Gresham College Lectures

The Maths of Coins and Currencies

October 14, 2022 Gresham College
Gresham College Lectures
The Maths of Coins and Currencies
Show Notes Transcript

People have used money – and made counterfeits - for thousands of years. Archimedes came up with a clever way of finding out if you’ve been cheated by a goldsmith. Making coins with the right proportions of the right metals presented a huge mathematical challenge for Fibonacci and other mathematicians in the middle ages.

This lecture will discuss mathematical elements of coin design, denominations, and why former Gresham Professor of Astronomy Sir Christopher Wren favoured decimal coinage.


A lecture by Professor Sarah Hart

The transcript and downloadable versions of the lecture are available from the Gresham College website:
https://www.gresham.ac.uk/watch-now/maths-coins

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- Hello everyone. Today's talk is on "The Maths of Coins and Currencies." And maths can help us with lots of questions about coins and currencies. How do you detect a counterfeit coin? We know that bank notes are rectangular, why aren't coins rectangular? What's so great about circles, if they are indeed the best solution? And finally, why is the money in Harry Potter's pockets so implausible? We'll get onto all of those things, but let's begin with the fakers. There's a very old mathematical problem about detecting a fake, and it's the story of Hiero's crown. This was recounted by the Roman architect Vitruvius, in his book, "De Architectura," which had lots in it, apart from architecture as well. And one of the things was this story about King Hiero II of Syracuse. And he had commissioned a goldsmith to make a votive crown out of gold, he gave him a certain amount of gold to do this with, and the goldsmith presented him with this beautiful thing. But Hiero had heard a rumor that the goldsmith may have cheated him, that he may have kept some of the gold for himself and replaced that amount with silver. So Hiero wanted to know if this was true. So he asked nearby mathematician Archimedes, who lived in Syracuse very conveniently, to help him with this. How can we work out if this thing that's meant to be pure gold actually has been mixed with silver or something else? And the problem is, you could check quite easily, because gold and silver have different densities, they take up different volumes of space, the same mass of silver would take up more space than the same mass of gold. But it's very difficult to measure the volume of some, you know, fancy artifact like a crown, without damaging it. So the problem that Archimedes is faced with, is how can I test this thing to see if it's pure gold or not? How can I measure the volume of this very delicate object, without damaging it? So we all know the legend, right? Archimedes is sitting there in the bath having a think about this. There's the crown over there in this old picture. And he's just sort of musing, like all the best theories are come up with in the bath, right? He suddenly has this great idea. He realizes that when he gets into the bath, the amount of water that's displaced is exactly equal to his volume. So this, he could do this with the crown as well. So eureka, runs through the streets, great story. There's a postscript to this though, comes quite a while later. Over a thousand years later, Galileo wrote a little treaties about this story, and he said, Vitruvius, he thought, had got the story a bit wrong. He perhaps had heard that Archimedes had solved this problem using something to do with water, and had made the assumption that he'd solved it basically by chucking a crown in a bath. That didn't seem very elegant to Galileo. He thought it would be too inaccurate and it wouldn't quite work. So Galileo wrote this little treaties on what he thought Archimedes had really done. I mean, we'll never know, but I really like Galileo's idea, so I'm going to share it with you. So there's Galileo. The basic underlying solution that Vitruvius said that Archimedes had done, was based on comparing or knowing that the densities of gold and silver are different. And, so using modern units for ease, we can see what's happening, the density of gold is high. It's sort of 19.32 grams per centimeter cubed. The density of silver is much less, about 10 1/2 grams per centimeter cubed. So density is mass divided by volume, right? So you can see from that, from the units. So then the volume would be, just flip that upside down, mass divided by density. So that tells you something that's very dense will have, for a given mass, a smaller volume. So gold will take up less space per kilogram than silver will. And you can work out exactly how much in modern units. So a kilogram of gold, you can just put the numbers in, a kilogram of gold will take up a volume of 51.76-ish cubic centimeters, whereas the same mass of silver takes up much more space. So that's all fine. So you could compare the volumes and have a look. The issue is that of course we're not saying that the goldsmith has replaced all of the gold with silver. That would be easy to see with the naked eye, the color would be wrong. So what's probably likely to have happened, if there has been cheating, is that a bit of the gold has been replaced, but not lots. So maybe let's say about 10%, just to have something to work with. If this crown, let's say it weighs a kilogram, is supposed to weigh a kilogram. If actually 10% of it is silver, not gold, so it still weighs a kilogram, but it has a slightly different composition, then that's 900 grams of gold, and you can work out the volume of that. And a hundred grams of silver, and you can work out the volume of that, based on those densities. And so the total amount that you would arrive at for volume is about 56 centimeters cubed. That's only just over four cubic centimeters discrepancy. So how are you actually going to measure that? The crown in a bucket method, which is sort of the crude, the crude thing you could do. There's your vessel, you take a, you know, something that's big enough to hold a crown. So how big is a crown? I mean, it's going to have to be sort of at least like 20 centimeters wide, let's say this vessel. So you get some kind of, let's say a bucket, you're going to put your crown in, you fill it with water, or you put some water in it. Then you want to know how much volume the crown takes up compared to how much volume the equivalent