# 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.

# Building some Platonic Solids to take home at the Curious Minds Club (St Thomas of Canterbury Primary School, 7 February 2020)

This week at the Curious Minds Club the children built models of the Platonic Solids to take home. In the previous two weeks I had given the children Polydron Frameworks and what I call a generic version of Geomag to make the Platonic Solids. These materials are expensive and I took them home at the end of the sessions. I wanted to give the children an opportunity to make models they could take home with them.

The materials I used were simple: plastic drinking straws and pipe cleaners. (I did ponder the ethics of plastic straws, as they are due to be banned in the UK in 2020; my conclusion was that I should use up the ones that currently exist before switching to paper). Feel free to contact me for details of the construction method. Once I showed the children how to build the models they took to it quickly. They enjoyed commenting on what the models looked like as they went along. When they were building a cube: “it looks like a laptop” and “I have made a table”.

Here are all five Platonic Solids that I made. (The session overran and there was no time to take photos of the children’s models). I think they look good in black. See how the pipe cleaners form the vertex:

Here is each one by itself:

They all look good …. until you get to the Dodecahedron above. This was a real struggle. My first attempt is below. The edge length is 75mm, the same as the others. It was hard to make every face look like a regular pentagon.

I thought I would try making each edge half the length (75mm to 37mm) to see if this was easier. This is how it looked during construction:

The end result is below. It was the best I could manage. I manipulated a lot of the vertices by switching the pipe cleaners around. It goes concave in places but the whole thing should be convex.

My conclusion is that drinking straws and pipe cleaners are a cheap and easy building material for building the Platonic Solids, if you can tolerate an imperfect Dodecahedron.

I also made a whole set using neon straws, but I don’t think they look as good as the black. You can see some gaps between the pipe cleaners inside the straws.

# 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.

# Starting to build the Platonic Solids at the Curious Minds Club (St Thomas of Canterbury Primary School, 24 January 2020)

This week at the Curious Minds Club we focused on three dimensional space by starting to build the Platonic Solids.

I gave the children some Polydron equilateral triangles and showed them the net of a tetrahedron. They were able to build one quite quickly, then form it into a three dimensional object. Next up was the cube. Before handing them the pieces I asked the children if they could work out the cube’s alternative name, based on it having six faces. With a little prompting to think about which two dimensional object has six sides, one girl correctly suggested ‘hexahedron’. The children then constructed their Polydron cubes. I did not show them all 11 nets of a cube (an activity for a later date?) but let them figure it out for themselves. The next was the octahedron. I gave each child eight equilateral triangles and told them that four triangles meet at each vertex. They found this a little harder and I showed them one I had made earlier as a guide.

I decided to leave the final two Platonic solids (the dodecahedron and icosahedron) to next week as I thought it would be too much to attempt all five in one session. I got out some magnetic rods and balls (similar to Geomag but a generic version) and asked the children to make the tetrahedron, cube and octahedron in this material. The Polydron pieces are good at bringing out each face of the solid, but the generic Geomag are better at bringing out the vertex and edge.

Some of the children built quite large cubes with three rods forming an edge. They soon discovered that this made a wobbly and unstable cube, due to the degrees of freedom in a square. I encouraged them to use one rod for an edge, and they had more success this way.

Below are some of the children’s creations:

I had prepared some material on the symmetry of the Platonic solids. I showed one girl how to look at each object in three ways: face on, edge on and vertex on. I gave her two dimensional pictures of each object and a piece of mirror card, and asked her to find the lines of reflective symmetry for each object. She did really well at this activity and seemed to really enjoy looking at the objects in different ways.

We finished with a quick game of Dotty Dinosaurs. This week we played the colour matching version as a memory game.