Muzak to math by

We are in the throes of initial planning for a series of “Music and Maths” sessions aimed at 16-19 year old students, to culminate in a public performance. Using a mix of computing technologies and Blue Peter style building from scratch, the idea is to start from rediscovery of the twelve note scale and build up through construction of instruments.

The first problem we have encountered is an apparent dearth of devices or software which will listen to a note and read out its frequency. There are plenty of them (aimed at instrument tuning) which will do it the other way round, reading out a note name (C, F#, G, etc), but not a frequency. And although we did work out an alternative approach based on these guitar tuners, the interference from a building full of computing equipment, hearing aid loop generators, WiFi networks, several hundred cellphones etc, swamped them and made them useless.

A microphone attached to an oscilloscope is too unwieldy for our purpose: first introduce the oscilloscope, then explain the setting of time bases, learn to disregard noise … a one hour session would be over before anything useful had even stared. It will be useful and interesting further in, but not at the beginning.

Plan C involves auditory comparison of a tone generator signal to played keyboard and guitar string notes, by tweaking the frequency specified in the generator and deciding by consensus when a played note has been matched. This looks initially promising. We have started with NCH’s tone generator, which works well; the synthesiser at National Taiwan Normal University’s physics department also looks promising:

An alternative, offering sequential playing of different frequencies will be needed for subsequent work; a purpose made interface for preference, though it could be done using a mathematics package or even BASIC at a pinch. Ivor has written one as a Java Applet, but security measures in the browser environment where it will be used are raising barriers which have still to be resolved.

More as the idea progresses…

[contributed by Ivor McGillivray and Felix Grant]

The joy of equations

Part of my summer holiday was spent in trying to learn something about stuff outside the textbook areas of maths I’ve been looking at. They are fascinating, but because I’m still an arts and humanities girl at heart I needed something more romantic to lighten them up a bit.

My history teacher showed me some examples of how models can be used to try out ideas and see whether they fit what really happened in the past - for instance, I’ve played with a set of equations for the expansion of the Mongol empire mentioned in Sunstorm, and the spread of the Black Death in fourteenth century CE Europe. He also introduced me to sociology, where equations describe the behaviour of large numbers of people.Anyway, to get back to scientific computing, I find the way equations are written very beautiful but the way they go into a lot of software programs is ugly (especially spreadsheets). I often need to write them out myself before I can relate to them. Mr Grant lent me a computer with several programs which just write equations, the way you would by hand but typing them on screen. I’ve also been given a school copy of a free one (supplied by the government education ministry) to use on my own computer. I’ve had a lot of fun with these programs, and they have made the final connection between the excitement I feel about physics models and the “aesthetic me” that loves poetry and drama and painting.

The free program is Formulator Express, and is part of a set of programs given to teachers. I am very glad to have it for my own, but I hope to get my own copy of either MathType or Equations! (both of them have to be bought, but my uncle is talking about getting me one for my birthday). They are both very good, and do more than the free program, but I think different people would buy them. MathType appeals to the part of me which likes to write words, and Equations! pleases the bit of me that likes pictures - equations are both descriptions and pictures of something I can’t see with my eyes, only in my head.

All of these programs come down to picking and combining symbols, then letting the computer take care of drawing, spacing, arrangement and so on. The result is wonderfully sensual, with all the curves of a proper font setting off the beauty of the equation itself. They give you all sorts of ways to control and fine tune the way the equation looks, but won’t let you break the rules which control how an equation is supposed to look. They are magic. They have all the best bits of hand writing equations but let you adjust everything until it’s just right.

Equations! and Mathtype both help you to do a techie language called LaTex as well. I don’t think Formulator does, or if it does then I haven’t found it. I’m only just starting to figure this out, but it’s a way to describe equations. I’ve sort of got my head round the basic idea, but I don’t think I’d ever have the patience to get good at it - so it’s a good thing that these equation editors do a lot of it for you. For myself I found Equations! best for this part, it seemed more like the way I think, although MathType does whole pages of stuff at once.

I said before the summer that I had started painting equations. The equation editors have encouraged me to develop that work, and I have several sketchpads filled with arrangements of equations and graphs combined on the same page. (Apart from being beautiful, this is also useful. I tried to make sense of Einstein’s relativity stuff from a book, and got closest to understanding it through my montaged watercolour sketches.)