weight of pure gold will take up. So you take some gold, the water level will rise a bit, you can mark that off, then you replace it, take out the gold and put in the crown, and the water level, if it's different, then you'll be able to see that there's been a problem. But the issue is, it's not going to be very much different. And let's think about how different it's going to be. We've got this, let's say it has a diameter of 20 centimeters, because then the radius is 10 centimeters, and we can all do pi r squared in our heads and get the surface area of this. The surface area of the water will be a hundred pi, right, 314 centimeters squared, roughly. So that's the surface area of the water at the top. And then if our difference that we're looking for between the pure gold and the potentially slightly dodgy crown, the difference is only four point, whatever it was, three six centimeters cubed, we've got to spread that out over the whole surface of the water. So when you do that, you find that the actual difference in the water level is tiny, tiny. So that, you know, would be completely outweighed by losing a few drops here and there. You're not going to be able to detect that. Now this is kind of a false problem really, because this isn't a good way to measure the difference in volume. If you just did it like this, crown in a bucket, that's, you know, this isn't the Archimedes that is famous. He wouldn't have done it like this, even if he was using this method. So there is a way to do this better. A better approach to this would be to use something, and this is sort of the kind of technology you might have used with the water clocks of the time. So you have your vessel, but you have a little hole, a little opening, some of the way up, and you fill it up to that level, and then you put your lump of gold in, your one kilogram of gold, and, you know, the water level will rise and will overflow. Don't worry about that. Then before you put the crown in, you take out the gold and replace with the crown. What you're going to do is put a very narrow tube, like a test tube, at the opening there. And because it's very narrow, then it'll be much easier to detect what's going on. So you do that, and you put the crown in, and there we go. And so some extra bit of water will now be displaced. 4.36 centimeters cubed of water will be displaced. But because now we're putting it into a narrow tube, if it was a centimeter wide, you would get, I mean 4.36 centimeters is clearly enough to detect, and that would've been possible at the time to do, you could get that kind of accuracy. So that would be a way to deal with this. But I still like Galileo's suggestion because it, and we'll come onto that now, because it incorporates more things that Archimedes knew. So he knew about the volume, his eureka moment was about the volume being displaced is equal to the volume of the object that you're putting into the water. But he also said things about buoyancy, and he said things about levers. So Galileo's suggestion involves all of those things. So it's pretty cool. So what Galileo says is, and we'll do a little thought experiment. Imagine we've got some scales, sort of balance the scales. And we've put, just to start us off, we're going to weigh equal amounts of gold, equal masses of gold and silver. So say you have a kilogram of gold and a kilogram of silver, and you know that they're equal 'cause the scales balance. So now, thought experiment. Imagine if we take this whole setup and put it underwater, we submerge it. What's going to happen? So I've put it underwater, I haven't let go yet. Let's think about what's going to happen. So now that the scales are underwater, imagine if they had nothing in the pans, then you've got the same amount of water pressing down on each side. So they're going to balance. Then you put the gold in the left hand pan, and you put say a kilogram of gold in the left hand pan. Now, that is pressing down, a kilogram is pressing down on the left hand pan. But, but remember, we have displaced its volume of water. So we've got a kilogram of gold, that has volume, we worked out earlier, 51 point whatever it is, 76 centimeters cubed. That amount of water will be displaced, pushed out of the way, and will therefore not be pressing down on the scales. So how much does 51.76 centimeters cubed of water weigh? Well luckily, it's very easy because one centimeter cubed of water by definition weighs a gram. So that's just, replace centimeters cubed by grams and that's how much it is. So the apparent weight of the gold now, because we've pushed some water out of the way and so that's not pushing down the scales, the gold will appear to weigh 948.24 grams underwater. Whereas the silver, now silver, a kilogram of silver takes up more space. It's more like 95 centimeters cubed, and so its apparent weight will only be 904 and a bit grams underwater. So this is really cool that the less dense thing will appear to weigh less when you put it underwater. So actually the scales won't balance. The gold, the true gold, will look like it weighs more, because of this buoyancy effect. And this is a really clever and sort of an instant thing that you could say, okay, look, yeah. So if you've got your crown, which has a bit of silver in, it's going to appear underwater to weigh less. So that's a very nice thing. I don't have practicalities to really do, but I don't care, I really liked that solution. And Galileo actually went on to explain how, by moving how far away the pan is from the central fulcrum, and that's the levers bit, you can, if you make them balance underwater, you can then work out actually how much silver has been added into the crown. So this is a really cool solution, I really like it, but it also, it struck me suddenly, while thinking about this, that actually air doesn't weigh nothing. Air, I mean of course it's much less dense than water, but even air weighs a little bit. And so actually, if you've got a pair of scales like this, and you want to truly compare the different masses of things, you'd have to do it in a vacuum to get the right answer. Because actually, if you think about, you know, the old puzzle, the old trick question that we ask kids, which weighs more, a ton of bricks or a ton of feathers. And of course, you know, the unwary person will say, oh well it's the bricks, 'cause bricks are really heavy. And you say ah-ha-ha, and you laugh at them. But no, but no, because if you weigh