Further fun with the Archimedean Solids at the Curious Minds Club (St Thomas of Canterbury Primary School, 13 March 2020)

This week at the Curious Minds Club we continued to build the 13 Archimedean Solids, with Polydron Frameworks and Magformers.

My Y2 girl who got half way through a Truncated Icosahedron two weeks ago (then had to miss last week’s session) was happy to finish it. She then got her first go at Magformers. She made several of the Platonic Solids from looking at a picture of the net, then made a lovely symmetrical pattern on her own initiative.

I asked my two Y1s to build a Truncated Cube in Polydron. Once they had got the alternation correct at the start (put a triangle on one edge of the octagon, miss one edge, add another triangle etc) they were able to bring the whole solid together. They needed a little help snapping it together at the end.

My Y5 boy completed the Icosidodecahedron in Polydon, having made it in Magformers last week. My Y4 girl made the Cuboctahedron in Polydron first, then in Magformers. She built the Rhombicuboctahedron in a Polydron net really quickly, then needed quite a lot of help bringing it together. With a few minutes left at the end I gave her the Tangram puzzle. She solved it in a few minutes with no help. My Y6 girl and Y6 boy attempted the Rhombicuboctahedron in Polydron. It didn’t go quite to plan. Bob was born instead.

Here are the photos:

Y2 girl. Truncated Icosahedron (you may know it as a football); three Platonic Solids (can you name them?)

Y1 girl. Truncated Cube; half an Icosidodecahedron (to be continued).

Y1 boy. Truncated Cube; Truncated Octahedron; a Heart.

Y4 girl. Rhombicuboctahedron; Cuboctahedron; a completed Tangram.

Y5 boy. Icosidodecahedron. Really pleased with the angle I took this at: you can really see the line of reflection symmetry.

Y6 girl.   Meet Bob. Apparently he doesn’t have a best side. He looks good from every side. Hard to disagree.

Continuing the Archimedean Solids at the Curious Minds Club (St Thomas of Canterbury Primary School, 6 March 2020)

This week at the Curious Minds Club we continued to build the 13 Archimedean Solids. We used Polydron Frameworks and a new material – Magformers.

My Y6 girl and Y6 boy sat together and used the Polydron. They started with the Truncated Dodecahedron. I then asked them to make the Cuboctahedron, explaining that it uses the six squares from the Cube and the eight triangles from the Octahedron (hence the name). Continuing with this theme, I asked them to make the Icosidodecahedron: it uses the 20 triangles from the Icosahedron and the 12 pentagons from the Dodecahedron. They needed very little help from me to work through these.

Meanwhile I introduced my Y1 girl, Y1 boy and Y5 boy to Magformers. The younger children can find it hard to snap together the Polydron pieces so I wanted to try them on a material that uses magnets to join the pieces together. I started them on all five Platonic Solids, asking them to use a picture of the net to make the shape in two dimensions, then lift it up and join the edges together. This worked really well. Even the Icosahedron comes together well, provided you work from one end to make one half, then switch to the other end, make the other half and bring them together.

My Year 1 girl was then determined to have a go at the Truncated Icosahedron in Polydron. She had seen an older girl make one last week and must have thought “I can do that”. With some reminders that every pentagon is surrounded by hexagons she was able to complete this one. She took it out to show her mum at the end and looked very proud.

I asked my Y1 boy to make a Cuboctahedron in Magformers, from a picture of its net. He cracked the net and just needed a little help lifting it up and joining the edges. He then made some fun shapes: a fish, hourglass and small star.

My Y5 boy also made the Cuboctahedron in Magformers. He then asked for ‘something harder’ so I showed him the net of the Icosidodecahedron, which he cracked. Not content with this, he went on to make a copy of the Compound of Two Tetrahedron (not an Archimedean Solid but I brought my model with me again as it is such a nice thing to look at). He worked really hard to figure out where each of the 24 triangles should go.

Here are the photos.

Y6 girl. Truncated Dodecahedron; Icosidodecahedron.

Y6 boy. Truncated Dodecahedron; Cuboctahedron.

Y1 girl. Truncated Icosahedron.

Y1 boy. Cuboctahedron; Fish; Hourglass; two views of a Small Star; one of the Dodecahedrons.

Y5 boy. Cuboctahedron; one of the Dodecahedrons; Icosidodecahedron; Compound of Two Tetrahedra.

At the end of the session it is irresistible to build some towers.

This one gets bigger and bigger. It ended with a Tetrahedron on top, then threatened to topple over so we had to stop.