Now my English teacher (who started me on this stuff in the first place) and the art teacher have suggested that I work up some of my sketches into background scenery for a German play called Die Physiker, about Newton and Einstein. I am worried about this, as I don’t want to get known as a geek, but the idea does make me feel excited. I have done some experiments in the drama studio after school this term, putting Equations! equations and Autograph curves from the computer onto large sheets of calico to see how the forms and dynamics work together on a large scale.

[contributed by Lakshmi]

• Autograph and Equations! were supplied by Chartwell Yorke (who also stock MathType).
• Formulator (from Hermitech Laboratory in the Ukraine) is licensed for educational use as part of a standards pack from the UK Department for Education and Skills. The free version used by Lakshmi, Formulator Express, can be downloaded, or a full version purchased.
• MathType was supplied by Design Science.

Cabri3D: building big models on small beginnings

Over the summer, I spent a lot of time getting to know Cabri3D better, after the success with a simple net demonstration.

Truancy work has to continue through holidays - not at the same level as term time, perhaps, but there must be some continuity or the youngsters disappear you simply lose all that you’ve done. So, there have been drop ins and workshops at intervals over the summer. I used some of this time to get my young clients exploring Cabri on my behalf, letting them teach me - something which engages them in a way that a lesson the other way around can rarely do.

They particularly liked the “models” class of packaged examples, and that led to a lot of impromptu work in which I hastily learned about some of the ideas embraced by Lakshmi in earlier posts. They were fascinated by the basketball example, in which a single bounce through the hoop is repeated and rotated through 360 degrees. They also made the link for themselves between this sort of mathematical modelling and the animation of computer games - in fact they commented, without my prompting, that movement in video games is generally less realistic than the Cabri3D bounce or “Claude on a swing” and “Claude on a Trampoline” which cracked them up. The GPS system model appealed to the boys (though not the girls) as a techie toy.

Several of the girls were fascinated by “Escher’s stairs”, and that was their way into the actual works of Cabri3D - they wanted to know how it was done, and set about finding out. The boys were then challenged by macho pride into exploring how to do it as well. So now all of them are conversant with the Cabri3D innards, and are making progress with teaching me. Models have also, as a result, become a regular talking point, and basic maths is improving visibly in consequence.

All of which I call a worthwhile result

[contributed by BobTheBumbler]

• Cabr3D was supplied by Chartwell Yorke

Polaris and me

I was going to review Polaris, a science fiction novel by Jack McDevitt. I’ve also been asked to write about what has happened to me since I reviewed Sunstorm as well. They have a lot to do with each other and I don’t think I can do them separately. So am doing them both together, and I hope it makes sense.

Before my English teacher recommended Sunstorm I was not interested in maths or science at all. In this essay I am going to save a lot of explanation by just using bold type to show things and ideas which are new to me since I started reading Sunstorm. I am glad that I was told to use a pen name, because if my friends knew I was writing this I would be socially dead forever.

After I reviewed Sunstorm, I read Donna’s review of Seeker. The thing that I liked most about Sunstorm was the idea of a planet being fired across space to hit a sun, like a stone being fired at a target with a catapult. Then my maths teacher showed me how to model this on a computer, and I realised that it’s actually more like firing the stone from a catapult in London and hitting a melon in Australia or somewhere. Anyway, Donna’s review mentioned that something similar happened in Seeker, so I read that as well.

I found that Seeker is the last book in a set of three about the same characters (the first is A Talent for War and Polaris is in the middle). So then I read the other two as well. All of the books have the same pattern: there is a mystery, the main characters discover it through something to do with the antiques trade, historical research gets them close to solving the mystery, and the mathematics of moving bodies finally gives them the answer. The mysteries are all different, and make you want to read to the end, but I won’t spoil them by describing them here - and anyway, it’s the maths bits that interest me (I never thought that I would hear myself say that). The historical research interests me too.

In Seeker the maths was about how a stellar system is affected by a brown dwarf star passing close by. In A Talent for War, it’s where a spaceship would be after two hundred years. And in Polaris it’s sort of like a cross between Sunstorm and Seeker because a small but super dense star called a white dwarf hits an ordinary G class star like our sun (not deliberately, it just happens) and goes straight through it and out the other side and destroys it.