them, if you actually have genuinely a real, genuinely the same mass of these things and you've weighed them in a vacuum and you're really sure, and then you weigh them in the open air, I hope the feathers don't fly away, but weigh them in the open air, then the feathers take up so much more space than the bricks, that they would displace a lot of air, and so actually maybe the bricks do weigh more than the feathers. So if you ever accidentally fall for that question, you can now come back and say yes, but actually they would appear to weigh more than the feathers. So there we are, it even is true for air. We don't need to worry about that too much. But it's a nice little solution to the question. So that's one way of catching the counterfeiters. I believe history tells that actually the goldsmith had been cheating. I do not know what his fate was. It probably wasn't a good fate. So yes, do not cheat the king, if you're a goldsmith. Now if you are thinking about coins, it could be a little easier to check, or to show that you've got a counterfeit coin, because, of course, a counterfeit coin has to look like the real thing. So it would have to be the same size and shape, and therefore the volume should be the same as the real coin. So then you can just weigh them and the counterfeit is likely to have a different weight. So that's one way of finding counterfeits. There are various other methods that have been used over the centuries. You can measure things, and to stop people snipping bits off coins, we have things like milling round the edge of coins. We've put some sort of markings or words round the edge of the coin, that stops people clipping coins, which used to be a real problem in Tudor times, for example. So you can have a look at the coin and check has it been tampered with in that way. And then if you've got a whole batch of coins, or you want to analyze, you know, a foreign currency or something like this, you can do an assay. So you can actually sort of chemically, first you melt it, you melt away the impurities, then perhaps you're left with a mix of gold and silver, and then you can dissolve the silver away with nitric acid. That leaves you with just the gold. So then you can weigh that and then you can sort of work back, reverse engineer, this is how much silver there must've been, this is how much impurities there must've been all along. So there are various ways of analyzing coins, to check whether they're counterfeit or not, or what their composition is. But, so that's sort of testing to see if people are trying to cheat you. But kind of the other, well the other side of the coin is, if you are trying to genuinely make coins. If you are, you know, master of the mint, and you have been instructed, we want coins of this particular grade of silver, this particular level of purity, how do you do that? And that's what I want to turn to next. And we're going to meet this chap, Leonardo of Pisa, who wrote a very influential book in the early 13th century, which had many things in it. And one of the things it had in it is a sequence of numbers that we know now, not as the Leonardo of Pisa sequence, but by this guy's nickname, which is Fibonacci. And so the Fibonacci Sequence, as it's called, he didn't discover it, but he kind of popularized it in the West. It'd been known in India and other countries for hundreds of years. That's one of the things that's in his book. But another thing that's in his book is a really, really useful trick for how to blend different qualities of metal that you have, to produce a desired quality. And this is a real genuine problem that you would be faced with, if your monarch says, I decree that silver coins have to be precisely this level of, this amount of silver, and this purity, and you have different kinds of levels of quality of your bullion, you've got to work out how to actually produce the desire amount. This is a tricky mathematical challenge. So just an example of this that Fibonacci gave. Imagine you've got two kinds of silver bullion, and one is too impure and one is too pure, it's very good quality. So your bad bullion has fineness, let's say, four. We don't need to worry about what the units the fineness is measured in. It might be ounces of silver per pound of metal, or something like that. But the fineness is four, by some measure. This is sort of the purity level. And then you've got some good bullion with fineness nine. And your desired fineness, the decree says that your coins must have fineness of seven. So how do you mix these? In what proportions should you mix these different bullions to get the desired number of seven? And what Fibonacci says is that, so this process is called allegation, by the way, this process of deciding how we're going to mix quantities together. So his allegation solution is the following. He says, okay, so the bad bullion, the impure one, that has fineness of four, that's three worse than you want. And the good bullion has fineness of nine, and that's two better than you want. So you've got that three to two proportion. And what Fibonacci says is you just swap that over, and mix them in the opposite ratio. So he says, therefore you mix in the ratio two bad to three good. And if you sort of think intuitively that makes sense. I'm not going to write down a detailed proof, although I have put one in the transcript which you can pick up afterwards, or download if you're watching this online. I have sort of done the working out in the transcript, if you're interested to see that. But intuitively, we can see that it kind of makes sense to do this because the bad one is kind of three bad. So three times two is six, right? And the good one is three, is two good plus two, and you have three lots of that. So you've got a two times three is six there. And those six plus six and minus six kind of cancel out. So intuitively it makes sense that you, this feels plausible, and indeed it does work. So this is a very simple trick. A bit of algebra proves that it works. But what Fibonacci does so well, and if you want to read about this, Fibonacci and the history of this whole development, the development of this whole kind of process or techniques for solving this kind of problem, you should check out a paper that I've linked to in the transcript by Norman Biggs, who was a former Gresham professor of geometry. You can generalize this idea. So you can have lots of different kinds of bullion that you can mix. And what you