“In preparing a lecture I find I always have to work hardest on the things I do not say. The things I am sure to say I can easily get up. They are obvious and generally accessible. But they, I find, are not enough. I must have a broad background of knowledge which does not appear in speech. I have to go over my entire subject and see how the things I am to say look in their various relations, tracing out connections which I shall not present to my class.

One might ask what is the use of this? Why prepare more matter than can be used? Every successful teacher knows. I cannot teach right up to the edge of my knowledge without a fear of falling off. My pupils discover this fear, and my words are ineffective. They feel the influence of what I do not say. One cannot precisely explain it; but when I move freely across my subject as if it mattered little on what part of it I rest, they get a sense of assured power which is compulsive and fructifying.”

The Teacher: Essays and Addresses on Education, page 17. Written in 1908. I find this quote deeply relevant and stimulating, as a classroom teacher who is currently preparing for the return of school next week!

I’m a maths teacher, and this next quote could easily have been written about the current trends in the teaching of my subject:

“Among the many changes in mathematical education during the last twenty years, and among the many and often conflicting ideals which have directed these changes, one element at least appears throughout; a desire to relate the subject to reality, to exhibit it as a living body of thought which can and does influence human life at a multitude of points… Our children must learn to think.”

This is from page 35 of Essays on Mathematical Education, written in 1913.

These are just two of more than 8000 results that appear when you search archive.org for texts mentioning ‘education’.

Places like archive.org allow us to correct the notion many people have about the way people were taught in the past. Vocational education; project-based teaching; differentiation; learning styles; curriculum content; the importance of the physical education of youngsters — all of these and more have been considered by teachers for many generations.

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“Is he married or single?”

“Oh! Single, my dear, to be sure! A single man of large
fortune; four or five thousand a year. What a fine thing for our
girls!”

(Pride And Prejudice, first published 1813, by Jane Austen.)

If you’re like me, you’ve wondered from time to time exactly how large this ‘large fortune’ is, in today’s money. What about someone strugging on a couple of hundred a year? Several years ago I stumbled across an interesting UK government research paper which attempted to answer that question, by tracking the purchasing power of the pound through the last 250 years. It could easily be used as a rich resource for a variety of activities in the classroom.

The most recent edition of the document was published in 2006, and is called ‘Inflation: the value of the pound 1750-2005‘. It provides a series of tables which enable us to convert prices between any two years from 1750 onwards.

From the introduction to the research paper:

This paper presents a price index covering the period 1750 to 2005 and illustrates the changing purchasing power of the pound over the long-term. No attempt is made to measure changes in the external value of the currency as a result of movements in exchange rates, but changes in the prices of imported goods are reflected in the price index.

It must be stressed that, for a number of reasons, such an exercise is very approximate. Expenditure patterns have changed dramatically over the past 250 years. Many products now commonly purchased (cars, electrical appliances, processed foods, etc) simply did not exist in 1750 and, conversely, goods that consumed a large share of household budgets in the eighteenth century - candles for instance - are now an insignificant part of most families’ expenditure.

It is, however, possible to compare price levels over the long-term by linking price indices covering relatively short periods into a single series.

The heart of the document is a table giving a relative price index for each year from 1750 to 2005 (artibrarily setting the value of the pound on January 1974 to be 100), as well as the inverse of this, which they call the ‘purchasing power‘

Let’s take the ‘large fortune’ I quoted at the start as an example of how to interpret the values.

The price index for 1813 was 16.3, and that for 2005 was 757.2. So in 2005 the average price level was roughly 46.5 times the 1813 level. This means that to have the same purchasing power in 2005 as £5000 had in 1813, we would need an income of around £230,000 a year. Not too shabby!

We’d get an even larger result if we’d used the value at other years close to 1813 — the pound in 1813 was at its weakest point since 1750. It soon recovered, and wouldn’t return to that low level until 1917 — over 100 years later.

Purchasing power and inflation

This research paper highlights a key way in which the character of money has changed over time. As the charts (which are reproduced below) show, the world we live in today, where prices rise year on year, is a fairly recent phenomenon. The first world war triggered a large jump in inflation, but we find significant deflation through the 20s and 30s — a pound was worth 50% more in 1935 than in 1920. The second world war triggered another bout of inflation, and we’ve been on the inflationary roller-coaster ever since — to the extent that a pound today is worth less than 10% of a pound in 1970!

