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Fitting Flat Shapes

Stage: 5 Challenge Level: Challenge Level:1

 

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?
  1. Circular disks of radius 1cm.
     
  2. A mixture disks of radius 1cm and disks of radius 2cm.
     
  3. If you are packing up a set of disks of two different sizes, are there any particularly good combinations of radii?
     
  4. Hexagons, pentagons, squares, or triangles.
     
  5. 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.