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
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