As the research paper points out, prices have risen every year since 1945. Positive inflation is now built into modern economic theory (it encourages you to spend, rather than hoard, as the hoarded money will become less valuable over time), and ‘deflation’ is now looked upon as something to be avoided at all costs. Indeed, the Bank of England is instructed to keep inflation within a band around 2.5% per year, and is required to act when inflation is ‘too low’ just as much as when it gets ‘too high’.

We can see the contrast between the periods of price stability and price inflation very clearly if we graph the price index:

The price index is essentially flat for hundreds of years, and then ‘takes off’ after the second world war.

This linear graph does, however, hide some of the quite severe price fluctuations that occured even during the period of price stability. We can get a better idea of the relative changes in the value of the pound if we use a log scale, rather than a linear scale, for the price index:

Remember that both of these charts show the value of the price index — the higher the number, the less the currency is worth. In a sense, the value of the pound is the inverse of this price index. A (logarithmic) plot of the value of the pound over the last 250 years (scaled to make the value in January 1974 equal to 100) looks like this:

We can see very clearly here the fact mentioned earlier — that the value of the pound in 1813 was at a low point which would not be reached again for over 100 years. We can also see the deflationary period after the first world war, and the constant loss in value of the pound ever since 1945 (with a particularly steep loss in value during the 1970s — the period of double-digit inflation rates).

Further thoughts

In the middle of the 19th century, according to this site,

Servants, who had all living expenses taken care of, earned as little as £10/year, and the sign of being (or having become) a member of the middle class was having at least one servant. Some poor vicars at mid-century earned as little as £40-50/year.

Would you be able to survive on the modern equivalent of the poor vicar’s wage?

An interesting further investigation would be to get hold of some information on the wages of various occupations, and try to translate them into modern equivalents — or, equivalently, to try and convert modern salaries into those of Victorian England. Many interesting difficulties await!

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(graph generated by Graphviz, from this source file.)

Continue reading for an explanation.

Background

Euclid’s Elements is one of the classics of world literature, with an importance that transcends mathematics. Various editions of it were a vital part of western education for literally thousands of years. And then, in the first half of the 20th century, it slowly disappeared from the curriculum.

By the time I went to school, in the late 1980s, Euclidean geometry, and the definition-proposition-demonstration-proof style was seen as distinctly old-fashioned. We had to memorise a couple of circle theorems and apply them in a trivial way to pass GCSE Maths, but that was about it… and even those weren’t needed at A-level (or for the IB, which is the examination I took). It was equally neglected during my University maths degree.

The year after I graduated I rediscovered Euclid’s Elements though my involvement in Project Gutenberg and Distributed Proofreaders. Copies of old books were beginning to be scanned and made available for free on the internet, on sites which were the forerunners of Archive.org and Google Books.

Browsing through the Project Gutenberg archives, I was amazed that they didn’t have an edition of Euclid’s Elements, and decided that I would try and produce one for myself. Although I had never looked at the Elements myself, I knew of its reputation as one of the founding texts of mathematics. I found a public domain edition, and started to type it up. I soon found that it would be a much bigger task than I expected — and never completed this task (although I later helped with this edition, which was processed through Distributed Proofreaders). It did spur me to actually read the Elements for the first time, though.

The Pleasure of Euclid

Folk knowledge about Euclid’s Elements:

• Everything in the Elements is derived from five postulates.
• Four of these postulates are ’self-evident’ — the fifth (the ‘parallel postulate’) is more convoluted, and exploring alterations to it led to the non-Euclidean geometry of the 19th century.
• The main focus of the first book of Euclid is the proof of Pythagoras’ Theorem.
• It’s a boring, old, dusty and hard-to-read book that no one needs to read any more.

This is what I knew about Euclid before reading it.

The first two points were mostly correct (there are also definitions and common notions), and I’ll discuss the third point later. The last point, however, is one I disagree with — many people (and particular teachers of mathematics) would profit from reading the Elements, if only to get a deeper perspective on the history and development of mathematics.

From the very first proposition:

To construct an equilateral triangle on a given finite straight line.

one is challenged to think, and to be an active participant. This proposition is one of fourteen in the first book which involves finding a construction — and also, crucially, proving that the construction is valid, using only more basic knowledge. It is very easy to let the proofs ‘wash over’ you (something which I imagine schoolkids who studied Euclid did for hundreds of years!), but by doing this you miss one of the main points of the Elements — that all the statements and constructions made can actually be justified.

I found reading the propositions rather like reading a murder mystery novel: you knew the protagonists (the people at the mansion / the given points and lines), and you knew the goal (finding the murder / proving the statement). The Elements without the (sometimes difficult and convoluted) proofs would be like an Agatha Christie novel which revealed the murderer on page twenty.