do is, according to Fibonacci, you can pair them off. You take a good one and a bad one. You work out this process for all those pairs, and then you can mix them in any way you like. So you can combine the solutions in any way you like. What's called a linear combination of the solutions. So you can generalize it to more than two kinds of bullion, but you can also apply this kind of reasoning in other situations beyond coins. And that's another thing Fibonacci's really great at. For example, you can apply it, say you are a winemaker and you've got lots of different wines of different kind of unit costs, and you want to blend to produce a wine that you will charge a certain amount of money for. How do you do that? Or if you have a certain budget and you want to buy different amounts of things, what are the possibilities for doing that? If each thing has a different price, how many of each thing should you buy? And Fibonacci included, he had lots of puzzles in his books. He liked a puzzle, he liked a sequence, he liked a pattern. He's got this lovely puzzle in his book, the puzzle of the birds, which, oh yeah, we all like birds, right? So this is a really cute puzzle. It seems to have nothing to do with blending bullions to make coins, right? But you can apply exactly the same kind of reasoning to solve it. So the puzzle says, I bought 30 birds of three kinds, so it's sparrows, pigeons, and partridges, for 30 pennies. Well denari, of course, it said in the original. But for 30 pennies. So you get the price of each one. Per sparrow, half a penny. Per pigeon, 2p, and so on. And the question is, what, how many of each bird did I get? So 30 birds for 30 pence, how many of each bird did I get? And the answer to this, the solution to this that Fibonacci gave, is he converted it to an allegation question. So what's the, you know, what's my bullion, what's my fineness here? Well if we think, the final outcome is that I've got 30 birds for 30p. So the sort of price per bird, the fineness is one penny per bird. So what we're aiming at is to produce that. And what we do is to pair them off, as I said, so you take, you have to pair something that's worse, worse fineness, so cheaper than one penny per bird, which in this case is just the sparrows, with something that's better, more than one penny per bird. And then you work out the proportions that you must have just for that pair. And then you combine them as you like. So, for example, right, sparrows, we want to end up with a fineness of one, one penny per bird. Sparrows are worse than that, they're half worse than that. And pigeons are one better than that cause they cost 2p, that's 1p above what we're aiming at. So sparrows are kind of badness of half, pigeons are goodness one. So that ratio, a half to one. What's the rule? We flip it. So for every sparrow, we buy half a pigeon. And you can't buy half a pigeon, wouldn't be very nice to the pigeon, so let's just double that up. For every two sparrows, you buy one pigeon. So that's a total of three birds. Two sparrows cost you total of a penny. The pigeon costs you 2p. So you've spent 3p, you've got three birds. So just that pairing, two to one sparrows to pigeons, that will give you a fineness of one. So you can have as many of those as you like in your final solution, and you'll get a fineness of one. You can do the same thing with the sparrows and partridges, sparrows have badness half, partridges have goodness two more than you want. So you have that two to a half ratio. So you should buy two sparrows for every half a partridge, or four sparrows for every partridge. And that gives you five birds there, four sparrows and a partridge. And you can check that it would cost you 5p. So then, any combination of these threes and fives is going to give you a price of one penny per bird on average. So the only remaining bit is to work out the ways that you can combine these threes and fives to come up with a total of 30. And if you think about it for a bit, you can see there's actually only one way to do it that really involves all three birds. And that is, if you have five lots of the sparrow pigeon combo, so five pigeons, and three lots of the sparrow partridge combo, so five partridges, and then you just work out, you know, from that how many sparrows are required. And if you check, 22 sparrows, that costs you 11p, five pigeons will cost you 10p, three partridges cost you 9p. If you add that up you get 30p. So it's a nice little solution, and it's one of those examples where you've got a mathematical technique, that if you have a kind of a little bit of inspiration, you can apply in what looks like a completely different area. And this idea, this allegation thing, and the techniques for solving problems like this, have come to be a mathematical subject really in their own right. It's now called linear programming. And so this is, again, it's come from working with money and how to produce correct amounts, but it has lots of applications beyond bird puzzles to real life. And so it's really kind of interesting there. So we now know, we're master of the mint, we know how to make the desired alloys for our coins. We know what to make our coins of and how to produce the desired purity. So the next question for us is what shape should our coins be? Most coins are round, circular. Not all but most. Now one reason for this is kind of, you know, the organic development of using bits of metal for currency. Originally you might have just like a little lump of metal, and like a pebble, it might get smoothed down or whatever. But then at some point people started stamping them with pictures of emperors or something like this, or amounts or writing. So then if you take a sort of vaguely round ball of something and you squash it, by impressing it with some kind of image, then it will become a flat disc, and essentially a circle. Some coins have holes in them. It kind of makes it easier for threading them onto strings or necklaces for carrying them around. Not all coins are circular, but quite a lot are. There's a more modern reason why the circle is a good solution for the shape of a coin. A good solution, not the only one. And that's vending machines. Because when you, often vending machines will have, they'll have a single slot where you put your money in, and then the machine has to somehow decide what's the value of that coin, what coin is it, so that it can then give you change or whatever, and know if you've paid