# Air Nets

You will see our mathematician attempt to assemble the net into a solid 3D shape, sometimes successfully and sometimes unsuccessfully.

Before watching the mathematician fold each net, consider these questions:

- Can you imagine folding the net up into a solid shape?
- Do you think that the net will fold into a shape with all sides clicked together?
- Can you imagine the shape of the final solid if the net does indeed correctly fold together?

- Were you correct? Was the result a surprise in any way?
- Try again to imagine how the shape folded together.
- Draw an accurate drawing of the net. Can you see which sides joined together? Can you indicate this clearly on your diagram?
- If you have access to Polydron, try building each net and replicating the final solid, where one was created. Could you make a solid shape from the net in the cases where our mathematician failed, or is it actually impossible to make the net into a solid shape?

- How might you be able to look at a net and be
*certain*that the net*will not*fold up into a solid? - How might you be able to be
*certain*that the net*will*fold up into a solid? - In what cases might you be unsure as to whether or not a net will fold up correctly? Can you give a good set of conditions for a net being a
*good possible candidate*for folding up into a solid?

In order for the net to be able to folded into a polyhedron, there must be an even number of unattached edges! We must also be able to match them up into pairs each of the same length that will join together when the net is folded.

For example, here's net 23. We can see there's an odd number of edges, so it certainly won't be possible to fold it. If the yellow triangle numbered with sides numbered 7 and 8 were removed, it would fulfill our requirement, as the 'new' edge of the red triangle with numbered side 6 could match up with the yellow side numbered 7. However, it's still not certain whether this net would fold up, as the triangles have to be put together to ensure it's possible for the squares to fit in.

There's lots of interesting ideas at http://nrich.maths.org/1381, particularly the fact the angle deficiency of a regular polyhedron (i.e all vertices look the same) add up to $720^{\circ}$. We could use this to predict whether a net will form a polyhedra. Usefully, this also works for concave regular polyhedra like some of the examples in this problem.

You can find the 24 videos individually on our YouTube channel.