Fractals

A fractal is built by repeating a simple rule. In the Sierpinski triangle, each triangle is replaced by three smaller triangles, again and again, so the whole is made of copies of itself.

In real life: A fern leaf, a coastline and the branching of a tree all show fractal, self-similar patterns in nature.

Worked example 1

Question. In the Sierpinski triangle, one triangle becomes three smaller ones each step. How many after two steps?

Answer: After two steps there are 9 small triangles.

Nets, faces, edges and vertices

Unfolding a solid into its net helps us see all its faces at once. A cube, for example, has 6 square faces, 12 edges and 8 vertices, and its net is six squares joined in a cross.

In real life: A cardboard box is made from a flat net that is folded and glued into a solid cuboid.

Worked example 2

Question. How many faces, edges and vertices does a cube have?

Answer: A cube has 6 faces, 12 edges and 8 vertices.

Worked example 3

Question. A net has 6 equal squares joined in a cross. What solid does it fold into?

Answer: It folds into a cube.