St Thomas of Canterbury, 6 December 2019

This week at the Curious Minds Club we explored Borromean and Brunian 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 over-under-over-under 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.

200px-BorromeanRings.svg

I asked the children to turn their flat ring into a three-dimensional structure, almost like a gryoscope.

220px-3d_borromean_rings_by_ronbennett2001

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.

200px-Brunnian-4loops.svg

Here are some photos of the children’s work.

Year 2 girl:

 

Year 4 girl:

Y4girl Borromean

Year 5 boy:

Y5boy Borromean

Year 6 boy:

Y6boy Borromean

Year 6 girl:

Y6girl Borromean

St Thomas of Canterbury, 29 November 2019

This week at the Curious Minds Club we started by trying some more real world knots. I used the NetKnots website to learn these knots which I showed to the children: Clove Hitch, Transom Knot, Constrictor Knot, Highwayman’s Hitch, Barrel Hitch and Cat’s Paw.

Some children played a game called Knotted. The person who created this game put it online to download for free. I did this and printed it on thick card. I tried the rules but decided to adapt them to my version, in which the winner is the first player to lay a continuous path from their side of the board to the opposite side.

knotted2

All the children had a go at making a Celtic Knot pattern. I had created some patterns of my own which I asked the children to copy to get them familiar with the different pieces and how they fit together. I then challenged them to come up with their own design. Here are three examples:

Two strands interlinked (Year 2 girl):

Y2girl_rotate

Two separate strands (Year 6 boy):

Y6boy

One continuous strand (Year 6 girl):

Y6girl

St Thomas of Canterbury, 22 November 2019

A move away from mathematical knots this week at the Curious Minds Club, and a look at real world knots. We started with a look at the symmetry of shoelaces. I made some ‘shoes’ out of felt and threaded them with one blue lace and one yellow lace. The instructions asked the children to make the base knot with the lace on the left leading first, then make the actual shoe lace knot with the lace on the right leading; this resulted in a neat and tidy, balanced knot with two types of rotational symmetry. I then asked the children to take the other shoe and make the base knot with the lace on the right leading, then make the actual shoe lace knot as above (right lace leading). This lack of alternation resulted in an ugly knot with one part facing the toes and the other part facing the heel; this is the knot which is much more likely to come apart. Ian’s Shoelace Site has more information and really good diagrams.

I then showed the children how to make some actual knots, specifically some loops and some bends.  I used the NetKnots website to learn these knots. The loops we tried were: Slip, Bowline, Hanson and True Lover’s. The bends we tried were: Square (Reef), Flemish and Fishermen’s.

To end the session, one group of children played the Knot Game. Another group had a go at solving some of Adrian Fisher’s Celtic Knot Playing Card puzzles:

adrian fisher

St Thomas of Canterbury, 15 November 2019

Another week exploring Knot Theory. The first activity was to look at a page of knots and work out which ones were actually unknots. Some children could do this by looking at the knot, and other children made the knot and manipulated it. The second activity was to learn the Reidemeister moves then look at some pairs of knots and decide which Reidemeister move had been used to turn picture one into picture two. The last activity was to learn about crossing numbers, and to look at pictures of knots and see if they could be simplified in order to decide what their true crossing number was.

reidemeister

We ended the session by playing the Knot Game.

knot game

St Thomas of Canterbury, 8 November 2019

We started Knot Theory in this week’s Curious Minds Club. The first activity was the Handcuffs Puzzle: in pairs both children had a length of rope attached to their wrists in such a way that they were linked to their partner; the challenge was to untangle themselves. It looks like the length of rope forms a closed loop for each child, but the solution lies in realising that there is a gap under each wrist: if a loop is threaded through in the right direction, passed over the fingers and back over the wrist again then the two pieces of rope do become detached. The children had fun trying this out, but I did need to show them the solution.

The next activity was to challenge the children to tie a knot in a length of rope without letting go of either end. I demonstrated that holding a length of rope in each hand forms a closed loop and that if you do not let go you end up tying your arm into the knot. The children were pleased when I showed them there is a solution which involves tying your arms into a knot before picking up the rope, then unknotting your arms. This works on the principle of the transference of curves: when you unknot your arms you transfer the knot that was on your arms onto the rope.

The final activity was to start to explore some mathematical knots. I had attached velcro to the ends of some shoelaces so that the ends could be joined. The children made the 3,1 (the trefoil) then the 5,1 and the 7,1. They made one crossing change and saw how this allowed them to transform their knot into the previous one in the series.

St Thomas of Canterbury, 18 October 2019

We did some perplexing things with paperclips and rubber bands this week. I showed the children how to bend a long strip of paper into an ‘s’ shape, then squash it together and add two paperclips at the two places where the paper touches. When I pulled on each end of the paper the children were surprised to see the two paperclips land on the table linked together.

The children then followed a set of drawings I had made, each one getting progressively harder and involving adding in one rubber band, then two rubber bands, then changing the positions of the bands along the strip of paper. The results were a combination of the paperclips and rubber bands being linked together in lots of different ways, some falling off the paper and some staying on. Some children made it to drawing 10, which results in an amazing Borromean link!

If you want to see how the paperclips become linked together, here is a slow motion video:

This is what the table looked like at the end! Lots of mess and lots of fun.

Table after Perplexing Paperclips 18 October 2019

St Thomas of Canterbury, 11 October 2019

An investigation of that classic problem in topology, the Three Utilities Problem. The children made their first attempt in flat, two-dimensional space i.e. on a mini whiteboard, so they could rub out their attempts and keep trying. After a few minutes I explained that the problem cannot be solved in two-dimensional space as there is not enough space! I asked the children what we could do in this case, and one boy was very quick to suggest trying it in three-dimensional space.

I produced four big tori, in the form of an inflated swim ring which I had wrapped in white duck tape and labelled, so that the dry wipe markers could be used and rubbed off as needed. After a few attempts, and some hints from me that they needed to use the whole length and circumference of the torus, we had some correct solutions.

three utilities

I then reminded the children that a torus is topologically equivalent to a coffee mug, and that if they could solve this problem on a torus they could also solve it on a mug. I then produced four white mugs which I had labelled. The dry wipe markers rub off very easily on a mug. The children spent the rest of the session exploring how to solve the problem. They needed a few hints about using the handle and the base, and going under the handle, but most of them got there in the end. The children seemed to really enjoying solving the problem using such unusual materials.

three utilities mug