Posts Tagged ‘visualisation’
Exploring Euclid’s Elements
A recent post at God Plays Dice, which linked to this intriguing visualisation of the 2008 Olympics medals table as a partially-ordered set, reminded me of a diagram I created a while back, while reading through the first book of Euclid’s Elements:
(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.
Tags: euclid, Maths, project gutenberg, visualisation
Visualisation: Six moving lines
As promised, here is an animated Geogebra visualisation. Click ‘Start’, and six moving lines will appear.
- What behaviours do the lines exhibit?
- What relationships are there between the different lines?
- How are these lines defined?
(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.
Tags: animation, geogebra, lines, visualisation
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: 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