A Sliceform Zonohedron

See also the posts on how to make and assemble the models and Sliceform templates and Downloads

This Sliceform is available in two versions, a simple one which takes about 10 minutes to cut out and assemble since it has only six slices and a more advanced one which has 16 slices and takes about an hour to make. They are both made of the same shaped slice, and all slices are the same.

You should make the 10 minute one before attempting the advanced one.

How the model is designed

If you make a cube from drinking straws, connecting the straws with string threaded through them, then the cube is all floppy. If you distort it so that sides of the cube remain parallel by pulling two opposite corners away from one another, then you create a zonohedron. This is a rhombic prism and is the simplest of the zonohedra.

Questions

  1. In this model all the faces of the zonohedron are identical. Do they have to be in a zonohedron?
  2. The model is made up of identical slices, not only in shape but in the way the slots are cut. Why is there only one template slice required when the slots point up in one direction and down in the other?
  3. Which Archimedean polyhedra are zonohedra?

The zonohedron templates

Download the file you want for printing and use that to create the model. The following graphic is to help you understand what the slices look like in each model.

Each slice looks the same. The only difference in the two models is that the simple one, because it has only has six slices, has three slots in each slice and the advanced one has nine. There are two ways the slots are cut in the simple model. For the central slices they are not all on the same side. This helps to keep the slices from fallling out:

A single page of the 10 minute zonohedron template contains slices to make four models.

A single page of the advanced zonohedron template contains slices to make one model.

See the post on how to make and assemble the models for more about printing or copying onto card.

Make copies on different coloured card and make all slices in each direction a different colour.

Assembling the 10 minute zonohedron

See the post on how to make and assemble the models for details about cutting out and cutting the slots. Take extra care with the slots.

Since there are only a few slices, cut out all the pieces and cut the slots on each one.

The simple model only has six slices. Start with the two slices that have opposite slots in the centre. These are the central slices. Then add the two outside pairs of slices in each direction, making sure that you insert them the same way as the central ones in each direction.

When you have made the model, fold it flat in two directions.

Assembling the advanced zonohedron

This model has many more pieces and takes up to an hour to make. With more slices the zonohedron looks more solid and it generates more interesting patterns when it is flattened.

Make the simple model first so that you can see the structure when you are making this one.

Take extra care not to cut the slots too wide or you will find that the pieces fall out in the early stages of assembly. You will probably find it easier to cut pairs of slices and then fit them onto the model.

All slices are identical, so cut any two out and their slots and fit them together. Count along the slots to fit the fourth slots together symmetrically. Then cut the next four slices (two in each direction) and add them to your model. Continue cutting and adding four slices at 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.

Activities

  1. Create your own zonohedra with different shaped rhombi. You could make a number of ones like the simple, 10 minute, one or you could work as a group in a class or with your friends with the advanced one.
  2. When you have the set, compare the shapes and patterns they make when they are flattened. What is the relationship between the shape and the angle of the rhombus?
  3. With the simple (10 minute) one, investigate possibilities using mixtures of colours.
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