# Category Archives: science/maths

## How to abolish Mondays

Yesterday, Twitter was full of tweets about two vaguely interesting features of this month. One was the nice recurrence of the number 10 just after 10 am, at 10:10:10 on 10–10–10. The day was variously referred to as Binary Day; 42 Day or Meaning of Life Day; and various more suggestive names based on the idea that 10 10 10 in Roman numerals would be X X X. (I myself thought the day should be celebrated ten 99ths of a second later, at 10:10:10.10101010 . . . )

The event was disappointing, though: not only did it only last a fraction of a second, but it wasn’t even the same fraction of a second, because 10:10:10 arrived at different times according to what time zone people were in.

The other “amazing fact” was that there are five Fridays, Saturdays and Sundays in the month, coupled with a claim that this only happens once every 823 years. (The claim puzzles me: it only takes a moment’s thought to realise that October does this every time it starts on a Friday.)

Sadly, having more weekends in a month doesn’t make them any more frequent. The days of the week just plod along as normal. Despite appearances, we haven’t actually conjured up any more Fridays, Saturdays or Sundays. Even worse, the week still contains its most basic flaw: the Problem of Monday. But the 10:10:10 time-zone fiasco hints at a solution.

### The Problem of Monday

The Problem of Monday is simply stated:

• Monday exists;
• it is universally hated;
• there is one in every week, and it lasts a full 24 hours.

Fortunately the problem is easily solved.

### The solution to Monday

Why does every week contain a Monday? The answer is simple: there is a Monday in every week because we persist in living the whole week in one time zone. Yet this is quite absurd in today’s world. When I log in to Twitter, I regularly converse with people in Australia (currently something like 9 or 10 hours ahead of UK time), the US (anywhere from 5 to 8 hours behind UK time), Scandinavia and mainland Europe (both currently 1 hour ahead of UK time). When it’s midday here, it can be anything from 4 am to 10 pm for the people I’m talking to. And even when the clock says it’s midday here, it’s not really midday: we’re on British Summer Time at the moment, so the clock says midday when the position of the sun in the sky shows that really it’s 11 am.

Now, when everyone was waiting for 10:10:10 to arrive, there were 24 hours’ worth of times for it to happen. And Monday is only 24 hours long. Given the wide range of times which 10:10:10 could mean, the beginning and end of Monday should be equally movable . . . If we can move them closer together by 24 hours, then Monday will be gone. Is this achievable?

It turns out that it is. We can completely abolish Monday merely by making an appropriate choice of time zone each day. In fact there’s even flexibility to build in specific requirements such as a longer weekend or a nice long Saturday.

Below are several possibilities. They all work on the same principle:

• Between the beginning of Tuesday and the end of Sunday, periodically move the clock back by a specified amount, thereby making some or all of the days longer.
• Arrange for these changes to add up to 24 hours.
• At the end of Sunday, put the clock forward by 24 hours so as to both compensate for the added hours, and remove Monday from the week.

In what follows, GMT is Greenwich Mean Time, GMT+1 means 1 hour ahead of GMT, and so on. In each case, we arrange for our clocks to be set to GMT-12 at the end of Sunday, meaning that changing to GMT+12 will put them forwards by 24 hours and thus eliminate Monday.

#### Basic solution: equal days

This is the simplest solution to work out, but has some disadvantages. The main one is that although it makes the weekend longer, it only increases it by 8 hours,  from 48  to 56.

All we do in this solution is move our clocks back by four hours each day. After six days, we’ve moved the clocks back by a whole day. We’ve effectively shared out Monday’s 24 hours among the other days of the week, which are each 28 hours long. Monday is no longer needed, so we simply skip over it. Here it is in more detail.

• Our Tuesday begins at what other people think is midday on Monday. Our clocks are set to GMT+12, synchronising us with friends in the mid-Pacific.
• At midnight (our time) on Tuesday night, we move our clocks back four hours. We’re now on GMT+8, synchronised with parts of Australia and Asia. Don’t worry about the fact that it’s still light outside; as far as we’re concerned, it’s midnight. Other people will claim that it’s only midday. They’re wrong.What’s important is what our clocks say, not what other people say. Four hours later, our clocks reach midnight for the second time, and Wednesday begins.
• At midnight on Wednesday night, we move our clocks back another four hours. We’re now on GMT+4. Thursday starts at what would be 8 pm GMT.
• On Thursday night, we move our clocks back again. We’re now on GMT, meaning our Friday starts at the same time as everyone else’s.
• On Friday night we move back to GMT-4. Our Saturday starts at everyone else’s 4 am, letting us get up four hours later than them.
• On Saturday night we move back to GMT-8. Our Sunday starts eight hours later than everyone else’s, giving us time for a real Sunday lie-in.
• On Sunday night, we first move our clocks back to GMT-12, postponing midnight by four hours. If we left it at that, our Monday would start twelve hours after everyone else’s, at what they call midday on Monday. But this is of course when Tuesday starts. Monday must therefore end as soon as it begins, to make room for Tuesday. So when midnight arrives for the second time, we make our 24-hour jump forwards from GMT-12 to GMT+12. This has the effect of removing Monday, and we’re back where we began: 00:00 on Tuesday, with the clocks set to GMT+12.

