A slice of the action

21st January 2000 at 00:00
Move into another dimension with sliced polyhedra. John Sharp shows how your class can produce these fascinating models which prompt many questions in the making.

If you have made polyhedra using kits such as Polydron, or even glued polygons together, you have probably not thought about what happens when you start to cut them up. This activity, which makes a tetrahedron as a sliceform model, gets right inside the polyhedron. It also allows you to see what happens when a tetrahedron is deformed, which a rigid model cannot.

Sliceform models are created by slicing a solid in two directions and recreating the solid by slotting the slices together. The resultant sliceform is continuously deformable from the extremes of two flat shapes which form interesting designs in themselves. The pictures do not show their full beauty. You have to make them and play with their endless shapes and see the way light plays on them as you move them about.

Cutting correct slots It is very important that you cut slots and not just slits. You must cut just enough for the card making the slice to fit neatly in the slot. If the slot is too much wider, the model is likely to fall apart easily, especially in the early stages of assembly. If you just cut a slit along the line, then the forces acting sideways on the slices will cause the model to buckle.

To practice cutting slots, use some bits of card from the edge of the template with a short pencil line drawn on them. Cut the slot with scissors making a pair of cuts either side of the line. This will give a sliver of card which will often curl up like a hair. Pinch it off at the end of the slot by grasping it with your fingernails.

Copying onto card The templates are for copying onto photocopy card (160gsm). Do not try to make them bigger; they will not work as well.

If you make copies on two cards and use one colour in each direction, you will get much better models; they are also easier to make.

As you move the sliceforms they then change colour dramatically at times. This is apart from the constantly varying and interesting shadows they cast.

Designing the model Imagine a tetrahedron in a cube like this: If you cut the tetrahedron in a series of horizontal slices, the shape of the slices will be rectangles. Why?

This is a net of a tetrahedron cut in half Draw two of them on paper, cut them out and place them together so that they make a tetrahedron. Look at the dotted lines that form the rectangular slices to see how the shapes of the rectangles vary. Each type of dotted line matches one set of slices.

Draw an extra version of the net and plot the sides of the rectangles against the height of the slice, or measure the slices for the advanced model. Why is the curve the shape it is? Plot the sum of the two sides of the rectangle against the height. Explain your result.

The pyramid-shaped sliceform models pictured here are essentially the same design; the advanced (blue) one has many more slices. The simple (red and white) one has slots cut in different directions to make the slices less likely to fall apart, whereas the advanced one has them cut regularly.

The slots are cut at an angle of 45 LESS THAN to the edge of the slice. Why?

Assembling the simple model The simple model only has six slices. The two central squares that correspond to the squares in the design model and two other slices in each direction. This model should take about 10 minutes to cut out and assemble.

Cut out all the pieces and cut out the slots as described above. Fit the centre squares together, then add the other two slices in each direction. Use the two half tetrahedra models to judge how they are orientated. Think symmetrically. When you have made the model, fold it flat in two directions.

Assembling the advanced model This model has many more pieces and takes up to an hour to make. With more slices the tetrahedron looks more solid. Make the simple model first so that you can see the structure. Take extra care not to cut the slots too wide or you will find that the pieces fall out in the early stages. Cut out the central squares and their slots and fit them together. Then cut the next four slices (two in each direction) and add them to your model.

Continue cutting and add four slices at a time until you reach the outside. If you make the model with two colours of card it will not only be a more interesting model, it will also be easier to make.

When you have made the model, fold it flat in two directions. With it folded, tap the long flat ends gently on a flat surface to distribute the slices evenly in the slots. Look at the patterns it creates.

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