# Fitting flat shapes

NOTE: This is a very broad, open problem raising many issues. It is best approached intuitively with a low level of mathematics and plenty of discussion and diagrams. It should provide much food for thought.

Certain flat shapes can fit together on a flat surface efficiently (i.e. using the least amount of space), whereas others leave lots of gaps when we try to pack them together.

- Circular disks of radius 1cm.

- A mixture disks of radius 1cm and disks of radius 2cm.

- If you are packing up a set of disks of two different sizes, are there any particularly good combinations of radii?

- Hexagons, pentagons, squares, or triangles.

- What are the possibilities if you select a mixture of hexagons, pentagons, squares or triangles?

#### Notes and background

Whilst it might seems relatively simple, the problem of 'shape packing' is very difficult mathematically to solve with certainty. Intuitive visualisation often works just as well as a strict mathematical analysis, and often is the only sensible possibility with packing together complicated shapes

In reality, complex molecules such as proteins pack, or fold, together in very intricate ways.

Use your common sense and intuition as detailed calculations are exceedingly difficult and not really what the question is about.

Keep your eyes open for shapes in nature where flat fitting is exhibited.

Once you are aware of fitting of flat shapes you will see that examples abound in nature. It is fascinating to ask: why does this fitting of shapes occur?

### Why do this problem ?

Approximating physical quantities by idealised mathematical shapes is a commonly used tool in mathematical biology. Working with these shapes requires a good degree of skill at geometrical visualisation. By consider packing problems, students will develop this skill and see how important packing is in nature.### Possible approach

### Key questions

- How reasonable is the mathematical idealisation that you make?
- Are there any objects which are particularly well represented by a certain shape?
- Do any sorts of packing occur particularly often?