Visualising the Relationships Between Propositions

As well as trying to understand each individual proposition, I was also trying to understand the relationships between the propositions — was a particular proposition an end in itself, or was it only used as a stepping stone to greater things?

Most editions of the Elements record in a margin when a proposition makes use of prior propositions — proposition 16 (“in any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles“), for example, uses propositions 3, 4, 10, and 15 (as well as two postulates and a common notion). Keeping track of the dependancy relationships can be hard, though, and is obscured by the obvious need to write the propositions in a linear order.

In an effort to understand the relationships between the propositions, I created the diagram given at the start of the post.

The graph connects two propositions together if one is used in the proof of the other.

For example, the top portion of the graph,

shows that:

• Proposition 2 relies on proposition 1;
• Proposition 3 relies on proposition 2;
• Proposition 5 relies on proposition 3 and 4;
• Proposition 6 relies on proposition 3.

Actually, it’s a touch more complicated. Technically, I’m showing you the transitive reduction of the dependancy relationships of the propositions. All this means is that, for example, I don’t draw an arrow from ‘4′ to ‘16′ (even though proposition 4 is used directly in the demonstration of proposition 16), because there is a chain of arrows that connects ‘4′ and ‘16′ indirectly through a path of other propositions.

Graphing the relationships between the propositions in this way reveals some quite interesting things:

• For a start, there are some propositions proved very early on, which are never used again — like proposition 6 (”if in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.”) and proposition 12. In both these cases, the propositions are used in later books of the Elements. Others, such as proposition 40, are used nowhere else in the Elements, and are conjectured to be later additions (or ‘interpolations’) by other authors.
• It demonstrates how everything in book 1 is proved from two basic propositions: 1 and 4.
• It highlights the relatively convoluted relationships between the propositions in the second half of the first book.
• It demonstrates just how long the chain of reasoning is behind some of the propositions. Proposition 25, for example, relies on a chain of 15 other propositions.

A diagram like this can also help us to investigate the third item of folk knowledge I mentioned above — that the point of the first book of Euclid is to prove Pythagoras’ Theorem.

Pythagoras’ Theorem and the Elements

It’s certainly true that, in one sense, the culmination of the first book is Pythagoras’ Theorem — there are 48 propositions, and ‘the square on the hypotenuse…’ is proposition 47. In another sense, though, Pythagoras is only one of several threads.

To explore this a little, let’s shade all the propositions relied upon in Euclid’s proof of Pythagoras’ Theorem

Although almost all the propositions from the first half of the book are shaded, the majority of the propositions in the second half have nothing to with directly proving Pythagoras’ Theorem. While some are just corollaries or converses of previous propositions, and so would logically be placed close to the original proposition, the dependency diagram clearly shows that a significant portion of the propositions are there to lay the groundwork for proposition 45.

Proposition 45 is a construction, the aim of which is

to construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.

The importance of this construction, and of the other propositions which it depends on, is that it means that given any figure made up of straight line segments, we can construct a parallelogram of exactly the same area, which has any specified angle — and so, in particular, we can construct a rectangle with the same area. The ultimate aim is to be able to quadrate shapes: to be able to construct a square with the same area as a given figure. This is done (for rectilinear shapes at least) in proposition 14 of book 2.

We can now see that Pythagoras’ Theorem, which is about the areas of squares on the sides of triangles, is just one part of a more comprehensive study of area in the first book of the Elements.

Exploring Euclid for Yourself

I don’t claim that everyone should read Euclid’s Elements, and I certainly don’t want to return to the days where 11 year olds were forced to memorize and recite propositions and demonstrations. With that said, I think exploring the first couple of books of the Elements is a great way to improve understanding of a facet of mathematics which is often overlooked these days.

The most commonly seen book version of Euclid’s Elements you find these days is the academic and heavily annotated 1923 translation “The Thirteen Books of Euclid” by Sir Thomas Heath. While a great resource, it is very dry, and not the best way to approach Euclid if you’ve never read it before.

I’d recommend instead this interactive version of the Elements, created by David Joyce, a Professor at Clark University, which I have linked to several times already in this post. Every diagram is dynamic, and there are extensive notes on almost every proposition.

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• What behaviours do the lines exhibit?
• What relationships are there between the different lines?
• How are these lines defined?

Geogebra applet (enable Java to see it).

(Source: movinglines.ggb and movinglines.html.)

