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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Packing 3D Shapes

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

 

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?

  1. Spheres
  2. Circular-based cones of a fixed base radius and height. Do the relative values for these two measurements affect your packing strategy?
  3. 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..