Archimedean AND Logarithmic Spiralling!!!

I was at a meal this evening with a Mathmo from Trinity - he was really lovely, and definitely stereotypically a Mathmo!!! We had a GREAT time! It was a meal with a family from church; because we're freshers and new to the area, it's really nice to be in a family environment sometimes! Anyway, so they had circular table mats a bit like (not exactly but I felt it would be slightly frowned upon to take out my phone to take a picture of their table mats during dinner) the picture above. And me and this mathmo suddenly decided it would be normal - having said that, who wants to be "normal" anyway?! - to determine what kind of spiral the pattern on the mats was making. 

Now you can see that it is an Archimedean spiral. It spirals out from the centre such that each "strand" is the thickness and each would reach the same radius line (if you drew it on) the same width from the last one as the next one does. I mean, if you want to be REALLY technical, the above illustration doesn't show this as well as the ones at dinner because the individual strands in the ones at dinner seemed more twisted themselves and so the whole thing was tighter - whereas this one looks tighter in the centre than it does as this spreads out. I think it's a different kind of weaving; but still the best example I could find!

Now if you look up what an Archimedean spiral is, you will soon come across it contrasted against a logarithmic spiral. This is because whereas an Archimedean spiral reaches a radius line at equally spaced intervals; with a logarithmic curve, the spacing of this gets wider and wider as the spiral goes on. So how, I hear you ask, can it be both?!

Well... If you actually tried to trace your finger round the spiral, following the path of least resistance or just in a way that meant you allowed the pattern to guide where your finger was going, you quickly found that your finger was actually spiralling out much more like a logarithmic curve than like the Archimedean curve you expect from looking at it.

Which is REALLY cool. I mean, who doesn't like surprises?! 

The reason for this is because, for any point in the spiral, if you drew a line from the radius out to that point, and then another line showing the gradient of the spiral at that point, the angle between those two lines would be constant all the way round the circle.

Now this is VERY unlikely to have been an intentional design idea but more likely to be just an effect of the tightness in which the strand of wicker had been twisted - but all the same it's a pretty awesome conversation starter at the dinner table!!! Me and this Trinity Mathmo managed to get EVERYONE holding up their place mat tracing round the spiral. A great bit of interactive maths which not only highlights the difference between an Archimedian spiral and a logarithmic spiral - but also shows that sometimes, actually discovering/doing/finding out for ourselves, is so much more powerful than just looking at something - or as a teacher, just showing something - and expecting to grasp it completely in all it's beauty and fullness!

After meeting the lovely family I shared the meal with, this was definitely one of the major highlights of today!!! :D

Rediscovering Integration!!!

In order to keep up my maths and leave my options for next year as open as possible, I’ve been teaching myself the modules for Further Maths A-Level. So far, it’s going really well and I’m doing 1 module per month, meaning I should have completed all 6 by the end of February and can focus on past papers from that point until the exams at the end of May/throughout June (it’s a sad thing there are no January exams any more!).

Anyway, having done Decision D1 (in my mind, not mathematics, but helpful for anyone wanting to go into computer science of logic) and Mechanics M1 as my applied modules last year, I had to discover Statistics for the first time this year!

Now Stats was always going to have a hard act to follow, as I absolutely ADORED mechanics! My mechanics revision is what you will see written all over the window both still in the Maths Workroom of my school last year, and also as my cover photo if you wander over to my Google+ Profile Page (+María Norlov)! I LOVED it. It was just so much fun. So then Stats… well, I’m sorry statisticians, but it just doesn’t have the same draw for me. I mean, FP01 has by far been the best module I’ve done so far and I am definitely more interested in Number Theory type questions than applied mathematics, but Stats just isn’t so fun even within that bracket.

Or that was until (dun dun duuuuuuuuun) we reached Chapter 3 of S2 and we hit integration!!! I was like, what on earth is integration doing here? Turns out continuous random variables give us probability density functions and cumulative distribution functions which give us a graph, the area under which, of course, equals 1 (as it is probability), but also gives us the opportunity to integrate to obtain the area for specific sections, or to integrate and equal to one to find any constants we don’t yet know. And it all go me to thinking; ‘What actually IS integration? How does it work?!’

Right, Left, Maximum and Minimum methods for
calculating Riemann sums as shown in the link below!

I mean, we integrate an equation of a line between two points and it gives us the area under the curve. So I thought back to when I was first introduced to integration, and sadly - I think now - I believe it was just as a “Fundamental Theorem of Calculus”. Just as the opposite to differentiation really. Which is sad, considering it is actually such a beautiful mathematical phenomenon. Integration gives us the area under the curve, right? Well this is because it is like splitting the area under the curve into infinitely many and infinitely small strips which we can then find the area of and add to get the area under the entire curve. That’s why we integrate ‘with respect to x’ or ‘dx’. Imagine ‘dx’ as relating to ‘x’ meaning a ‘small part ofx’. And this is a small part of x where the width of this strip is getting infinitesimally close to 0.

And then just think about that! How amazing is integration?!?!?! That it can contain within it infinitely many strips to get an accurate estimate for the area under the curve! The coloured picture above actually shows estimates made by various ways of calculating ‘Riemann sums’ [another wonderful thing that came from the world of Bernhard Riemann - his famous hypothesis is one of the things that inspired me to a love of mathematics in the first place!!!]. And it shows how these converge to give closer and closer to the right answer, obviously, the more strips the space is divided into. But if you look at all the complexity that goes into Riemann sums (http://en.wikipedia.org/wiki/Riemann_sum), you will very quickly begin to see that the relative simplicity and beauty (and contained-ness if I might push the limits of English vocabulary!) of the humble integration. I mean, can you IMAGINE having to calculate Riemann sums every time you wanted the area under a curve?!?!?! And apparently it is defined by being merely the opposite of differentiation.

I encourage you, see integration for what it is today - it’s not merely the undoing of differential equations, it is something all of itself which contains within it the mystery of infinity. And that shouldn’t be something that is over-looked! Having not used integration for a while, I forgot how immensely wonderful it is; thankfully, today I rediscovered it! :D

Little Addition to my Tribute to Pi!!!

I've just realised. I never actually explained what I meant in the title of my last post when I called Tau ‘adolescent’! It is merely that from my point of view, it looks as though Pi is the legend - is the genius and the well-known star from history, and Tau is coming in as though it’s the new teenager ready to show off what it’s all about. It just has that “typical teen” characteristic of needing to PROVE it’s great. Whereas Pi doesn’t need to prove it’s great, that comes with the territory of being Pi. Obviously, I realise this anthropomorphism of numbers has a way of possibly going too far, but then, they do seem to have their own personalities sometimes!

I also just wanted to share a really cool photo! For a while, before I came to uni, I had a Saturday job working at a beautiful little Tea-Room in my home village - not that I was very good at my job, but the people that I met there were all great! Anyway, one day Saturday morning, it was quiet and so I was learning to make a pie from the chef and she had a bit of leftover pastry at the end, so I said she should definitely make a picture out of it to put on the top. Then I had a brainwave! If she put the symbol for Pi on the top, it would be a Pi pie!!! And, amazingly, that’s exactly what she did - just for me! To my utter delight; I spent the rest of the day showing everyone who came in the Pi pie and we sold all of it!



BEAUTIFUL!!!