Lessons taught; Lessons learnt

(Update: This post is featured in the 39th Carnival of Mathematics. Check it out!)

Last summer I developed a bit of an obsession with hexagons. More specifically, with the patterns you can produce from tiling multiple copies of a single, simply decorated hexagonal tile:

I cut out and laminated around 100 of these tiles, and spent several happy hours moving them around into new patterns, trying different types of symmetry, and experimenting with different layouts.

If you’d like to follow me down a similar path, here’s a sheet with six of these hexagon tiles to print and cut out: hexagon-tiles-122.pdf (svg source), made using the free vector drawing program Inkscape.

Background

Although I wasn’t aware of precursors at the time, I later found this shape in several places: it is one of the tiling generators you can buy from the ATM, and it’s one of the tile shapes in the game of Tantrix. I recommend you browse the ATM’s store if you are a maths teacher — many excellent things await!

I also recall reading a very interesting web page which found this exact pattern being used as a paving slab in Spain, and developed quite an in depth mathematical discussion, but sadly this page now seems to have disappeared into the æther.

Although less common than they used to be, hexagons were often used for tiling and paving. I particularly like this hexagon paving tile designed by Gaudi, and used all over Barcelona.

Returning to the particular hexagon above, we get the following when we tile it:

The fun comes when you rotate some of the hexagons. Even with just six patterned hexagons you can find a range of different interesting patterns… but it’s much more fun to explore with a large pile of them!

In My Classroom

At the start of last year, the tiles appeared on my classroom walls in the place now occupied by the Pascal’s Triangle wall display. As with Pascal’s Triangle, the tiling wasn’t static, but slowly changed form and shape over time, and several interesting questions were raised by some of my groups (several of which couldn’t believe at first that all the different patterns were generated by a single type of tile).

Eventually the hexagons migrated over to the other side of the classroom, and took the form revealed in the very first post I made on this site:

Read more for: more examples of patterns you can produce from these tiles; some questions for you to explore; my thoughts on the mathematical content of this ‘pattern space’; and source files for all the diagrams.

I suggest you pause here, print out the hexagons, and have a play before continuing.

Some Patterns

Here are eight patterns made with the hexagon tile, on a 6-by-6 grid:

(click on the images for a better view). These by no means exhaust the patterns you can generate with this tile.

Questions Raised by the Patterns

I believe that these tilings form a rich and non-trivial environment for pattern spotting, and the generation of mathematical questions and conjectures in the classroom. Here, for example, are some questions which occurred to me as I was exploring the ‘pattern space’:

• Several of these patterns have rotational symmetry of order three. What other rotational or reflectional symmetries are possible? What about if we allow ‘holes’ (i.e. we don’t have to place hexagons in every grid position)?
• The last of the 6-by-6 patterns demonstrates glide reflection. There are seven possible frieze patterns; can we generate examples of all of them?
• Various motifs recur in different patterns: a small circle using three hexagons; an oval using four hexagons; several braids; a large circle and a trefoil using six hexagons. What other closed loops can we make using only a small number of tiles? Also, are there any forbidden lengths, with no examples of loops of that size?
• Fix a small grid size (for example, a 2-by-3 rectangle). How many distinct patterns can we make?
  The meaning of ‘distinct’ here is something that would need to be negotiated with the students. If we take a pattern and rotate or reflect it, should we count that as a new pattern?

I think all these questions could be generated, explored, and altered quite productively by students at a range of levels. They also give students an opportunity to demonstrate several key reasoning skills, such as systematic exhaustion: how can we be sure that we’ve found all the possible patterns of a given grid size, or loops of a given length?

Complexity

In some sense, all the questions in the previous section were combinatorial. We had to count the number of ways that something happened, or generate examples.

Here’s a deceptively simple question which leads into an investigation of another sort:

Which of the patterns above is the most complex?

We might perhaps say that simplest patterns are those which we can describe in the shortest amount of space. The very first pattern, for example, could be described by saying ‘all the tiles are this way up’. How could we describe how to generate some of the other patterns?

Given this view of complexity, what do complex patterns look like?

What do simple patterns look like?

This seemingly naive and simple view of complexity, when formalised, leads to algorithmic information theory, and in particular to the concept of Kolmogorov complexity. This also has connections to an immense body of work in computer science, including compression algorithms.

Transformations

Moving away from complexity, let’s now consider what happens when we start with a pattern, and want to alter (transform) it in some way.

It is obvious that we can transform one pattern into any other pattern by rotating each tile in turn — but what happens when we impose constraints on the ‘moves’ we are allowed to make? For example, suppose we had a rectangular grid of tiles, and were only allowed to rotate the tiles a row/column at a time.

Starting from the basic pattern we saw right at the start:

we could rotate the second ‘column’ one step anticlockwise:

and then the third row:

and then the fourth column:

and then the second row:

Could we generate every possible pattern this way?