Click on ‘Show Values’ and three values will appear — all six of these lines are generated from these three values. If you click on ‘Stop’, and move the sliders, then the lines will be automatically updated to reflect those new values.

• Can you make any conjectures now about how the lines are defined?
• What values would we need to make one (or more) pair of the lines
  • perpendicular?
  • parallel?
  • vertical?
  • horizontal?
• Can we generate any line we wish by setting the values appropriately?

I managed to captivate both a lower school class and an A-level class with this visualisation for quite a while last year, and the A-level class made good progress in making and justifying conjectures about what was going on.

Technical Note

One of the few downsides of Geogebra compared to other dynamic geometry software, is that it contains no in-built facility for animation. Thus, to create any animations using Geogebra, you have to create the animations manually in Javascript. It’s not particularly difficult to do, particularly if you know some Javascript, as you can see if you view the source of the .html file above. Putting the ability to animate inside Geogebra itself is, though, my one big wish for future versions.

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(Source: mean-and-frequency-in-excel.swf)

This was created using the excellent (and free!) piece of software Wink.

Wink is basically a free version of software like Turbodemo (or for the more Web 2.0 people out there, Jing), which allows you to create ’screencasts’, capturing screenshots and turning them into standalone animations (in Wink’s case, Flash .swf files). These are used in the help files of many pieces of software, like the graphing package Autograph. As well as capturing screenshots, and keypresses, you can add annotations, and link forward/back to different sections of the recording.

Screencasts like this have great potential for demonstrations, not just of technology, but also as ways of recording how to solve maths problems. They also let a teacher run through a problem without having write material on the board constantly. The downside, of course, is that they do require a significant amount of effort to produce, but sites like MathCasts are beginning to offer a number of premade screencasts, which I need to look through at some point!

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This is an way of developing the use of the chi-squared distribution, which can also be used to test whether your students can remember what the Binomial distribution looks like!

Start by challenging everyone in the class to generate some data which they feel could be modelled by a Binomial(5,p) distribution, for some value of p. Calculate the value of chi-squared for the frequencies entered (which involves reviewing how to estimate the mean, and what the formula for Binomial is), and then compare that with the critical value needed for the data to be a ‘good fit’ (to, say, a 5% level).

Interactive Binomial Fitness Calculator

I thought I would set myself the challenge of converting this activity into a form which could be directly placed on a Webpage, like this one. After a few days messing around (and a morning wondering why Wordpress didn’t like my Javascript), I can present the following:

(Source: testbinomialmodel.html.)

Note that the condition it is using for goodness of fit is the 5% critical value for chi-squared with four degrees of freedom (6 - 1 because we know the total frequency - 1 because we’re estimating the probability). Note also that it does not combine cells.

Uses

Beyond an initial check of how good students are at modelling a Binomial distribution, this interactive tool can also be used as a tool to explore the Binomial and chi-squared, by systematically altering values and seeing what happens. We can also do something similar, but fix a particular value for the probability of success — this makes it easier to improve by ‘trial and error’ toward a fixed destination, and would also allow us to discuss whether a fit can be too close.

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Dudeney was responsible for one of the most well known hinged dissections, of which this is a simplified example. It converts a square into, well, something else.

The white circles are the hinges — move the coloured circles to move the corresponding parts of the square.

Can you predict what the end result of the transformation is? Can you prove it?

Geogebra applet (enable Java to see it).

(Source: dudeney-dissection.ggb.)

The Haberdasher’s Puzzle

Dudeney’s classic dissection, published in his ‘Canterbury Puzzles‘ in 1907, is a slightly altered version of this, which allows you to transform a square into an equilateral triangle. You can download a program which will allow you print out a template for this here.

Constructing Dudeney’s dissection takes a touch more effort than the dissection illustrated above, but the process is described incredibly well in this lesson plan, which demonstrates how to make a model of the dissection using foam rubber. I haven’t tried it yet, but it may make a tempting break from lesson planning next week!

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Find the locus of the stationary points of these quadratics.

The following Geogebra worksheet might help. If you select “Show stationary point” and move the blue point, it will trace out the locus.

Geogebra applet (enable Java to see it).

(Source: quadratic_stationary_points.ggb.)

Thinking about this question kept me from going to bed until far too late on Thursday. My answer, and other thoughts, will appear tomorrow.

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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.]

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Geogebra applet will be here if you enable Java.

(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!

Geogebra applet will be here if you enable Java.

(Source: rotating_hexagons.ggb)

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