different areas of mathematics is a clue to the deep connections between them.

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