Interpenetrating solids
This problem provides training in visualisation and representation
of 3D shapes. You will need to imagine rotating cubes, squashing
cubes and even superimposing cubes!
Age
16 to 18
Challenge level
Problem
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.
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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?
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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 projection of the cubes.
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 original 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 and octahedrons.
Getting Started
For the first part of this problem try to view the cubes from above to turn it into a simpler 2D problem to begin with.
The second part of this problem will be greatly simplfied if you work with a physical model of a cube, preferably wire framed which you can see through (such as with polydron) and the bigger the better. You may wish to attempt to create frames using wires or straws to help to understand the problem.
Instead of trying to visualise the entire solid 3 dimensional cube, you may like to consider the cube as defined by its 8 corners. If you work out where the corners go then the rest of the cube will follow by joining up with straight lines and planes.Visualising the objects by looking directly at one of the original faces will help you to understand the problem.
Finally, don't forget that if you can work out the view of the interpenetrated object in the direction of one of the faces, then the view in the direction of the other faces may follow by symmetry.
Teachers' Resources
In mathematics, projections are often used to help understand the properties of 3 dimensional objects. Formally if an object is created from a set of points (x_n, y_n, z_n) then the projection in the z-direction, for example, will be the same set of points with the z coordinates squashed to zero (x_n, y_n, 0).
An obvious use of projections is in computer games where the 3D virtual world is shown on a flat screen.
It would be an interesting activity to attempt to construct real 3D models of the interpenetrated objects. Drinking straws would enable the creation of a framework and then the surfaces could be tiled with coloured paper cut to an appropriate size. It would be very interesting to receive photos of any such constructions!