Archive for August, 2008
The Joy of Hex
(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.]
Tags: geometry, hexagons, investigations, Maths, patterns, tiling, wall-display
Visualisation: Yet more rotating squares!
I opened my computer this morning intending to post something new, but soon got caught up in a further exploration of these square-to-square linkages. You soon notice, when creating these, that you are quite constrained in the points you can use to link the squares together. You are also limited in how ‘fast’ you can get the squares to rotate. Trying to use the least different speeds of rotation, I created this:
(Source: rotating_tesselation.ggb -- apologies for the spelling mistake :) )
Moving the slider, you get images like this:
This can easily be extended to a tessellation of the plane by squares and rhombuses.
Question: What does it look like if we do something similar with hexagons? You can see my attempt below. Also, are there any other regular polygons we can do something similar with?
No matter how tempting, I promise I'll move on to something else tomorrow!
(Source: rotating_hexagons.ggb)
Tags: geogebra, Maths, visualisation
Visualisation: More rotating squares
While we’re rotating polygons, here’s another nice visualisation, this time of a linkage of squares that make up a larger square, and rotate around without self-intersecting as you move the slider.
(Source: rotating_square_linkage.ggb)
Just as with the previous visualisation, you can see this just as a source of pretty pictures, like
or you can start asking yourself questions. For example:
- What other ways can we find to connect the small squares together so that they will expand and collapse back into a square without any of the small squares intersecting? How many of these have rotational or reflectional symmetry?
- How can we describe the movement (the locus) and the amount of rotation of the different squares?
- Can we do something similar with other polygons?
- More generally, can we do something like this to move smoothly between different shapes? And what does ’something like this’ mean?
For the second question, we can use the ‘trace’ feature of Geogebra (and all other dynamic geometry programs) to follow the path of particular points, which might give us some idea about how to derive the equations of the points.
On the last point, there are a large category of dissections which involve taking an object, slicing it into a finite number of shapes, and reassembling them into a different shape. There are several in Amusements in Mathematics, a puzzle book by Henry Dudeney from 1917 which I had a part in digitizing for Project Gutenberg. See this section, for example, for a great introduction to dissection puzzles.
Moving further afield, we can do something very similar to this example to demonstrate Pythagoras’ Theorem. That visualisation may appear later this week :).
Tags: Books, dissections, geogebra, visualisations
Visualisation: Rotating polygons
As mentioned in my last post, I frequently use Geogebra to create interesting little visualisations which can be used as starters, or just to fill in a transition gap. Several of them are animated using javascript, but I can’t yet figure out how to get javascript into Wordpress in a consistent manner, so no animations yet!
I like playing with these without initially setting any questions, and wait to see if any interesting questions arise from the students. If not, I let them know some of the questions which have crossed my mind. I didn’t pursue any of these questions last year, but may well next year.
So, in that spirit, have a play with these:
(Source: rotating-triangles1.ggb and rotating-squares1.ggb.)
Can you see how these were generated?
Pretty Pictures
The first thing I’d do with something like this is to play around with the slider, and see what happens, to try and understand what’s going on. You soon find that changing the number alters the picture, and some of the numbers produce ‘nicer’ pictures in some sense than others.
Here are some interesting configurations for the triangles:
And for the squares:
The Mechanics
Each file is created by starting with a shape (either a triangle or square) at the centre, and one parameter which you can change: an angle. The angle represents the amount by which each triangle (or square) is rotated anti-clockwise to create the shapes at the next ‘level’. Each level is given a different colour.
If you want to explore in more detail how the files were created, then (as with all my Geogebra worksheets) you just have to open them in Geogebra, and play — unhiding everything and looking at the definitions at points is always a good start.
Questions
- What other angles produce ‘interesting’ patterns?
- What angle would I need to have the shapes surround a pentagon, and what would the pattern look like?
- Can we calculate which angles result in exactly overlapping triangles/squares?
- What would a similar pattern look like using pentagons, or non-regular polygons?
- What would it look like if we added another ‘level’ of shapes?
- What about if we alternate clockwise and anticlockwise rotation?
I’d be interested to hear any answers to these questions… and (more importantly) any further related questions.
Tags: geogebra, polygons, visualisation
The wanderer returns
Preface
I’ve now returned from my holiday in Cumbria — default weather setting: rain. It would have been very relaxing if I hadn’t had my very active son with me, but I did, which at least meant I got out the house every day. We visited the Lake District, two castles, the highest cafe in the country, climbed several hills (mostly with him on my shoulders!), travelled on a steam train, had a climb over a Roman fort on Hadrian’s Wall, did a lot of playing on swings, and stroked a zebroid at Ostrich World.
Near-Term Posting Plan
I spent most of yesterday browsing through the Geogebra resources I’ve created in the last year, most of which I used as visual ‘thinkers’ — interesting visuals which would be on the board at the start of a lessons, to raise interesting questions with my students while waiting for everyone to arrive so the main lesson could get under way.
I should be posting at least one of these every day this week in interactive form, together with some interesting questions raised by them. Look for the first later on today.
Tags: a-levels, holiday, the future