You might think moving the clock back four hours at a time is a bit drastic. Maybe it is. But it isn’t strictly necessary; all that’s required is that by the end of each day, the clock has been moved back four hours. So maybe you’d adjust it at strategic times during the day. For example just after the alarm goes off in the morning, to give you an extra hour in bed; then at the end of lunchtime, to create a two-hour “lunch hour”; then two hours at night so you can don’t have to abandon the conversation which was just becoming interesting. Or maybe only one hour at night, so you can add an hour to your afternoon tea break. The 28 hours of the day are yours to arrange as you see fit.

The supreme advantage though—present in all the solutions I’ve considered—is the timing of Tuesday, the day after Sunday. While other people are suffering Monday morning, you’re enjoying Sunday evening; while they’re having their Monday afternoon, you’re fast asleep because it’s the small hours of Tuesday morning.

#### The long weekend solution

Suppose, instead, you want is a nice long weekend (and still no Monday). This too is achievable, but you’ll need to nocturnal on weekdays and able to tolerate some rather extreme clock changes. In this version the four weekdays have their normal length of 24 hours, but Saturday and Sunday are extended to 36 hours each. The time zone settings for this option are as follows:

• Tuesday, Wednesday, Thursday, Friday, Saturday: these all use GMT+12. Tuesday starts at what GMT people think is midday on Monday. Saturday starts at what they call midday on Friday.
• on Saturday night (or at intervals during the day), we have to move the clock back by twelve hours. By the end of Saturday, it’s set to GMT. Sunday starts at the same time as for everyone else.
• on Sunday night, we again adjust the clock by twelve hours, to GMT-12. We enjoy the extra twelve hours as we see fit, then when midnight arrives again, we switch to GMT+12, skipping over Monday and arriving at Tuesday.

Our weekend is now 72 hours instead of the usual 48, and runs from what other people call midday on Friday to what they call midday on Monday.

Or maybe you don’t much like Sundays, but you do want a long Friday in which to finish off your work, followed by a long Saturday in which to relax. In this case the pattern might be

• Tuesday, Wednesday, Thursday, Friday: clock set to GMT+12
• Friday night: move back to GMT (gaining 12 hours)
• Saturday night: move to GMT-12 (gaining 12 hours)
• Sunday night:  don’t move the clock back at all, but instead move it straight forward to GMT+12, thereby skipping Monday.

Clearly there are trade-offs to be made. There are disadvantages as well as advantages to each solution. But what disadvantage can possibly outweigh the complete abolition of Monday, which is achieved merely by choosing an appropriate time zone each day?

(Ironically . . . this post was written on a Monday!)

## Why did people think the Sun went round the Earth?

Here’s something which puzzled me for a long time, until I happened to come across the answer a couple of years or so ago. I think the book Episodes from the Early History of Mathematics by Asger Aaboe (Cambridge University Press, 1997) provided me with the revelation, but I wouldn’t swear to it.

#### The puzzle

When people began watching the motions of the Sun, Moon and planets in the sky and measuring them properly—which happened many centuries ago—their movements turned out to be quite complicated, especially if you assumed (as most people did) that the Earth was stationary at the centre of the universe. Everything circled the Earth once a day, but with various wobbles superimposed on the motion.

This led to a rather unsatisfying and complicated picture involving epicycles, in which a planet’s motion in the sky was a combination of two or more movements: its circle around the Earth plus the various wobbles.

Before I go any further, let me point out that the the idea of a flat earth wouldn’t have entered into this: the realisation that the Earth was a sphere happened around 300 BC and Eratosthenes was the first to measure its size—with remarkable accuracy—in about 240 BC.

It was occasionally suggested that things might make more sense if actually everything went round the Sun, with the Earth rotating once a day. Yet, there was great resistance to that explanation. Why? What was the problem? Why not switch to the obvious answer? Why not just do it? Why was the “Copernican revolution” so revolutionary?

One reason was that Copernicus’ system, which based everything on combinations of perfectly circular movements, ended up being even more complex than what had gone before. But there’s another, very simple and logical reason.

If we look at the night sky we see the Moon, and lots of little points of light. Mostly these keep the same positions relative to each other, simply circling the North pole (or South pole if you’re south of the equator). They behave as though they’re attached to a rigid sphere, making them all rotate together. That was called the sphere of the fixed stars, and was thought of as the outermost part of the universe.

