Tag: tessellations

Tangram square with 3 pieces

I Solved Every Single Tangram Square (So You Don’t Have To)

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I bought some wooden Tangram sets to use at my Curious Minds after school club. Inside a wooden frame fit seven differently-coloured pieces: two large triangles; one medium triangle; two small triangles; a square; and a parallelogram. The frame is a square, and all seven pieces fit together to fill the space with no gaps and no overlaps.

“No gaps and no overlaps” is the univeral requirement when tessellating shapes. A tangram is not a tessellation, but the two concepts are closely related in geometry. It seems plausible that the study of tessellations helped inspire the creation of geometric puzzles like the tangram. My blog on two dimensional tessellations has turned out to be one of my most popular posts.

Playing around with the seven tangram shapes, I quickly saw it was possible to take a subset of the seven pieces and make different sized squares. Despite my best efforts, I could not find a website, blog post or activity sheet which showed every possible square. There are a wealth of resources for animal, bird or flower outlines, plus plenty of exotic shapes. But not for the humble square. The challenge was obvious: to create my own post to fill this gap for others.

I spent a happy hour sitting on the floor figuring out every combination. Here they are, leaving out simple rotations and reflections. I have also left out images where I could have swapped a small orange triangle for a yellow one, or exchanged the large blue triangle with the pink one: these are just colour swaps, not a new configuration of the pieces.

1 piece

Tangram square using 1 piece - the square

2 pieces

Tangram square using 2 pieces. Either 2 large triangles. Or 2 small triangles.

3 pieces

Tangram square using 3 pieces. 1 medium triangle and 2 small triangles.

4 pieces

Tangram square using 4 pieces. 1 large triangle, 1 parallelogram and 2 small triangles.
Tangram square using 4 pieces. 1 large triangle, 1 square and 2 small triangles.
Tangram square using 4 pieces. 1 large triangle, 1 medium triangle and 2 small triangles.

5 pieces

Tangram square using 5 pieces. 1 medium triangle, 1 square, 1 parallelogram and 2 small triangles.

6 pieces

No solution is possible. See below for an explanation.

7 pieces

Tangram square using 7 pieces. 2 large triangles, 1 medium triangle, 1 square, 1 parallelogram and 2 small triangles.

For those of you who like their data summarised:

Number of piecesNumber of solutions
11
22
31
45
52
60
72

If you are tempted to spot a pattern between the number of pieces and number of solutions, I didn’t get anywhere either!

Why can’t you make a square with 6 pieces?

For each piece, we can express its size as a unit of area:

The small triangle = 1 unit

The square = 2 units

The parallelogram = 2 units

The medium triangle = 2 units

The large triangle = 4 units

For any square, the total area must be either 2, 4, 8 or 16 units. Did you notice these are the successive powers of 2? That’s not a coincedence. Every time you step up to the next allowed square size, you are exactly doubling the area of the previous square. This is scaling symmetry, a beautiful, repeating geometric pattern. I will explore this further in a future blog.

The biggest square we can make with all 7 pieces has a total area of 16 units (1 + 1 + 2 + 2 + 2 + 4 + 4).

The squares with 5 pieces have a total area of 8 units (1 + 1 + 2 + 2 + 2).

The squares with 4 pieces have a total area of 8 units (1 + 1 + 2 + 4). (Note that any of the 2-unit pieces work).

The squares with 3 pieces have a total area of 4 units (1 + 1 + 2).

The squares with 2 pieces have a total area of either 2 units (1 + 1) or 8 units (4 + 4).

The square with 1 piece has a total area of 2 units (2, or the square by itself).

For any combination of 6 pieces you choose, there is no way to make a square with either 2, 4, 8 or 16 units. Lets look at what happens if you try to make a square using exactly 6 pieces. You have to leave one piece behind, which ruins the required “power of 2” area:

Leave out a large triangle (4 units): Remaining area = 16 – 4 = 12 units.

Leave out a medium triangle, square or parallelogram (each 2 units): Remaining area = 16 – 2 = 14 units.

Leave out a small triangle (1 unit): Remaining area = 16 – 1 = 15 units.

Because areas of 12, 14, and 15 cannot geometrically form a square using the standard fixed angles of a tangram set, a 6-piece square is mathematically impossible! Play about with 6 pieces so you can see how they stubbornly refuse to fit into a square.

Every possible Tangram square

For the sake of completeness, here they are in one image. Does it look like a giant tangram person? But which way is it facing?

All 13 possible Tangram squares.

When I took my tangram sets to my after school Curious Minds club they were very popular. I had three boys at that session. We got into a beautiful groove. For example, I told them to take the purple piece, the yellow piece and the orange piece and make a square bigger than the one before. They were really good at focussing on their own pieces and not sneaking a look to see someone else’s solution. As we progressed to 5 and 7 pieces they did need some assistance, but not as much as I expected. By this stage, they had developed a feel for how the pieces fit together. It was helpful that with 7 pieces they got to make the square inside the wooden frame, which naturally guided them to the total area of the solution.

Feel free to contact me if you think I successfully filled a gap in your tangram knowledge.

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

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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.

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

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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:

regular tessellations

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.

semi regular tessellations

Here are some examples of completed tessellations:

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

Y4 girl tessellation 3 4 6 4

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

Y6 boy tessellation 3 3 4 3 4

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

Y6 girl tessellation 3 12 12

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:

 

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