Fitting flat shapes
How efficiently can various flat shapes be fitted together?
Age
16 to 18
Challenge level
Problem
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.
For the following types of shape, try to visualise how best to cover a surface with them, without overlap, so as to leave the smallest amount of gap. Are there several equally good ways of fitting together each type of shape, or is there a clear winner? Can you draw a clear diagram or give a clear explanation of your packing mechanism in each case?
- 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?
Can you think of biological situations in which the surfaces 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?
Bear in mind packing as you study more biology. Where is packing clearly manifested? Are there good scientific reasons why this would be the case?
Supplementary activity: Try the NRICH problem semi-regular tessellations . It has a lovely interactive tool which allows you to experiment with tessellations of regular shapes
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 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?
Teachers' Resources
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
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 packing 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
- 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?
Possible extension
Can students think of good evolutionary or chemical reasons
for the shapes and packings that certain organisms take?
Are there situations in which efficient packings might be
particularly helpful or particularly unhelpful?
How is symmetry important in packings?
Possible support
Some students might need cut-outs in order to experiment with
the packing possibilities. Some students might also struggle with
the 'open' nature of the question, as there is no 'complete'
answer. To begin, they might like to read the
Student Guide to Getting Started with Rich Tasks.