I have got totally into this moving bodies stuff. I find the ideas exciting. My maths teacher has shown me how to find information about it and I have done a lot of reading. He has also shown me how to use a spreadsheet and a program called Autograph to set up and investigate my own models. I have learnt a learnt a lot but the the biggest thing I’ve learnt is that I have gone as far as I can without learning some pretty scary maths.

I have started studying some AS maths modules on my own. Well not really on my own because my maths teacher is helping me before school and my uncle is helping me at home but I mean not in a class or anything. I have completed module M1, which is the first mechanics module, and started on M2. Mechanics is what they call the sort of maths that will eventually let me cover orbits and trajectories and stuff (M1 and M2 don’t get that far, but I need to understand the basics). To understand some of the mechanics I need other maths, called pure maths, which doesn’t have anything necessarily to do with mechanics but you use it as a sort of way to describe things - my English teacher pointed out that it’s like I can only enjoy poetry if I can already read. So I’ve done quite a bit of P1 as well (that’s the first pure maths module).

I am using some software called Derive to help me with understanding the maths I am doing. There’s a lot of other software as well and none of it would be so exciting without the models which they let you build to try things out.

I’ve done a little bit of calculus with my maths teacher and my uncle. Calculus is when you imagine very small bits of a problem so you can get your head round it, then imagine that small bit happening over and over again, forever, to make it back into the big problem again but now you understand it. I haven’t explained that very well, but it’s important and it works. Its how you can start with the velocity of something, and the gravity of a star pulling it, and see where it will go, or the other way round.

By September I think I will have finished all three AS modules. My uncle says I could take the AS exam, even though I won’t have done my GCSE yet. But that would totally blow my cover and everyone would think I was a geek. My teacher says he’ll see if I can take it somewhere else that nobody knows me. I don’t know. I’ll see.

Doing all this other stuff has made me better in ordinary school maths and science too. I used to be rubbish at algebra, but now it seems easy. I know now that when you do experiments you do them lots of times and then look at all the results, not just one, and now the handling data part of maths makes sense too (but I don’t want to do the S1 statistics module cos that looks really scary).

My maths teacher has set up some experiments for me, like rolling a marble across a rubber sheet on a frame. You can poke your finger into the rubber, or put a lead weight on it, and pretend the dent is a gravity well and see what happens when the marble (which is supposed to be a lump of rock in space) passes near it at different speeds. And we tried firing an air gun through an egg in front of a video camera to see what might happen when the white dwarf goes through the G type star in Polaris, which is a physical model instead of the mathematical models which you do with pen and paper or with software.

I’ve started to think about what I want to do in my life. I am still most interested in literature and drama but I’m interested in other things too. I’ve been doing paintings and models from the shapes that all the trajectory models make, and imagined using them for stage sets - weird or what? I just tell my friends they’re abstracts. Because of these novels by Jack McDevitt I’ve got really into history as well, and I’ve seen the same sort of graph shapes in history books as in mechanics, like the way population grows looks like the way a rocket’s height changes as it takes off.

It would be nice to do everything, but I’m not sure you can. People seem to do one thing or the other. Mr Grant who organises this site and asked me to write about this stuff says he did literature as well as maths and sciences when he did his A levels but he’s quite old and I think things have changed since his day. He says that people who write books like Sunstorm and Seeker need to understand the maths and science as well as being able to write, and Jack McDevitt must understand history too, and I suppose that’s true. But A levels are a long way yet. I don’t even start my GCSE subjects until September.

Well, that’s a little bit about Polaris and quite a lot about what’s happened to me since I read Sunstorm. I hope it wasn’t too boring. And I hope nobody I know ever realises who I am.

[contributed by Lakshmi]

McDevitt, J., A talent for war. 1989, Sphere. 0747403333.
McDevitt, J., Polaris. 2004, New York, Ace Books. 0441012027.
McDevitt, J., Seeker. 2005, New York, Ace Books. 0441013295.
Clarke, A.C. and Baxter, S. Sunstorm: A time odyssey. 2006, London, Gollancz. 0575078014