Walking Randomly

Internet Explorer 6 sucks…a lot! Despite this, many people continue to use it and Walking Randomly has always looked rubbish in it tahnks to some bugs in the rendering engine of IE6. Well, I mentioned this to the guru who supplies the hosting for the site and he said he’d take a look. I have no idea what he has done but it seems to have done the trick – WR is now much easier on the eyes for IE6 users. It’s still not as good as in Firefox but there is only so much one can do with a sow’s ear.

For those of you stuck with IE6 – I hope you enjoy the improvement. For those of you with a choice – go get Firefox and use that instead. You’ll thank me if you do.

Some discoveries in science and mathematics can be rather technical to say the least, understood and appreciated by relatively few people who have worked for years in some esoteric subject. Other discoveries can touch the imagination of pretty much every one of us. While reading through a collection of news articles on the train this morning I came across one of the latter which caused me to be come so excited that I temporarily forgot that I was British.

I turned to the random woman sat next to me and said in an excited voice “They’ve found water on Mars – beyond a shadow of doubt. They’ve as good as tasted it. Look…” and I presented my phone to her with the news article on it. To be fair to her she did make a show of reading it and nodded in encouragement before moving to a seat far, far away from me.

Over the last few years scientists have tantalised us with various snippets of evidence that point to the possibility of there being water on Mars. For example, back in 2002, NASA Space probe Odyssey discovered large quantities of Hydrogen under the surface of Mars via Gamma Ray Spectrometry. From this data it was inferred that there might be large quantites of water under the Martian surface but no one could be totally sure until a sample of it was scooped up and tested in a laboratory More information concerning the history of discoveries concerning the discovery of water on Mars can be found at this link.

Recently, NASA sent a laboratory to Mars in the form of the Phoenix Mars Lander which scooped up a sample of Martian soil, tested it and confirmed that it did indeed contain water. So now it’s official – Mars has water beyond any doubt which is a very cool scientific fact in my humble opinion.

This news has been spreading through the web like wildfire – and rightly so – but before we get too carried away we need to take a step back and think about what we actually know. (Take the following with a handful of salt – I am not a planetary scientist and, therefore, may be talking rubbish but I have included the links that led me to making these statements)

They have found water but they do not know its chemical composition yet. It may need a lot of processing before it’s drinkable due to a high concentration of salt and acid or it may be as pure as (Earth) snow – we simply don’t know. The answer to this question is important if humans are ever going to visit Mars – they’ll need a drink!

NASA have not detected the signs of life (check out their twitter feed to see what they have to say about this) but the presence of water makes it more statistically likely. That’s more likely not almost certain. If I buy 2 lottery tickets this week then I am more likely to win compared to only buying one (twice as likely in fact – unless I am stupid and choose the same numbers twice!) but I still probably won’t win. Little green men there are not!

Oh but imagine if we did discover life there – how wonderful would it be? We have to face facts – any life we might find would be no more complicated than viruses or bacteria but even that would almost certainly open up a whole new field of biology. They might be completely different to anything he have on Earth – with different biochemistry and everything…or they could look very similar to life here. Who knows? It’s going to be fun finding out though.

For more information – check out Popular Mechanics.

We live in interesting times.

The 37th Carnival of Mathematics was posted over at Logicnest last week with 13 (if I have counted correctly) articles for your reading pleasure. There is some really good stuff to be found in this edition of the carnival but I won’t spoil the show by telling you all about it here.

The next Carnival will be hosted by catsynth, although the carnival submission form doesn’t seem to know that yet. If you are a blogger and have recently written a maths based post then please think about sending in a submission.

No integral this time – I’m saving those for another day. This week’s problem is

Give a proof of the relationship between the Fibonacci Numbers and the determinant of the matrices discussed in this post.

As always, this is just for fun as there are no prizes (unless, of course, some kind sponsor would like to help me out).

Feel free to give submissions via the comments section but, be warned, it doesn’t support latex. Probably the best way to submit your solutions would be to send them to my email address. I’ll take just about any format – even Microsoft word – but pdf or Latex is probably best. Solutions that I think to be correct will be published here at a later date (subject to obtaining your permission of course).

Have fun.

Update: Click here for the solution.

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Many years ago (way WAY before the web), at the tender age of 10, I did a school math project about the Fibonacci numbers and got rather carried away with writing about the many different areas of mathematics and everyday life where this sequence popped up. Although I didn’t have the internet to help with my research, I did have a wonderful maths teacher called Ron Billington who taught for many years at Birchensale Middle School in Redditch. Mr Billington had a personal library of maths books that he collected over the years which was a treasure trove of material for someone like me who had significantly more enthusiasm than talent. He would have loved hearing about the little discovery I made while browsing through The College Mathematics Journal the other day.

First, a bit of background. The Fibonacci sequence starts off like this:

1 1 2 3 5 8 13 21

Each term in the sequence is formed from the sum of it’s two predecessors so the next term would be 13 + 21 = 34. What fascinated me as a child (and continues to fascinate me now) is the fact that this incredibly simple sequence of numbers, and others like it, seems to appear all over the place from the distribution of sunflower seeds to the study of photonic crystals.

There really is an astonishing amount of mathematics around the Fibonacci sequence as you will be able to verify with a quick google search. There is even an academic journal dedicated to the mathematics around it – The Fibonacci Quarterly – which I, unfortunately, have no access to at the moment (might have to have a word with the University librarian about that).

In the article Fibonacci Determinants, by Nathan Cahill et al (The College Mathematics Journal vol33 p221-225), the authors demonstrate the fact that you can obtain the nth term in the Fibonacci sequence by taking the determinant of a n x n tridiagonal matrix of the form

What’s more, if you change just a single entry (row 2 col 2) from 1 to 2 then you will obtain the Lucas Numbers instead. I thought that this was fun and so knocked up a Wolfram Demonstration for it which you can get to by clicking on the image below.

So Mr Billington – I have found yet another branch of Mathematics where the Fibonacci numbers turn up – Linear Algebra. I know it’s been 20 years but is there any chance of upping that B- to an A :) ?

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