enough. So how does it do that? Well, the most often, the most commonly usual way to tell what value of coin you've put in, is to measure its width. So if you've got a coin that has very varying widths, depending on, you know, where it is as you're moving it along, like if you had a long thin rectangle or something, if you have a coin whose width keeps varying, that can make it very difficult for a machine to work out actually, what is the coin I've got? And of course, circles, you know, archetypically, they are the things that have a constant width. The width of a circle, if you just kind of put it between two lines, it's going to be the diameter, and the diameter of the circle is always the same. Wherever you measure it, it's twice the radius. So circles have this property of having a constant width, but they are not the only shape with that property. I'm going to show you another example. Not this. Equilateral triangles very much do not have constant width. But we can modify this triangle. So what I'm going to do is I'm going to replace each edge by the arc of a circle, like that. So this circle, it has center, the top vertex there, and it has radius, whatever the side of the triangle is. And so you just make an arc of a circle there, and you do that for all of the edges. So you get this sort of slightly inflated triangle. And this triangle, well let's have a look at it, this, whatever it's called. Let's have a look at what the width of this thing is, when we measure it. So we're going to roll it along the bottom blue line there and see what happens. And I'm saying that we'll call the side length s, and then the edges of the triangle were replaced by these new edges, which are arcs of circles. And the radius of these circles is also s. So to begin with, the top blue line and the bottom blue line, they just have an edge of the triangle between them. So the distance between them is s. Then we set the thing rolling, and it starts rolling along, and at the top you've still got the vertex of the triangle. And at the bottom you're rolling along this arc of a circle. So at all points along that arc, the distance from top to bottom is still going to be s. That's the radius of the circle. So then we reach this point again. So the end of that edge, the end of that arc, and again, we've got a vertex at the top, vertex at the bottom, distance between them is still s. And we keep going, and now, you know, now we've got an arc at the top that's rolling along. Again, the distance is s. And you can keep doing this, and you find it's always, it's always going to have this width s. This is a shape of constant width. And so you could roll this along, and it really does fit nicely between these two parallel lines that do not have to move up and down. So this could potentially be a shape for a coin maybe. Shapes like this, where you take a polygon and you replace edges with arcs of circles, are called Reuleaux polygons after this guy, Franz Reuleaux, a German mechanical engineer, who investigated these shapes. This trick works as long as your polygon has an odd number of sides. And the reason is, if you had an even number of sides, like a square, you don't, you always have vertices opposite vertices in a square, or a hexagon, or something like that. If you have an even number of sides you have vertices, opposite vertices, and that distance is always going to be greater than anything you could achieve by curving the sides a little bit. So this works for polygons with an odd number of sides, these Reuleaux polygons can be produced, and you may well find you have one in your pocket, because lots of currencies have Reuleaux polygons as coins. In Britain we have the 50p and the 20p coin. They are seven-sided Reuleaux polygons. Shapes of constant width. So you see these little curved edges, not just, you know, flat heptagons. They have these curved edges, they're Reuleaux heptagons. Canada has got an 11-sided Reuleaux polygon for its, they call it the loonie, 'cause of the bird that's shown. The Canadian dollar is an 11-sided Reuleaux polygon, shape of constant width. Perhaps my favorite one though is the Bermuda Triangle. Look at that. What a great, great name, great polygon. I mean it's just, they have won the competition there. Fantastic, fantastic choice. So that you do get these things as coins. You also see Reuleaux triangles around and about the place. So if you don't have one in your pocket, you may have one at home, because a lot of guitar picks or plectrums, plectra, those are often Reuleaux triangles. Not always, but often. And the reason, I think, I mean you get, you've got a mixture there. You've got your pointed, you've got three points and they give you sort of a sharp, bright sound, or you could use the more gentle curved parts if you want a softer, warmer sound, and you could sort of move it around in your hand and it has constant width, so it's nice to do. So that's one use. Perhaps, how am I going to say it's more important? Well, it's a different use, and a really nice one, 'cause it involves triangles which have something in common with circles, but they're being used to make squares. Oh, just, this is amazing. So this thing is a drill bit, and it can drill a square hole. You can rotate a Reuleaux triangle around a little circle, and it will trace out almost a whole square. You do lose a tiny little bit at the corners, you get slightly curved corners. But this is a really, really good approximation to a square. This isn't just theoretical. This thing was patented by Harry Watts in 1916. The Harry Watts drill bit, still available, still on sale. And it can produce very nearly over 98% of the area of the square is covered. So that's a, I just love these kind of things that link triangles, circles, squares, in unexpected ways. So that's another place you might have seen Reuleaux triangles. Why use them for coins? Well, answer one, which is the correct answer, is they're super cool and we love them, and why not? But there may be another, if you really want a practical answer, there may be something about areas. If we think about shapes of constant width. So circles are one example, and these Reuleaux polygons are another. There are others, but we won't go into detail about that. But there's a theorem called Barbier's theorem that says if you have a shape of constant width and that width is s, then its perimeter is always pi times s, whatever the shape is. So we can see that's true for