A few of them, though, seem not to be attached to it. They  move around according to  rules of their own, and were given spheres of their own to move them. But apart from that—and from not twinkling—they look pretty much the same as the other points of light. They just happen to be attached to their own spheres instead of the outer one.

How far away were the fixed stars?  Nobody knew, but they couldn’t really be much further away than the moving ones or they’d be too dim to see.

So far so good. But why can’t the Sun be at the centre instead of us? What difference would it make, apart from a little bit of wounded pride?

Putting the Sun at the centre of this picture creates a huge, glaring problem.

The closer you are to something, the bigger it looks. If the Earth went round the Sun, then any given constellation would change size in the sky as we moved towards and away from it. This would be clearly visible. Yet we don’t see it. And that, surely, proves that the Earth isn’t moving. It must be the Sun which moves around the Earth.

#### The revolution

So if you want the Sun at the centre, you have to explain why the fixed stars always stay the same distance apart in the sky. This entails making the sphere they’re attached to considerably bigger. And then you have to make them bright enough to see, so they’re not little twinkly things any more, but are like other suns in their own right.

But the Sun and its collection of spheres was the entire universe. Saying that the stars are really other suns is like saying that actually there are thousands of universes. It’s mind-boggling. It sounds somewhat insane.

It’s not a matter of demoting the Earth a little from its symbolic position as the centre around which everything revolves, and giving that honour instead to the Sun; it’s a matter of making both the Earth and the Sun utterly insignificant in the scheme of things.

Seen that way, the idea is indeed shocking.

## Diagram Prize update

A few posts ago, I wrote about the Diagram Prize, awarded for “the book carrying the oddest title of the year”. There was a shortlist of six titles, and members of the public were invited to vote on which should win.

Winner of the 2009 Diagram Prize

The 2009 winner has now been announced on The Bookseller‘s website, and is Crocheting Adventures with Hyperbolic Planes by Dr Daina Taimina.

If you read the article, be sure to scroll down to the comments—the first is from Dr Taimina herself and gives her response to winning.

If you want to know more about the the book and its author, visit her blog at http://hyperbolic-crochet.blogspot.com/.

Further update: When I wrote that, Daina Taimina had only just started her blog. Now that it’s been going a little longer, it’s showing signs of becoming a fascinating blog about art and mathematics, with a strong personal slant too. I do urge you to visit it.

Images © Daina Taimina and used with permission.

## Christians and the environment

#### The environment as a moral issue

As you’ll know if you’ve read my About page, one of my interests is the relationship between sane religion and honest science. By that I mean religion which lives in a real world, and science which is allowed to be itself and not bent to fit some religious viewpoint.

Many current findings of science, of course, concern human impact on the environment. Christianity hasn’t always done a brilliant job environmentally. All you need do is read the beginning of Genesis for its ideas—not as as the pre-scientific science it was never intended to be—to see the difference between its vision and the role we have acquired. The Earth is meant to be “fruitful” and is “very good”. Our position of power over other living things, recognised in Genesis, gives us an obilgation to look after them, delighting in creation’s goodness and living in harmony with it.

Historically the church has largely forgotten this, seeing the Earth as being there simply for human beings to exploit as we like. So we’ve become alienated from it (another theme of the stories!), becoming agents of destruction rather than creation.

Personally I see environmental damage as a major moral issue for followers of a religion which believes in the goodness of God and sees God as the source of all existence and of all life. Harming the Earth is wrong for the same reason that harming people is: it is created and loved by God. [1] We should should be giving life to our part of creation and building it up, not destroying it.

So it’s good to know that there are Christians—and members of other religions—who are taking this seriously.

One organisation working to bring such people together and encourage those in the church to care about the environment is Christian Ecology Link. I’ve been a regular recipient of their email newsletter since well before the world went green (or at least, wanted to look green).

In keeping with the organisation’s name, the newlsetter is largely a collection of brief news items linking to information about organisations and events. Below are the links from the latest issue. Hopefully this will give you an idea of the breadth of material it covers. The wording is mine, not taken from the newsletter:

And I’ve not included the items which had no web address but just a person to email, or the final item which gave links for help on public speaking. Also, since it wasn’t in the newsletter, I’ve not mentioned a joint event with the London Islamic Network for the Environment . . .

If this kind of news is of interest to you, visit the CEL website and sign up for their newsletter.

#### Note

[1] Creation is a word which sometimes carries misleading overtones. Not helped, in fact, by the existence of creationism. People think of a moment in time when God made everything.