This is an interesting question, but probably a little difficult to approach at the secondary school level. So, let’s be even more restrictive:

Suppose we rotate all the tiles at the same time. What happens?

To be more specific, let’s start with this pattern:

What do we get if we rotate all the tiles one ’step’ clockwise?

Now is an excellent time to print out some hexagons and find out!

What about if we rotate again? And again?

What happens with different starting patterns?

What is preserved by this transformation, and what is not? If one pattern repeats every three hexagons for example, then the other one will as well. But what about symmetry? What about the ‘loops’ formed by the lines?

Notice that we are exploring, in an accessible way, several advanced concepts: applying a transformation to something; preservation of features; iteration of transformations among others.

Where’s The Maths?

Usually, in school mathematics, we only consider functions which transform numbers into other numbers — even transformations such as rotation, reflection and enlargement are almost never talked about as functions which can be combined, or reversed, or iterated. This naturally makes it harder for students to ’see the maths’ in situations which don’t directly involve numbers.

One way to help students to become comfortable with these ‘non-traditional’ areas could be to improve the emphasis, throughout their school careers, on key concepts and questions, like iteration (what happens if we do something many times?), inversion (how do we undo what we just did?) and iso- & homo-morphism (what has changed, and what has stayed the same?). These are applicable at all levels of mathematics:

What happens if we add the same number lots of times?

How do I undo a multiplication?

What properties of my triangle stay the same when I enlarge it?

What happens when I differentiate a polynomial lots of times?

How do I undo exponentiation?

What properties of number stay the same if we allow the square root of -1 to be a number, and which have to change?

The same types of question recur throughout the mathematical development of a pupil.

Conclusion

Even something as simple as a hexagonal tile can lead to some really deep mathematical questions, and thinking about them can develop skills which are applicable in many more traditional areas of mathematics.

It also lets you use the pun in the title — and we don’t get the opportunity for puns often in maths!

[As promised, here are the source files for many of the above diagrams.]

I was browsing through the Nrich website yesterday, and found this interesting problem:

If you are given the coordinates of the midpoints of the edges of a pentagon, can you find the coordinates of the vertices of the pentagon?

This was, apparently, originally given as a question in an Oxford extrance paper from 1926!

If you follow the link, you’ll see that the Nrich site gets you to investigate this problem with pentagons, triangles, and quadrilaterals, using the excellent free dynamic geometry app Geogebra. I use Geogebra regularly in my teaching, so I’m happy to see it appearing in well-known sites like Nrich.

I’ll let you think about this problem by yourself (Nrich gives the hint that you should think about simultaneous equations), and will focus on just one aspect of it in this post…

Midpoints of a Quadrilateral

It’s very easy to convince yourself by messing around with the dynamic geometry that it’s always possible to find a pentagon or a triangle with any given midpoints, but that quadrilaterals are more tricky. Indeed, the default configuration of midpoints presented to you seems to be impossible.

Why is this? Perhaps there is a special property which needs to be satisfied by the midpoints of quadrilaterals. A great way to look at this is to set up the situation in a dynamic geometry package, and see if anything interesting suggests itself for further investigation:

Geogebra applet will be here if you enable Java.

(Source: midpoints_of_a_quadrilateral)

It appears that the midpoints always form a parallelogram, and that this holds even when we don’t have a ‘proper’ quadrilateral, but a bow-tie shape.

How could we prove this? One route is to use vector arithmetic.

Using Vectors

Let’s call the four points of the quadrilateral P₁, P₂, P₃, and P₄, and the four midpoints M₁, M₂, M₃, and M₄.

We have

M₁ = ½ ( P₁ + P₂ )
M₂ = ½ ( P₂ + P₃ )
M₃ = ½ ( P₃ + P₄ )
M₄ = ½ ( P₄ + P₁ )

Now, we want to show that the quadrilateral formed by the midpoints is a parallelogram. We can do this if we show that the vector that takes us from M₁ to M₂ (M₂ – M₁) is the same as the vector that takes us from M₄ to M₃ (M₃ – M₄), and that the vector that takes us from M₂ to M₃ is the same as the vector that takes us from M₁ to M₄. Both of these come from the observation that

M₁ + M₃ = ½ ( P₁ + P₂ + P₃ + P₄ ) = M₂ + M₄,

from which we can conclude that M₂ – M₁ = M₃ – M₄ and M₃ – M₂ = M₄ – M₁. M₁M₂M₃M₄ is therefore a parallelogram.

Alternative Proofs and Further Questions

This seems like the type of property which could be proved in many different ways. While I like using vectors, the traditional way to prove something like this would be to use synthetic (Euclidean) geometry. I’ve never been particularly great at this, but if someone has a simple proof in this style, I’d love to see it.

This also raises several further questions:

• When is the parallelogram a rectangle?
• When is the parallelogram a square?
• What happens for hexagons?

Next in this series

Midpoints (2): Pentagons