circles, right? Because a circle whose width is s, that's its diameter. We know the circumference of a circle whose diameter is d, is pi times d. So if it's s then it'd be pi times s. So we know already that's true for circles. It turns out to be true of all of these shapes of constant width. So that's nice. But, the area of these things, that gets a bit more complicated. So I'm just going to do a quick calculation of the area of a Reuleaux triangle whose width is s, and the way we can do it with pizza slices, so that's always an advantage. So here's the first pizza slice. You take that little, it's a sixth of a circle, 'cause the angle there is 60 degrees, it's got that equilateral triangle and then the curved bit. So you get one of those, and then add another one, and another one, that's three pizza slices. So that's kind of half the circle. But of course we've done a bit of overcounting. We've got a lot of overlap here. We've counted that equilateral triangle in the middle three times. So we should take that away twice. So what we'll end up with is three of those pizza slices, but then take away two equilateral triangles. So you can work out the, do a bit of algebra, you're doing it now in your heads, the area of an equilateral triangle, little bit more complicated 'cause you have to find the height using Pythagoras' theorem, but you do that and you work it all through, and you get, this is the answer. So in terms of s, s is the width. You get this pi minus the square root of three times half s squared, which is about .7 times s squared. If you work out the same calculation for a circle, so a circle with width s, and s is the diameter then, so the radius is half s, so you do sort of pi r squared on it, and you find that it has area much more, .785-ish s squared. So a circle has a really big area for its perimeter. The Reuleaux triangle much smaller, and it can be shown that actually it has the best possible, the least possible area, is the Reuleaux triangle of all these shapes of constant width. So, you know, if you're worried about how much metal you're going to use for your coin, maybe the Reuleaux triangle is the right answer for you. Okay? So, of course you're sitting here, every person in the audience who has used British coins of late, knows, I'm standing here saying, oh these things can only be done with odd numbers of sides, right? And you are sitting there thinking, but pound coins have 12 sides, which has been true since 2017. Yes they do. They do. So how can that be the case? Well, if you are a more mature person, you may remember that we once had other coins with 12 sides, the thrupenny bits used to have 12 sides, and they were just, you know, genuinely a polygon, a dodecagon with straight edges. When it was mooted, when the mint was thinking of bringing in a 12-sided coin again, they did some consulting with, you know, the manufacturers, about how would this work. Would this be all right? And the manufacturers said no, it would not be all right, because the difference between the maximum width and the minimum width is too big, and our machines would not be able to cope with it. So they made the edges a bit curved, of the new pound coin, when it came in, and they did it so that that difference between the minimum and maximum width was small enough that the machines could cope with it. So it's not a Reuleaux polygon and it can't be a Reuleaux polygon because it has 12 sides, but it was a near enough solution that was okay for the vending machines. So it sort of worked. It doesn't mean I have to like it. I don't like it. So yeah, I think the Canadians have done better than us in, just have 11 sides, or 13, I wouldn't mind 13. 12, no, okay. Unsatisfactory. But there we go, this is, we do have a 12-sided coin. So we've got metals to make our coins. We've got some good candidates for shapes for our coins. Circles, Reuleaux polygons. The final piece of the puzzle that I want to talk about is what actual values should we give our coins? What coins should we have? So in Britain we have 1ps, 2ps, and so on, like this. What should we choose? And this is the great debate. Denominations. Now, a few hundred years ago, almost every currency in the world was non-decimal, was not a decimal currency. Now, almost every country in the world has a decimal currency. Humans are the same as we were. So something, what's changed, right? Why was non-decimal good 300 years ago, and now decimal is good? What has changed in that time? Why did we change and what's going on? We are the same people using money, after all. So what, if you think what you want out of coins, and coin values, denominations. There are two things you might want. The first thing, you want it to be kind of easy to work with, easy to calculate with. But you also would like to be able to kind of divide money, amounts of money up into fractions, to commonly used fractions. If you and your friend do something and get paid an amount of money, you want to be able to share it equally between you, if you have any sense of justice. So you want to be able to have, you know, half of amounts of money, and maybe thirds and quarters and things. So it's quite nice to be able to divide up money into small fractions like this. So the smallest first few fractions are a half, a third, and a quarter, right? If you want to be able to have a unit of money that you can divide into all of those fractions, then you need it to be divisible by 12, right? 'Cause if it's divisible by three and by four, you get two for free. So it must be divisible by 12. And so that is one really good reason to have shillings, right? So in the old, old, olden days, we used to have shillings, and they, one shilling had 12 old pennies, which we write d, 12d. And a really good thing about them is that you can then divide them up into these parts in, you know, an equal way. So I'll show you a few old coins, I love old coins. We've got a sixpence up there, that's half a shilling, 'cause a shilling is 12 pence. Then on the bottom here, this sort of gold colored one, that's a thrupenny bit, that's a quarter of a shilling. You might not recognize the top one there. That little silver one, the round silver one, that's a groat. Four pence, that's a third of a shilling. So we've got a half, a third, a quarter, and of course you can have one penny, a 12th of a shilling. There was tuppence as well. So there were lots of subdivisions of a shilling. You could go further even. So the cutest coin