# 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.

# Regular and irregular pentagons at the Curious Minds Club (St Thomas of Canterbury Primary School, 17 January 2020)

This week at the Curious Minds Club we continued our exploration of shapes, in two and three dimensions of space, by looking at (regular) pentagons. The children were able to identify that pentagons do not fit together without leaving a gap or an overlap.

I showed them how to cut two irregular pentagons out of a regular hexagon. They made a set and explored how to get them to tessellate with all the edges meeting at a vertex. The solution involved flipping some of the tiles over.

I gave one group the puzzle called Pentamania to explore. This is a set of 54 so called folded pentagons. Two can be seen in the image above (one pink, one grey) when regular pentagons overlap. The children solved one of the three puzzles and enjoyed making a pattern with the pieces.

A group of younger children enjoyed a quick game of Dotty Dinosaurs. They played the shape matching version.

We then made our first three dimensional object, a tetrahedron. I wanted to see how the children managed with the Polydron pieces I had, as they can be a little tricky for small hands to clip together. The children managed the activity well. One girl asked if she could take it home, so she must have liked it. I asked them what the object was called. One boy correctly said a triangle-based pyramid. They were all intrigued to learn its second name, the tetrahedron. I explained that tetra meant four in Ancient Greek, and this object has four faces.

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We finished by getting out the pattern blocks from last week, and completing some of the pattern boards. Here are a selection:

# Two dimensional tessellations at the Curious Minds Club (St Thomas of Canterbury Primary School, 10 January 2020)

In this new term at the Curious Minds Club we started our exploration of shapes, in two and three dimensions of space.

I gave the children a collection of wooden triangles, squares and hexagons. I asked them to make a regular, edge to edge tessellation for each shape. It didn’t take long for every child to find the solutions:

I explained that each tessellation has a vertex notation. I started with the square tessellation, explaining that its notation is 4,4,4,4 (every vertex is surrounded by a shape with four edges i.e. a square). I asked the children to work out the notation for the other two tessellations. With a little help they were able to find the answers: 3,3,3,3,3,3 and 6,6,6.

We then moved on to the semi-regular tessellations, of which there are eight. I used the same wooden pieces and some pieces that I had to cut out of card (octagons and dodecagons) as they are not available in wood. I gave each child a different vertex notation (e.g. 3,6,3,6 to make the pattern in the top left corner below) and asked them to put the pieces in the right order. When I had checked they had got it right (or offered a bit of help) I encouraged each child to take more pieces and extend the pattern out in each direction. I then rotated the activity between the children so they all got to try as many of the eight tessellations as possible.

Here are some examples of completed tessellations:

3,4,6,4 tessellation by a Year 4 girl:

3,3,4,3,4 tessellation by a Year 6 boy:

3,12,12 tessellation by a Year 6 girl:

For our final activity I gave each child a sheet showing all eight semi-regular tessellations and a piece of mirror card, and asked them to find the reflection symmetry for each tessellation (some have more than one). I asked them to find the ‘odd one out’. One boy was successful in identifying that 3,3,3,3,6 has no reflection symmetry. I explained that this is because it is chiral i.e. there are two different versions of it: