Dan Q

Blog

Calculating Pi (when you’re ill)

So, I’ve not been well lately. And because a few days lying on my back with insufficient mental stimulation is a quick route to insanity for me, I’ve been trying to spend my most-conscious moment doing things that keep my brain ticking over. And that’s how I ended up calculating pi.

Dan, shortly before inpatient admission but already recovering from the worst parts of his hospital visit, last week.
When I say I’ve been unwell, that might be an understatement. But we’ll get to that another time.

Pi (or π) is, of course, the ratio of the circumference of a circle to its diameter, for every circle. You’ll probably have learned it in school as 3.14, 3.142, or 3.14159, unless you were one of those creepy kids who tried to memorise a lot more digits. Over the years, we’ve been able to calculate it to increasing precision, and although there’s no practical or theoretical reason that we need to know it beyond the 32 digits worked out by Ludolph van Ceulen in the 16th Century, it’s still a fascinating topic that attracts research and debate.

Graph illustrating the calculation of digits of pi over the millenia. Note the logarithmic scale on the left and the staggered scale on the bottom axis.
Our calculation of pi has rocketed since the development of the digital computer.

Most of the computer-based systems we use today are hard to explain, but there’s a really fun computer-based experimental method that can be used to estimate the value of pi that I’m going to share with you. As I’ve been stuck in bed (and often asleep) for the last few days, I’ve not been able to do much productive work, but I have found myself able to implement an example of how to calculate pi. Recovery like a nerd, am I right?

A "pi pie", from a Pi Day celebration.
Pi goes on forever. Pie, sadly, comes to an end.

Remember in school, when you’ll have learned that the formula to describe a circle (of radius 1) on a cartesian coordinate system is x2 + y2 = 1? Well you can work this backwards, too: if you have a point on a grid, (x,y), then you can tell whether it’s inside or outside that circle. If x2 + y2 < 1, it’s inside, and if x2 + y2 > 1, it’s outside. Meanwhile, the difference between the area of a circle and the area of a square that exactly contains it is π/4.

A circle of radius 1 at the intersection of the axes of a cartesian coordinate system.
Think back to your school days. Ever draw a circle like this? Do the words “Cartesian coordinates” ring any bells?

Take those two facts together and you can develop an experimental way to determine pi, called a Monte Carlo method. Take a circle of radius 1 inside a square that exactly contains it. Then randomly choose points within the square. Statistically speaking, these random points have a π/4 chance of occurring within the circle (rather than outside it). So if we take the number of points that lie within the circle, divide that by the total number of points, and then multiply by 4, we should get something that approaches the value of pi. You could even do it by hand!

Output of Dan's demonstration of the Monte Carlo method as used to approximate the value of pi.
I wrote some software to do exactly that. Here’s what it looks like – the red points are inside the circle, and the black points are outside.

The software illustration I’ve written is raw JavaScript, HTML, and SVG, and should work in any modern web browser (though it can get a little slow once it’s drawn a few thousand points!). Give it a go, here! When you go to that page, your browser will start drawing dots at random points, colouring them red if the sum of the squares of their coordinates is less than 1, which is the radius of the circle (and the width of the square that encompasses it). As it goes along, it uses the formula I described above to approximate the value of pi. You’ll probably get as far as 3.14 before you get bored, but there’s no reason that this method couldn’t be used to go as far as you like: it’s not the best tool for the job, but it’s super-easy to understand and explain.

Oh, and it’s all completely open-source, so you’re welcome to take it and do with it what you wish. Turn off the graphical output to make it run faster, and see if you can get an accurate approximation to 5 digits of pi! Or slow it down so you can see how the appearance of each and every point affects the calculation. Or adapt it into a teaching tool and show your maths students one way that pi can be derived experimentally. It’s all yours: have fun.

And I’ll update you on my health at some other point.

× × × × ×

One of my favourite hosting companies is recruiting using anonymous online interviews, in an effort to combat industry sexism

This link was originally posted to /r/girlsgonewired. See more things from Dan's Reddit account.

The original link was: https://blog.bytemark.co.uk/2015/05/20/bytemark-is-hiring-by-anonymous-interview

…

So, in May 2015, we made the huge decision to start an anonymous recruitment process. The biggest change compared to tradition recuiting is this – your first two interviews are truly anonymous. We conduct them over instant messaging and run our skills tests remotely too.

You won’t even have to give your real name or a CV in the initial stages. We don’t know anything about you that you don’t choose to present in the interview.

That makes us work hard for explicit goals. We want to know about your:

Most valuable skills
Ability to learn
Ability to work effectively in a team

We make decisions based on those factors. Avoiding the “X factor” of cultural fit, which we’ve seen as an excuse for all kinds of implicit and explicit bias across industries.

We also want to be respectful of your time, your enthusiasm and your interests – we’ll test not just what you know but what you can learn. Our focus is on letting you put your abilities to the fore, without fear that you’ll be judged on irrelevant things. We define the job, we define the skills, and we want to test those without bias.

Our culture comes from you, the best person for the job at the end of the process. Of course we still need to meet you, we want to meet you. But we will start our interviews on the solid foundation of anonymity. Only at the final stages will you be asked to come in for a face-to-face interview.

…

Repost posted at UTC on .

2 mentions

Is it possible to ‘blow’ a whistle or other wind/brass instrument underwater? How?

This self-post was originally posted to /r/askscience. See more things from Dan's Reddit account.

Wind and brass instruments (which I’ve just learned are collectively ‘aerophones’) work because since characteristic of their shape causes the air that is pushed or drawn through them to vibrate at particular frequencies. They work in air whether you’re down a mine or at the top of a mountain, so there’s clearly some tolerance to pressure variation. I presume, therefore, that you could play them using other gases too (e.g. by connecting a whistle to a pump in a nitrogen atmosphere, for example), albeit presumably with a change in pitch (we have whistles driven by an air/steam mixture on steam locomotives, for example).

But I wonder: can such an instrument be used with other fluids? How about a liquid, like water? Can I pump water through an underwater whistle in order to make a whistle-like noise? What are the physical constraints on doing so (e.g do I need a higher pressure difference in water)?

If it’s not possible, why not? What could be done, hypothetically, to construct an aerophone-style (“hydrophone”?) instrument that would work underwater?

tl;dr: Could I pump water through an underwater whistle to ‘blow’ it? If not, how can I ‘fix’ that?

Repost posted at UTC on .

No comments

Guess where I woke up this morning, MegaMasons?

This link was originally posted to /r/MegaMasonsLounge. See more things from Dan's Reddit account.

The original link was: http://i.imgur.com/vUxVG6Z.jpg

Dan in hospital

So yeah: that’s not entirely pleasant. A couple of days ago I was diagnosed with what was supposed to be a minor bladder infection and given antibiotics. Then yesterday I became feverish and collapsed. And now I’m in hospital.

But on the upside, they’ve spent all night pumping me full of some kind of intravaenous antibiotic that must be made from like unicorn spunk and leprechaun tears or something because it’s frankly magical: feeling so much better today than yesterday.

Just wanted to say hi. Hi, all!

Repost posted at UTC on .

No comments

[x-post /r/MegaMegaMegaLounge] [Graph] Tracking ‘engagement’ across the MegaLounges. Why are some Megas more-active than others? (interpretation in comments)

This link was originally posted to /r/MegaMegaMonitor. See more things from Dan's Reddit account.

The original link was: https://i.imgur.com/3aXKsxq.png

Engagement level by MegaLounge

In /u/10_9_‘s recent thread about gilding trends, /u/razerxs made an interesting observation: that the huge ‘drop offs’ in membership of MegaLounges after /r/MegaMegaMegaLounge does not correlate with their expectations, based upon the level of activity in /r/MegaLoungeV. /u/razerxs

observed that e.g. /r/MegaLoungeV is a highly-active sub, which isn’t necessarily what you’d expect if the activity level was based entirely upon the number of people permitted to have access.

I started wondering if there might be a better predictor of engagement levels. I experimented by looking at the ratio of how many people ‘subscribe’ to each MegaLounge to how many people are permitted into there. This isn’t a perfect measure of engagement, of course, but my thinking was that of the people who are invited into a MegaLounge, only some of those will add it to their front page… but that those who add it to their front page are more-likely to actively participate.

The graph shows three things. From the left axis, the blue and red lines show the number of people who are allowed into each MegaLounge and the number of people who are subscribed to each MegaLounge. As you might expect, there’s a gap between the two and the gap narrows in higher lounges, where there are fewer people.