that we have had is the farthing. It's got a wren on it because the wren is the smallest British bird, right, and the farthing was the smallest British coin. That was a quarter of a penny. So farthing comes from fourthling, right? A fourth quarter. So, because, as well, there are 20 shillings in a pound, if you think how much can I divide, how many different ways can I divide a pound up, it's divisible by a half, a third, a quarter, a fifth, a sixth, loads of things you can break it up into. That seems really convenient and good, so why did we switch? Why go decimal? And the reason is 'cause we're, 'cause of the way we count, right? We have 10 fingers, and because of that, most cultures have developed at some point a decimal counting system, where we have tens and hundreds and thousands and so on. And this is how we write down our calculations nowadays, and it's how we do our arithmetic. So the problem with the pre-decimal currencies is that it becomes quite difficult to do calculations using the kind of techniques we've learned for decimal calculations. So just as an example, in the, you know, the old pre-decimal system, a pound is 20 shillings, and it's 240 pence. So suppose you have some little maths problem that says you've got, how much will it cost to buy 40 items at one shilling and seven pence each? So you think, okay, first I have to do that seven pence, so multiply that by 40, that's 280 pence. But then I have to divide by 12 to get how many shillings. Well okay, so do that and I get 23, remainder eight, but then, oh no, it's 20 shillings and a pound. So I've got to remember that. So it's one pound three shillings and eight pence. And then, oh I forgot about the one shilling at the beginning. 40 shillings, okay that's two pounds, so I add that, da-da-da, and you know, you're there for a while. Now of course people could practice this and get better at it, but estimates people have made has said that actually it would take kids almost an extra six months in school to be able to learn how to do all this stuff and convert, you know, it's not just that it's all base 12, it's a mix of base 12 and base 20. And don't even get me started on imperial units of weight, like ounces in a pound and pounds in a stone. You know, it's really tough to be able to get adept at these things. Whereas, if you think what's one shilling and seven pence in new money, well a shilling is like 5p, 'cause it's a 20th of a pound. Seven pence is about three new pence. So basically around 8p. So now, 40 items at 8p, three pound 20, done. So it is much quicker to use the decimal calculation, because that's what we learned to use. Okay, so people have been saying okay, this system, we should switch to decimal. Before I talk about who had said that and how long ago they said it, you can be guessing while we wait, I just want to mention the worst currency ever in the history of real or imagined realities. Poor old Harry Potter, right? Those wizards, I mean they had to invent magic purely to be able to pay, I reckon, for things, because they've got, I dunno what the real, the currency's supposed to look like, but they've got Bronze-Knuts, there are 29 of them in a Silver-Sickle, 29. And then there are 17 Silver-Sickles in a Golden-Galleon. So if you need to give your friend half a Galleon, you can't, no, I'm sorry. This is, you know, I can believe in all the rest, but the currency is just dreadful. I'm glad that dragon destroyed the bank. It's never going to work. So that has all the worst things of non-decimal coinage, impossible to calculate with. And it also doesn't have any of the benefits, because you can't divide things by half or third or anything. 17 and 29 are prime, like nothing to be done. Let's move on. Who do you think was a very early person to suggest that we switched to decimal currency, and when was that? Was it in 1920? Was it in 1850? Was it perhaps in 1696? Christopher Wren, already back then, former Gresham professor of astronomy, as we probably know, was suggesting we switch to a decimal currency. And he even wasn't the first. People had started saying this sort of in the early 17th century, and from then on people had started saying this. And I think it was because more and more people were becoming literate, they were being able to write down calculations. It wasn't all just like, you know, I'll share with you and you share with me. People were writing things down, they were doing actual proper accounting more and more. And so then you really come up against this wall of we're using hundreds, tens, hundreds, thousands in our calculations, but we're using these weird twelves and all sorts of things in our coins. So he said we should have a silver noble divided into 10 primes, and a hundred seconds, and this centesimal division will be very proper for accounts. So he suggested that in 1696. It did take a while for Britain to adopt a decimal currency after that. I think the first country to adopt a decimal currency, as far as I can tell, was Russia in 1704. The Russian Rouble became 100 kopeks. The U.S. dollar, in 1792 now, of course, one of the main objections to switching to a decimal system is that people said, look, people just won't go for it. They're used to what they're used to, they won't want to change, it's too complicated, you'll never be able to do it. But if you're creating a new currency in a new country, you don't have that problem. You just say, well, yeah, let's have this wonderful new thing, have a hundred cents, a hundred cents in a dollar right from the off. So that happened in 1792. Britain. Oh, Britain. So we talked about it a lot for hundreds of years. In the Victorian period, there was really an awful lot of talk about it. In parliament there were lots of debates, there were royal commissions. They even started to bring in coins that would sort of help ease us into decimalization. In 1849, right, they brought in a coin that was going to be, this is sort of a step along the route. And I don't know if you can see, but on this coin, so it says one decade at the top, and at the bottom it says one-tenth of a pound. So they're sort of explicitly saying that's two shillings, that's a florin. But they didn't say two shillings, they said one-tenth of a pound. This is kind of, let's edge people towards decimalization. Still took many, many more years, took another 120 years after that, after the one-tenth of a pound, to actually go decimal in 1971 on D-Day, in 15th February. But