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'Interpenetrating Solids' printed from http://nrich.maths.org/
This problem involves
visualisation and rotation of cubes. A significant and important
part of any visualisation problem is understanding clearly what the
problem is asking of you and then thinking about the key features
of the problem, so please be prepared to spend some time thinking
about what the problem is about before putting pen to paper. Don't
forget that sketches, diagrams and models are to be encouraged as
they will help you as you work towards the solution
Imagine that you place a cube on a flat table. You rotate the cube
45 degrees about the axis joining the centre of the top face to the
centre of the bottom face.
Imagine that the original and rotated cube are superimposed.
Imagine also the the resulting shape was sliced along each of the
original and rotated faces. How many pieces would the shape fall
into and what shapes would they be? Describe the shapes as
accurately as possible.
Investigate the shapes you would get by rotating the cube by
different angles about this axis.
How many rotations give a result where the original and rotated
cube are in exactly the same location?
Now visualise the same process except with rotations of the cube
about an axis passing through directly opposite corners.
A light is shone directly down from above the intersecting cubes.
What shadow does it make on the table? This is called making a
Finally, suppose that the cube is rotated by 60 degrees about this
axis through opposite corners and the original and rotated cube
superimposed. The shape is sliced up along the six faces of the
cube What shape
are the pieces which would be cut off, and how many are there? If
possible give the dimensions of the shapes exactly.
Extension: You may like to
consider the problem of rotating other solids such as tetrahedrons