Lots of little triangles

As if talking my dear wifey into letting us have Mobius strip wedding rings wasn't already taking things too far, I got my very first tattoo on Monday. And it's a maths one! Hooray!

I've long wanted a fractal tattoo, and for a time was interested in the Mandelbrot set or something like that, but eventually decided that I wanted one that was more geometricky. The Sierpinski triangle was the winner, as of all of the classic fractals, I find it to be particularly attractive. It is also very easy to explain to people if needs be.

Furthermore, I found many examples of Mandelbrot and Julia set tattoos, but only a couple of Sierpinski triangles online: one to one less iteration than mine, and a large one that is very cool and spans both forearms. There must be many more out there, but I would imagine the relative rareness online is reflected in the real world, so I can be sure of something fairly unique to me.

Actually, the picture was taken from Unit C5 of M338 Topology. I downloaded the PDF version, zoomed it as big as I was allowed, then print-screened, pasted, cropped, and printed. As for the size, this was just what I got when choosing the 4 copies per A4 option in Windows.

I took down a smaller print (9 per A4 size) on Saturday, and asked about it in the very professional 'The Bleeding Art' in Huddersfield town centre. They were pretty positive, but told me that with that much detail (if you think about it, there are 81 small black triangles, after this many iterations), it should be a bit bigger. There was no specific reason for the size I'd taken in anyway. Fortunately, the size mentioned above turned out to be more-or-less perfect.

When we went down on Saturday, it was actually for Kim's benefit. She's been after a realistic tattoo around her wrist of the Morning Glory plant, with the vine winding around, and a few of the (blue) flowers in bloom. I never really knew about this realistic style, so was amazed at some of the examples they have on display, and showed on the computer. She gave them some photos, and they will make up the design in time for her appointment in a few weeks (there is a waiting list for the guy that does these kinds of ones). It was only really an incidental thing, that I would ask about mine too. But here I am a few days later with a big triangle on my arm!

I was done as a "walk in." You turn up in the morning, and on a first-come-first-served basis, they allocate you a time slot during the day. I was scheduled for 3.30pm, but the lady having Pingu on her leg overran somewhat, so I got started shortly after 4. One of the staff had made my stencil on the computer earlier in the day, based on the larger print out; his version was better than mine as it had gone a bit blocky with the zooming. Anyway, it was all done and filled in by about 5.50pm, and I was on my way home. Not bad at all!

I'm really happy with it, and I've been keeping up well with the after care. I am wondering what I should go for, if I do that is, next time. One idea I had was the inverse of this design on my other arm in the same place. It think that would look good.

Neil H

Pi equals 3.0226

Okay, it doesn't, but I just finally did an experiment that I read about years ago when doing a translation on Monte Carlo approximation.

You may know it, but still, it involves drawing a circle inside a square with sides equal to the circle's diameter. Then, by some random method, you put points on the circle-in-square and work out the ratio of points on the circle to points on the square, and thereby approximate pi.

Here are photos of my effort, in which I dropped grains of rice, and then (quite laboriously) counted them. I counted those outside first, then those inside. There were 227 grains outside of the circle, and 702 inside. This gave me 929 grains on the square and 702 on the circle. I had to make a judgement when they were on the line, but just went with the obvious choice of inside the circle if more of the grain is inside, and outside otherwise.

For what it's worth, the radius of the circle was 4", and so the sides of the square were 8". However, I treated them as being of unit length. Hence, the area of the square was 4 and the area of the circle was pi.
The ratio therefore was pi/4, and the ratio from the rice-approximation was 702/929, which to 5 d.p. is 0.75565. Times 4 and we get pi roughly equal to 3.0226.

It may not be that close (I was hoping for something that would round to 3.1 at least), but for what it's worth, if 28 of those that ended up in the square only had ended up in the circle, we'd have got (730/929) x 4 = 3.14316, which is pretty close.

In the hopes of getting a bit closer, I also checked the weights and got 14.85g for the square and 11.15g for the circle. This got me an approximation of pi (to 5 d.p.) of 3.00363. Oh hang on, I shouldn't be going to more decimal places than there are in the input values, should I? So actually, this just gets pi = 3. Oh well.

So there we have it. Another way of wasting time on a Saturday afternoon that should be spent studying. At least it's maths though.

The Turing Solution

Hi all,

I just heard advertised on Radio 4, a programme that might be of interest to those of the mathematical persuasion.

At 11:00am this morning is The Turing Solution, a half-hour show about the life and work of the great Alan Turing. It will be presented by someone described as "standup mathematician Matt Parker." Never heard of him myself, but regardless, the programme should be worth a listen. Coincidentally, I just last week, started thinking back to M381, and was pleasantly surprised that I could still generate URM programs for basic things like multiply and divide two numbers. Not particularly impressive, I know, but proof that I actually took something in of that course. If it weren't for the fact that the autumn start of MST209 will mean I don't have a break this time, I would be interested in going back through the logic part of that course without the distractions of number theory.

In other news, I was disappointed (but not surprised) to see that Radio 4 have not included Andrew Wiles among their list of "New Elizabethans" despite what I thought to be an articulate and well-argued suggestion. Oh well.

Right, bed time!

Neil