But what I was interested in is whether and where this gap changes proportionally: is the “subscription rate” among eligible people higher in particular lounges, and can this been seen as a predictor of activity levels and engagement rates in each lounge? That’s what the green bars show (against the right-hand axis: note that it doesn’t start at zero). In general, across the MegaLounges up to and including /r/MegaLoungeSol, there does seem to be a slight upward trend: i.e. the higher a lounge you’re in, the more-likely that eligible people are to add that lounge to their front page. Beyond /r/MegaLoungeSol the bars jump around all over the place, probably because of the small number of people ‘up there’, and I suggest that we ignore them: accuracy of this as a predictor would be expected to be better where there were more subscribers: say, up to about /r/MegaLoungeDiamond.

What this would predict would be a “lull” at /r/MegaLoungeVIII. I don’t know if that’s your experience or not. And, interestingly, the ‘subscription ratio’ at /r/MegaLoungeV and /r/MegaLoungeX are also unusually low, bucking the overall trend. What does this mean? I don’t know. But if /r/MegaLoungeV really is to be considered one of the more “active” MegaLounges, then I think that we can safely say that my hypothesis – that we might be able to predict activity hotspots by looking at the subscription rate – is not backed up by the data.

Still: interesting stuff.

Repost posted at UTC on .

No comments

Supermarket “price match” deals are marketing genius

It’s been almost five years since Sainsbury’s supermarkets pioneered the “brand match” idea, which rivals Tesco and Asda later adopted into their own schemes, and I maintain that it’s one of the cleverest pieces of marketing that I’ve ever seen. In case you’ve not come across it before, the principle is this: if your shopping would have been cheaper at one of their major competitors, these supermarkets will give you a voucher for the difference right there at the checkout. Properly advertised (e.g. not in ways that get banned for being misleading), these schemes are an incredibly-compelling tool: no consumer should say no to getting the best possible prices without having to shop around, right?

A Sainsbury's Brand Match voucher worth 62p.
This voucher implies that I’d have saved 62p by shopping elsewhere. But that’s not the whole story.

But it’s nowhere near as simple as that. For a start, the terms and conditions (Asda, Sainsbury’s, Tesco) put significant limitations on how the schemes work. You need to buy at least a certain number of items (8 at Asda, 10 at the other two). Those items must be directly-comparable to competitors’ items: which basically means that only branded products count, but even among them, the competitor must stock the exact same size or else it doesn’t count, even if it would have been cheaper to buy two half-sized products there. There are upper limits to the value of the vouchers (usually £10) and the number that you can use per transaction or per month. “Buy X get Y free” offers are excluded. And there’s a huge list of not-compared products which may include batteries, toys, DVDs, some alcoholic drinks, cosmetics, homeware, flowers, baby formula, light bulbs, books, and anything (even non-medicines) from the pharmacy aisle.

Tesco Price Promise voucher worth 11p off my next shop.
A whole 11p off my next shop? That’s absolutely worth me carrying this piece of paper around in my wallet and trying not to lose it for a week.

But even if it only applies to some of your shopping – the stuff that’s easy to directly compare – it’s still a good deal, right? You’re getting money back towards what you would have saved if you’d have gone up the road? Not necessarily. Let us assume that on average the prices of these three supermarket giants are pretty much the same. Individual products might each be a little more expensive here and a little cheaper there, but if you buy a large enough trolley-load you’re not going to notice the difference. Following me so far? What does this mean for the voucher: it means that it no longer remotely represents what you would have saved if you’d actually been “shopping around”. Let’s take a concrete example:

Photo courtesy Johnathan Harford, used under a Creative Commons license.
A typical basket at any one of these supermarkets will, on average, come to about the same price… even if individual items vary wildly.

Suppose that this is my somewhat-eccentric shopping list (I wanted to select a variety of comparable branded products), and I’m considering shopping at either Sainsbury’s or Tesco:

  • Mozzarella
  • Fish fingers
  • Clover spread
  • Whole milk
  • Crunch corner yoghurts
  • Fromage frais
  • Cadbury Mini Rolls
  • Frozen chips
  • Frozen petit pois
  • Goodfella’s deep pan pizza
  • Dough balls
  • Chocolate-dipped flapjacks
  • Dry white wine
  • Bagels
  • Multigrain wraps
  • Red Bull multipack
  • Angel Slices
  • Cheerios
  • Windolene
  • Cornettos

Not too unreasonable, right? I’ve made a spreadsheet showing my working, where you’ll see today’s prices for each of these items (along with the actual brands and package sizes I’ve selected), if you’d like to check my maths, because here comes the clever bit.

