This week at the Curious Minds Club we explored Borromean and Brunnian links. We started with the Borromean rings as they are the simplest Brunnian link. I showed the children how to turn coloured pipe cleaners into circles. They made a red one and a white one, then placed the white on top of the red one and threaded a blue one through, in an overunderoverunder sequence. I showed the children that as a ring of three this is a strong link, but if you remove any one ring the whole thing falls apart.
I asked the children to turn their flat ring into a threedimensional structure, almost like a gryoscope.
A drew a table to show that each ring is outside one ring and inside another ring.

Outside 
Inside 
White 
Red 
Blue 
Blue 
White 
Red 
Red 
Blue 
White 
I surprised the children by asking them how their Borromean ring is like the game ‘Rock, Paper, Scissors’. One girl was able to identify that each component has a different relationship with the other two components. I drew the table below and pointed out that white occupies the same positions in the table as rock; that blue corresponds to paper; and that red corresponds to scissors. I explained that the concept of ‘beating’ equates to being ‘outside’ and the concept of ‘losing to’ equates to being ‘inside’.

Beats 
Loses to 
Rock 
Scissors 
Paper 
Paper 
Rock 
Scissors 
Scissors 
Paper 
Rock 
By this point some of the children had discovered that they could turn their Borromean ring inside out. I encouraged all the children to practice turning their ring inside out and back again. I think this is one of the most fascinating aspects of the Borromean ring and is due to the fact that no two of the three rings are actually linked with each other.
We then moved on to making a Borromean ring of a Borromean ring! Yes, we went fractal! I asked the children to take three pipe cleaners of one colour, three of a second colour and three of a third colour. They made normal Borromean rings with their first and second colours and placed one on top of the other. They made another Borromean ring with their third colour to get the twists in the right place, undid the ends, threaded it through and retied the ends. When made three dimensional this is a very interesting object to look at.
Some children had raced a bit ahead, so I asked them to make triangles instead of circles and thread them into a ring. I showed them one I had made earlier. Two children thought the circles made a better three dimensional shape, but one boy argued in favour of his triangles. He had indeed succeeded in making something harmonious to look at.
The final task was a make a 4 component Brunnian link, with four different coloured pipe cleaners. This is trickier than the Borromean rings as the green link has to be made much smaller, but the threading is not too complicated. With a little assistance the children were able to complete theirs. I asked what would happen if one link was removed. Due to the effort involved in making it some of the children were reluctant to try this on theirs! I let them use my demo instead, and they were surprised to see the whole thing fall apart. I explained that as this happens with the Borromean rings we should expect it with the Brunnian link as they are part of the same class of links.
Here are some photos of the children’s work.
Year 2 girl:
Year 4 girl:
Year 5 boy:
Year 6 boy:
Year 6 girl: