Archive for geometry

The Problem of the Moving Triangle

Posted in Cute Problems, mathematics with tags , , on August 16, 2018 by telescoper

I found this nice geometric puzzle a few days ago on Twitter. It’s not too hard, but I thought I’d put it in the `Cute Problems‘ folder.

In the above diagram, the small equilateral triangle moves about inside the larger one in such a way that it keeps the orientation shown. What can you say about the sum a+b+c?

Answers through the comments box please, and please show your working!

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Puzzle Time: Playing With Matches

Posted in Cute Problems with tags , , on January 13, 2015 by telescoper

matchesPuzzle time! Move three (and only three) matches and position them to create just four, i.e four and only four,  (identical) triangles. No cutting the matches, either!

Click on to see the answer…

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Pseudospheres Corner..

Posted in Cute Problems with tags , , , , on November 28, 2013 by telescoper

I’m sure you have all seen a knitted pseudosphere, but this is a particularly fine collection made by the excellent Miss Lemon and briefly displayed in my office this morning.

IMG-20131128-00223

A pseudosphere is a space of negative curvature (whereas a sphere is one of constant positive curvature). There are various ways to realize a two-dimensional surface which has negative curvature everywhere; this knitted version is based on hyperbolic space. If you’re keen to have a go at making one yourself you can find some instructions here. I’m advised, though, that the better way to approach the task is to start out with a large circular ring onto which you cast about 400 stitches, gradually working your way in with fewer and fewer stitches (say 400,200,100,50 etc), which is much easier than working outwards as described in the link. The folds and crenellations are produced quite naturally as a consequence of tension in the wool.

Happy knitting!

A Paradox of Galileo

Posted in Cute Problems with tags , , , on June 1, 2011 by telescoper

Going by the popularity of the little physics problem I posted last week, I thought some of you might be interested in this little conundrum which dates back to the 17th Century (to Galileo Galilei, in fact). It’s not a problem to which there’s a snappy answer, so no poll this time, but I think it’s quite a good one to think about – and please try to resist the temptation to google it!

The above figure shows a large circular wheel, which rolls from left to right, without slipping, on a flat surface, along a straight line from P to Q, making exactly one revolution as it does so. The distance PQ is thus equal to the circumference of the wheel.

Now, consider the small circle, firmly fixed to the larger circle with the two centres coincident. The small circle also makes one complete revolution as the wheel rolls, and  it travels from R to S. Similarly, therefore, the distance RS must be the circumference of the small circle.

Since RS is clearly equal to PQ it follows that the circumferences of the large wheel and the small circle must be equal!

Since the radii of the large and  small circles are different,  this conclusion is clearly false so what’s wrong with the argument?

Whatever happened to Euclid?

Posted in Education, The Universe and Stuff with tags , , , , , , on May 24, 2011 by telescoper

An interesting article on the BBC website about the innate nature of our understanding of geometry reminded me that I have been meaning to post something about the importance of geometry in mathematics education – and, more accurately, the damaging consequences of the lack of geometry in the modern curriculum.

When I was a lad – yes, it’s one of those tedious posts about how things were better in the old days – we grammar school kids spent a disproportionate amount of time learning geometry in pretty much the way it has been taught since the days of Euclid. In fact, I still have a copy of the classic Hall & Stevens textbook based on Euclid’s Elements, from which I scanned the proof shown below (after checking that it’s now out of copyright).

This, Proposition 5 of Book I of the Elements, is in fact quite a famous proof known as the Pons Asinorum:

The old-fashioned way we learned geometry required us to prove all kinds of bizarre theorems concerning the shapes and sizes of triangles and parallelograms, properties of chords intersecting circles, angles subtended by various things, tangents to circles, and so on and so forth. Although I still remember various interesting results I had to prove way back then – such as the fact that the angle subtended by a chord at the centre of a circle is twice that subtended at the circumference (Book III, Proposition 20) – I haven’t actually used many of them since. The one notable exception I can think of is Pythagoras’ Theorem (Book I, Proposition 47), which is of course extremely useful in many branches of physics.

The apparent irrelevance of most of the theorems one was required to prove is no doubt the reason why “modern” high school mathematics syllabuses have ditched this formal approach to geometry. I think this was a big mistake. The bottom line in a geometrical proof is not what’s important – it’s how you get there. In particular, it’s learning how to structure a mathematical argument.

That goes not only for proving theorems, but also for solving problems; many of Euclid’s propositions are problems rather than theorems, in fact. I remember well being taught to end the proof of a theorem with QED (Quod Erat Demonstrandum; “which was to be proved”) but end the solution of a problem with QEF (Quod Erat Faciendum; “which was to be done”).

You can see what I mean by looking at the Pons Asinorum, which is a very simple theorem to prove but which illustrates the general structure:

  1. GIVEN
  2. TO PROVE
  3. CONSTRUCTION
  4. PROOF

When you have completed many geometrical proofs this way it becomes second nature to confront any  problem in mathematics (or physics) by first writing down what is given (or can be assumed), often including the drawing of a diagram. These are key ingredients of a successful problem solving strategy. Next you have to understand precisely what you need to prove, so write that down too. It seems trivial, but writing things down on paper really does help. Not all theorems require a “construction”, and that’s usually the bit where ingenuity comes in so is more difficult. However, the “proof” then follows as a series of logical deductions, with reference to earlier (proved) propositions given in the margin.

This structure carries over perfectly well to problems involving algebra or calculus (or even non-Euclidean geometry) but I think classical geometry provides the ideal context to learn it because it involves visual as well as symbolic logic – it’s not just abstract reasoning in that compasses, rulers and protractors can help you!

I don’t think it’s a particular problem for universites that relatively few students know how to prove the perpendicular bisector theorem, but it definitely is a problem that so many have no idea what a mathemetical proof should look like.

Come back Euclid, all is forgiven!

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