Sainsbury's Basics range baked beans and coleslaw
These products don’t count as comparable. And personally, I’m not sure that “basics”-range coleslaw is likely to even count as ‘food’.

Based on my calculation, taking my imaginary shopping list to Sainsbury’s will ultimately cost me £52.85. Taking it to Tesco will cost me £54.13. Pretty close, right, and I’m not likely to care about the difference because Tesco would give me a £1.28 voucher off my next shop which makes up for the difference (note that Sainsbury’s wouldn’t reciprocate in kind if it were the other way around, after a policy change they made late last year). But that’s not actually a true representation of the value of ‘shopping around’. As my spreadsheet shows, if I were to buy each item on my list at the supermarket that was cheapest, it’d only cost me a total of £43.75: that’s a saving of £9.10 (or about 17% off my entire shop) compared to the cheapest of these supermarkets. These schemes don’t give you a real “best of all worlds”. Instead, they give you, at most, a “best of all worlds, assuming that you’re still going to be lazy enough to only shop in one place”.

Ruth and JTA shopping in advance of Murder At The Magic College.
When you’re buying this much shopping, you’re unlikely to want to go to two different supermarkets to do it, however much money it might save you.

If you’re particularly devious of mind, you can exploit this. For example, suppose I went to Tesco but when I reached the checkout I split my shopping into two transactions. The first transaction contains the frozen goods, milk, wine, dough balls, flapjacks, and mini rolls. This comes out at £33.73, which is £10.38 more than Sainsbury’s would charge me for the same goods. Tesco therefore gives me a £10 voucher, which I immediately use on the second batch of shopping: the one which contains goods that are cheaper than their Sainsbury’s equivalents. The total price of my shopping: £44.13 – only 38p more than if I’d gone to both supermarkets and bought only the best-value goods from each (the 38p discrepancy comes from the fact that Tesco won’t ever give you a voucher worth more than £10, no matter how much you’re losing out).

Photo by 'alisdair' on Flickr. Used under a Creative Commons license.
“I’d like to run these through as two transactions, please.”

It’s not even that hard to do. Obviously, somebody’s probably written an app for it, but even if you’re just doing it by guesswork you can get a better result than just piling all of your shopping onto the conveyor belt together. Simply put the things which seem like a good deal (all of the discounted products, plus anything that feels like it’s good value) at one end of your trolley, and unload those things last. Making sure that you’ve got at least ten items on the conveyor, ‘split’ your shopping somewhere towards the beginning of these items. Then take any voucher you get from your first load, and apply it to the second.

It’s pretty easy, so long as you don’t mind looking like a bit of a tool at the checkout.

A Sainsbury's Brand Match voucher advising that my shopping was 1p cheaper than the competition. In total. Photo with thanks to Brett Jordan, used under a Creative Commons license.
Well that makes it all worthwhile then, doesn’t it?

But to most people, most of the time, this is nothing more than a strong and compelling piece of marketing. Either you get reminded that you allegedly “saved money”, on a piece of paper that probably goes into your wallet and helps to combat buyer’s remorse, or else we get told that we paid a particular amount more than we needed to, and are offered the difference back so long as we return to the same store within the next fortnight. Either way, the supermarket wins your loyalty, which – for a couple of pence on each transaction (assuming that the customer doesn’t lose the voucher or otherwise fail to get an opportunity to use it) – is a miniscule price to pay.

× × ×

Article posted at UTC on .

2 comments

Welcome to the planets, new astronauts!

This self-post was originally posted to /r/MegaLoungeNeptune. See more things from Dan's Reddit account.

I just wanted to give a wave to our three new astronauts, freshly arrived here in the planets just over the course of this last week. They are:

Happy space exploration!

Repost posted at UTC on .

No comments

Dan Q found GLHR2MNY Alleyways of Oxford – Lamb and Flag Passage

This checkin to GLHR2MNY Alleyways of Oxford - Lamb and Flag Passage reflects a geocaching.com log entry. See more of Dan's cache logs.

Rediscovered this cache while leading a party on their first geocaching expedition. The new container and hiding place are a huge improvement, but the description and hint are now inaccurate! For the benefit of anybody as lost as we were: you’re not looking for a magnetic nano, and if you go for the hint: it’s not going to be at the height that you might expect! Hopefully CO will update the page for this otherwise excellent cache!