As far as I’m concerned creation isn’t a moment in time. Neither is it an alternative process to the one science sees, with God bypassing the laws of physics and designing every little detail of, say, the human appendix. When I say God is creator, I mean that every part of space and time exists because God makes existence possible; the laws of physics, or any deeper laws that explain them, exist because of God; the process of evolution that produced life exists because of the way those laws are. If God controlled the process, it would no longer be a free one and the universe wouldn’t really be God’s creation, just be an extension of God. Neither would there be any room for free will, or for spiritually aware life (such as us) to respond freely to God. Back

## The bizarre similarity between money and quantum physics

I don’t know about you, but I’ve always found money very puzzling for various reasons. Especially when it comes to what the banks do with it. I’ve never been quite convinced that money actually exists: it’s a number that we do things with. If you pay me for something, your number goes down and my number goes up. If my number goes down to zero and my bank won’t let it keep going then I’m in big trouble, because people won’t hand things over to me in shops and so on. To do that, they want me to make their number go up.

And I’ve never really understood how “the money supply” can be increased. Coins and notes can be printed, sure, but whose are they and how do they get into circulation without somebody in effect “stealing” them? Is it actually valid to create it out of nowhere? All very strange.

The subject came up again when my friend PamBG, who once worked in finance and is now a Methodist minister, wrote a puzzled post about “quantitative easing”.

She didn’t get the answer to her question, some discussion ensued about the nature of money, the value of a company’s stocks, and the like. (The more I think about these things, the more convinced I become that money is really just a mental trick.)

As it happens, I had to study quite a lot of quantum physics at university, for my electronics degree. (This isn’t surprising, since quantum physics is the physics of the extremely small, electrons are extremely small, and electronics is based on their behaviour.) And in studying that, I had the same sense of things only just existing, or not quite existing. (I mean, an electron exists but doesn’t properly know where it is or how fast it’s going, or if it knows one it doesn’t know the other; that’s roughly what the Uncertainty Principle says.) It seemed very much like a game played with certain rules and numbers. A game which happened to predict very well what measurement you’d get if you did a particular experiment, but a game nevertheless, which simply dealt with the numbers and rules and was silent about the actual nature of the physical objects playing the game. All it said was that they obeyed the rules. (Scientifically, all that can be studied is the rules and numbers, since they can be observed; all we can say about any underlying reality is that if there is one, it’s one which fits them.)

And it also so happens that I find the ideas of quantum physics easier to “grasp” [1] than the ideas of finance, so the analogy that follows was a bit of a breakthrough for me. Here are some shortened extracts from the later part of the conversation:

Pam:

In some senses ‘money doesn’t really exist’ – which was people’s complaint when the West went off the gold standard in the 1970s. (Previously, all money printed had to have a certain value with respect to an ounce of gold.)

And markets are driven by psychology. Fear and greed. A very simple explanation: financial instruments are priced according what people will think that they will be worth in the future. So, if a particular company is expected to grow by 5% per annum over the next two years, and its assets are now £100, its stock price would be £110.25. (This is hugely simplistic for illustrative purposes.)

The problem, of course, is that you have to guess how much everything is going to grow [. . .]

Tim:

[. . .] I heard on a radio programme that the first time there was a real banking crisis after the Bank of England formed, people were very unhappy about paper money, complaining that it wasn’t actually money.

It’s interesting: atheists accuse us of basing our lives on something that doesn’t exist, namely God, but arguably modern society runs on something that doesn’t exist, namelly money! (I’m only half joking.)

ISTM you’re saying that the value of a piece of paper simply exists in the mind of the buyer and seller.

Pam:

Yes, I think it probably is. It’s a corollary of ‘something is only worth what a buyer is willing to pay for it’ [. . .]

Tim:

Bizarrely, there’s a parallel with quantum physics, too [. . .]

In quantum physics—the physics of the ultra-small—a quantity generally exists in an “indeterminate” state until it is measured. The act of measurement forces it to stop being indeterminate and have a definite value.

Similarly it seems to me that a house, say, doesn’t have a definite value until you measure the price by letting somebody pay for it.

So in a way, the financial value of something is always in the future and hovering on the brink of existence.

Hmmm…

Pam:

Tim – Not only does that seem quite correct to me with respect to money and financial markets, but you’ve just helped me to better understand that principle in quantum physics.

And maybe that’s why money is so confusing. In quantum physics, you mostly don’t know things like the position or momentum of a particle; you only know the probability of it being within a particular range. And, according to the most widely accepted interpretation, the particle doesn’t “know” either. It really doesn’t have a precise position or momentum until it’s “observed” in some way.

Similarly with money: your house doesn’t have a precise value except at the moment of sale. All it has is a particular probability of lying in a particular price range. And the same is true of the the things in which your money in the bank is invested.

And it seems to me that this might be why the economy gives us so much grief: we’re dealing with things which have at best a shadowy existence, but much of the time we treat them like the most concrete reality there is.

Thoughts?

#### Note

[1] I think it was either Heisenberg or Schrödinger who said that if you weren’t confused by quantum mechanics you hadn’t understood it; hence “grasp” in quotes. Back