# Packing 3D shapes

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

## Problem

For the following solids, try to visualise how best to pack lots of them together so as to use up the least space. Can you draw a clear diagram or give a clear explanation of your packing mechanism in each case?

- Spheres
- Circular-based cones of a fixed base radius and height. Do the relative values for these two measurements affect your packing strategy?
- Long chains of spheres connected flexibly by rigid rods 0-0-0-0-0- (like a long string of beads). How do the sizes of the rods/spheres affect the results

Can you think of biological situations in which the shapes of real objects are closely approximated by some of the shapes described?

Can you think of biological situations in which shapes pack together in the ways described here?

What about situations in which shapes do not pack together in these ways?

#### 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..

## Getting Started

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 3D packing is exhibited

Once you are aware of 3D packing you will see that examples abound in nature. It is fascinating to ask: why does this packing of shapes occur?

Keep your eyes open for shapes in nature where 3D packing is exhibited

Once you are aware of 3D packing you will see that examples abound in nature. It is fascinating to ask: why does this packing of shapes occur?

## Teachers' Resources

### Why do this problem ?

This problem will allow students to engage with visualisations of 3D shapes. By exploring the shapes present in nature, students will begin to develop their skills of geometrical reasoning in a concrete setting.### Possible approach

This question could be posed individually or for group discussion. This problem also works effectively when students are given time to reflect on the question and look for packings in nature. Ask the question and let students consider it over, say, a week. What shapes and packings have they noticed in nature? Could they find any images to share? Then consider the questions of efficient packings. This results might make an effective display.### Key questions

- Can you clearly describe the underlying mathematical shapes in words?
- Can you describe your method of packing the shapes clearly in words? Can you draw an effective diagram?
- How many different sensible packing methods might you try for different shapes?

### Possible extension

Can you consider the efficiency of the various packings? (i.e. roughly what percentage of space is taken up by the packed shapes)### Possible support

Provide physical shapes for the students to manupilate.

Students might struggle with the 'open' nature of the
questions. To begin, they might like to read the
Student Guide to Getting Started with Rich Tasks