Building a Sierpinski tetrahedron at the Curious Minds Club (St Thomas of Canterbury Primary School, 14 February 2020)

I started this week’s session of the Curious Minds Club with some geometric snacks. First up were some nachos. I asked the children what type of triangle the nacho is. We talked about the Isosceles triangle last week, but none of them remembered. I wrote it on the whiteboard this week, to aid their learning.

Next up were some snacks I made, using cocktail sticks and midget gems:

Tetrahedron snacks

I told the children they could eat one if they could name the shape. One boy said triangle-based pyramid. I said this was correct, but that this shape has two names. I gave a hint about the first letter, and a girl very proudly said tetrahedron. I then handed one round to everybody, but made them all say tetrahedron first.

I explained that this week’s activity was to build Sierpinski’s tetrahedron, a three dimensional version of Sierpinski’s triangle. I showed them one part of it I had made using the generic version of Geomag that we used recently to build the Platonic Solids. I asked them to use just one colour to make the first part, then repeat this with a different colour. I got them to work in teams to add their four parts together to make a two layered Sierpinski tetrahedron. It involved removing some of the vertices, which the children worked out.

Two cousins (Y2 and Y4) made this (there was not enough dark blue to complete one part):

 

A boy and girl in Y6 made this:

 

For the rest of the session some children used the wooden pattern blocks to complete some more pattern boards. Others played Dotty Dinosaurs, a game about shapes. I asked two children to test a new game I have invented, with the working title of Plato’s Polyhedral Dice Game. Each child had five dice in the shape of the five Platonic Solids, and a dice cup. They had to race to complete tasks, such as roll five odd numbers. They seemed to enjoy it, although there was not enough time to get detailed feedback.

At the end of the session, because it was the last week of this half-term, I gave each child a gift to take home. I made these 2020 Rhombohedron calendars at home. I was testing different methods of attaching the parts of the net: PVA glue, glue dots, magnets. The winner was …. glue dots! The pdf is here.

 

I took all of the Sierpinski tetrahedrons the children had made home with me. I wanted to see if I could make one with three layers. I rearranged the parts, added two of my own, and came up with this, photographed from different angles:

 

 

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:

20200213_111535

Here is each one by itself:

20200213_11180120200213_11181820200213_11175120200213_11195120200213_111844

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.

20200213_112133

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.

20200213_111844

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.

20200213_112115

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).

Y6 girl tessellation 3 12 12
The 3,12,12 by a girl in Year 6.

Y6 boy tessellation 3 3 4 3 4
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:

Set Platonic 31 Jan 2020

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

Table for blog

 

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.

pentagon-154320_1280

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.

Y6 girl and Y6 boy dodecahedra

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:

Y6 boy icosahedron

Y4 girl:

Y4 girl icosahedron

Y2 girl:

Y2 girl icosahedron

Y1 girl:

Y1 girl icosahedron

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).

Y4 and Y2 girl icosahedra

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

Set Platonic 31 Jan 2020

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:

Collection Platonic Solids1

Collection Platonic Solids2

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.