Starting the Archimedean Solids at the Curious Minds Club (St Thomas of Canterbury Primary School, 28 February 2020)

Before half term we built all five Platonic Solids, in various materials, so this week at the Curious Minds Club we made a start on the 13 Archimedean Solids. The material was Polydron Frameworks. I gave a brief account of who Archimedes was.

I explained that we would start by truncating all five Platonic Solids, and that truncation meant slicing off every vertex. I gave each child a piece of paper I had prepared: it had a triangle drawn in pencil, with each edge marked into thirds. I labelled the markings A,A; B,B; and C,C, gave each child a pencil and ruler and asked them to join the letters and shade through the area near the vertex. I asked them how many edges the new shape had. They counted to six and told me the new shape was a hexagon. I pointed out that we started with three edges and ended with six edges; one boy was quick to tell me this meant we had doubled the number of edges. I picked up a Tetrahedron and said we were going to truncate it: the four faces that were triangles would now be hexagons, and we would need four new triangles to replace the old vertices. I gave each child four hexagons and four triangles and put a model of a Truncated Tetrahedron on the table for them to copy if needed. Every child was able to complete this activity quite quickly: some studied the model a lot, while others just glanced at it.

We then moved on to the Truncated Octahedron. The Octahedron’s eight faces are also triangles, so these are replaced by eight hexagons. The vertices are replaced by six squares. I had a model available for those who needed it.

Two Y6 children moved ahead of the others so I started them on the Truncated Cube. I gave them a square drawn in pencil, again with each edge marked into thirds. The markings were A,A; B,B; C,C; D,D. After joining the letters they knew they had made an octagon. I reminded them that they had doubled the number of edges. I gave them six octagons and eight triangles and they happily built this solid. I was unable to make a model: I only had enough octagons in my collection for two Truncated Cubes to be made at the same time. I was able to provide a picture.

I started a Y1 girl and a Y4 girl on making their own Truncated Dodecahedron. I gave them both 12 decagons and 20 triangles. This had to be done from a picture. The Y4 girl only needed a little assistance from me. The Y1 girl needed a bit more helping snapping the Polydron pieces into place, but she had no problem working out which piece went where. At the end of the session she took it to show her dad, and he looked very impressed. I was not surprised, as this is the second biggest of the 13 Archimedean Solids.

Meanwhile a girl in Y2 made a start on the Truncated Icosahedron. She used a picture to help her, and my tip that a pentagon is surrounded just by hexagons. She ran out of time, so I promised she could finish it next week. A girl in Y6 was able to complete this solid, and looked intrigued when I pointed out that this is the shape of a football.

My Y6 boy had noticed that I had brought with me a model of a Compound of Two Tetrahedra which I was keeping to one side in case I wanted to use it. I had used yellow and red triangles to really bring out the fact that it looks like two separate tetrahedra that have been spliced together. He asked if could make his own copy of it and I was happy to agree. He needed very little help from me, and I could tell he was really proud when he had finished.

I welcomed two new boys to my club this week, brothers from Y1 and Y5. After they had completed the Truncated Tetrahedron and Truncated Octahedron I decided to take them back to the Platonic Solids as they had missed these sessions. I started them on a Cube as I knew they would have seen one before. I showed them a net and got them to build it from there. I followed this with a Tetrahedron and an Octahedron. The Y1 boy did need some help in snapping the Polydron into place, but he got stuck into the activities with enthusiasm.

Below are some photos to enjoy.

Y1 girl. Truncated Dodecahedron; Truncated Tetrahedron; she sneaked in a Cube when I wasn’t looking.

Year 1 boy. Truncated Tetrahedron; Cube; Octahedron.

Year 5 boy. Cube; Octahedron; Truncated Tetrahedron.

Year 2 girl. Truncated Icosahedron (to be completed next week).

Y6 girl. Truncated Icosahedron; Truncated Octahedron; Truncated Cube.

Y4 girl. Truncated Dodecahedron; Truncated Octahedron; Truncated Tetrahedron.

Y6 boy. Two views of the Compound of Two Tetrahedra.

What a collection! A Truncated Octahedron sitting on top of: Truncated Cube; containing a Truncated Tetrahedron; containing a Cube; containing a Tetrahedron.

A glimpse of a famous number sequence amongst shapes in two and three dimensions

I run an after school club at a primary school on the Isle of Wight. I call it the Curious Minds Club, and my purpose is to show the children that Maths is not just about numbers, it is also about shape and space. In the first term I introduced the children to topology and knot theory. This term we are exploring shapes in two and three dimensions of space.

The first activity was to use wooden pattern blocks to find the three shapes which tile a two dimensional plane by themselves. It didn’t take the children long to find out how to make the equilateral triangle, square and hexagon do this. Using the same blocks plus some shapes I cut out of heavy card (the octagon and dodecagon) I gave the children a vertex configuration for each of the eight semi-regular tessellations and asked them to fit the shapes together around the vertex, then extend out in all directions (given the limitations of the size of the table, the number of children competing for the number of tiles and the length of the session, we were unable to approach infinity).

The 3,12,12 by a girl in Year 6.

The 3,3,4,3,4 by a boy in Year 6.

Further sessions involved using Polydron Frameworks to build the Platonic Solids; next up are the Archimedean Solids. Between them the children made this set of Platonic Solids:

As part of my preparation for the sessions I drew this table of vertex configurations as I had not seen one elsewhere:

I then simplified my table by counting the number in each category:

 2 dimensions 3 dimensions Regular 3 5 Semi-regular 8 13

I thought it was interesting that if you add 3 + 5 you get the 8, and if you 5 + 8 you get the 13. It only took me a few seconds to realise I was looking at an early part of the Fibonacci sequence:

1,1,2,3,5,8,13,21,34,55

I was not expecting this link to the Fibonacci sequence, and I am not claiming it is very meaningful, but I put it out there for others to notice and perhaps enjoy.

It is worth noting (but not being too concerned) that of the 8 semi-regular tessellations in two dimensions, one (3,3,3,3,6) is chiral i.e. it exists in two different forms. Of the 13 Archimedean Solids, two are chiral – the Snub Cube and Snub Dodecahedron.

The faces of a Dodecahedron are pentagons. Linking each vertex inside produces a pentagram and a smaller pentagon. Repeating this process on the smaller pentagon produces lines, some of which can be traced to produce the two shapes of Roger Penrose’s P2 tiling, known as kite and dart. For both kite and dart, the ratio of the length of the long side to the length of the short side is Phi (the golden ratio). The area of the kite divided by the area of the dart is also Phi. Phi is in fact all over the pentagon. We can approximate to Phi by dividing a value in the Fibonacci sequence by the value preceding it (89/55 is appealing). In a future session I intend to ask the children to find the two shapes that make P2 inside a pentagon, then give them a set of P2 tiles and ask them to create their own aperiodic tiling.

While the National Curriculum includes cubes and other three dimensional shapes in its geometry section there is no specific mention of the Platonic Solids, let alone the Archimedean Solids. Some of the children in my club knew they had made a triangle-based pyramid, but had no idea it is also called the Tetrahedron. I wanted to give the children an opportunity to use materials to explore shapes and space and hold these beautiful objects in their hands.

Completing the Platonic Solids at the Curious Minds Club, and making a hyperbolic surface (St Thomas of Canterbury Primary School, 31 January 2020)

This week at the Curious Minds Club we built the final two Platonic Solids, the Dodecahedron and the Isocahedron.

We used the same materials as last week: Polydron Frameworks and generic Geomag. For the Dodecahedron I did not give the children a model to copy: I gave them 12 Polydron pentagons and said there were three around every vertex. They were able to figure it out for themselves, which I was impressed by. I don’t think this would have been possible without the previous three sessions on two and three dimensional objects.

It is impossible to build the Dodecahedron using the generic Geomag as there is too much freedom in how the rods can move around. Instead we went on to the Icosahedron. I had drawn instructions for this, based on making two identical caps then adding 10 rods around the outside of one cap (two on each vertex), then fitting the two pieces together. With a little help from me, but only at the final step, the children were able to make their Icosahedron:

Y6 boy:

Y4 girl:

Y2 girl:

Y1 girl:

The children then moved on to using some Polydron equilateral triangles to make more Icosahedra. They did require some help to make sure there were five triangles at every vertex. See how I have captured two different views in this photo, the second one really bringing out the vertex. (I will admit this was not on purpose).

Here is the whole set of five Platonic Solids, from last week and this week:

I explained to a Y6 girl why there are only five of these that meet the criteria of being regular, convex polyhedra. I put three triangles around a vertex, pointed out the gap in this net, then asked the girl to snap them together. This is the start of the Tetrahedron. We repeated this with four around a vertex, then five. The gap in the net got smaller, giving less height to the cap. Putting six around a vertex leaves no gap in the net: it is two dimensional and cannot make a polyhedron. We then took a look at putting seven around a vertex. I explained that this was a hyperbolic surface. The girl was inspired to see that would happen with eight, nine and then ten around a vertex. The results of 10 are below. I have promised to bring this back next week to continue this exploration of space.