the, you know, it took people a while to get the hang of it, but now our school children have a better time with trying to do calculations. So, okay. Maybe we accept that decimalization is a good idea just to help us do our calculations and our arithmetic. We still haven't decided what coins to have. So the next question is, what denominations within the decimal system. We should have things that go into a pound exactly, right. So you don't want to have a 3p coin, because you can't get a whole number of those in a pound if you have a hundred pennies. Similarly with cents and dollars. So there's a mathematician called Adam Townsend at the University of Durham, suggested that one way of determining kind of the efficiency or, you know, best possible choices of different coins to have, is to look at the average number of coins you would get in change. So, you know, if you get 40p change, then you're going to get, in Britain you'd get two 20p coins. In America, if you're having 40 cents change, you might get a quarter and a nickel and a dime, right? 'Cause they don't have a 20 cent coin, they have a quarter which is 25 cents. So if you look, if you sort of average out over all possible amounts of change you could get, up to some limit, then you can see, this might be an indication of which system is the most efficient. So he'd written a couple of articles about this, I've put links in the transcript again to those. And he'd also come up with a bit of code that you can try, and I modified it for, slightly, to do something slightly different, and I put the code I used in the transcript if you want to try it. But the basic idea is that you do this and you work out the most efficient change you could get for each value up to, so I went up to 99p or 99 cents, on the grounds that then you're comparing like with like. But Dr. Townsend actually went up to the smallest bills, bank note that you can get. So in America that's the dollar bill, but in Britain it's a five pound note. So you can, you know, there's no right answer here. We're just looking at different ways we might measure the efficiency of these things. So if we look at the coins in the U.K. that are less than a pound, you've got lots of them. Six coins, we have 1p, 2p, 5p, 10p, 20p, 50p. That's a lot of coins. In America they only have four coins. The cent, the nickel, the dime, the quarter. So you might expect that you're going to need potentially more coins. You might get more coins in change in the U.S., 'cause they have fewer different ones. And that, indeed, is what happens. So in the U.K. you get a mean of 3.43 coins, and a median of three. So kind of about half the time you're getting three or fewer, about half the time you get three or more. In the U.S. that's a lot more. A median of five. So you get more things in change in the U.S. So we might say, maybe our system's a bit more efficient, but on the other hand, we do have more different kinds of coins, and maybe that has disadvantages. You could also think we have a 20p, they have a quarter. Is one of those better than the other? You can test that, and actually doesn't make too much difference. So in Britain, if you swap 20ps for quarters, absolutely no effect. It would have no effect, of course everyone would get very cross, so we probably won't do it. In the U.S., things would actually get slightly worse. If you tried to introduce a 20p instead of a quarter, or 20 cent coin, because they have so few coins already, that actually would make things a little bit worse. So within their system, that's probably the best coin to have. I encourage you to try this and think of your own, what do you think is the best thing to measure, and have a go at it. One thing to mention, and this is something that Adam Townsend pointed out in his article, that self-checkout machines are the work of the devil. He didn't put it like that. He said, so these machines are kind of, usually come from global designs, and they, because we have quite a lot of coins in Britain, we have those six coins that are below a pound, but we also have a one pound and a two pound coin. We have eight coins, which is quite a lot. Self checkout machines usually only give up to six different coins in change. So you can put in any coin, but they'll only give you out six different coins. And the ones that were chosen are usually 1p, 2p, 5p, not a 10p, 20p, not 50p, and then a pound and two pound. Now if you only have those coins available to you, then for getting amounts of change up to 99p, as I checked before, it's suddenly awful. You suddenly get an average of five coins in your change. If you ever curse one of those machines for giving you a load of shrapnel, now you know, you're not just sort of being grumpy, they really do give you too much change. They're not as efficient as a human would be. They give you loads of 5ps, usually, typically, why have I got a whole handful of 5ps? This isn't even the right choice. If you have to limit yourself to six coins, then if you change, if instead of giving pound coins you give 50ps, then actually can bring that right down, almost back to the what a human would give you. So it's kind of interesting this affects our daily lives, but we don't necessarily realize what's happening. And actually all supermarkets should immediately change to this second system, right away, starting today. That's my order, as a Gresham professor of geometry. The final thing I want to mention. We've talked about having a lot of coins. The 1p coin, has it had its day? Is it time to ditch the 1p coin? 1p doesn't buy you a lot these days. Inflation is quite high. How can we decide when it's time to let it go? Well, one way is to compare when other small coins have been discontinued. And the Bank of England has this amazing inflation calculator that will tell you, you can put in an amount of money, and it tells you what it would be worth today for any year, going back to 1209. So, wow! So you can try this out, or you could tell your kids, you know, how lucky they are and how they should be grateful for anything you've given, because, you know, the amount you had when you were a kid. I do this to my children and they love it so much. Halfpenny coin discontinued, 1984. People didn't like it. They didn't care about it as a coin. Very important issues arose. So this is an irate letter to "The Times." You cannot, how dare you get rid of the 1/2p? "It's indispensable for leveling off pendulum clocks." So, you know, in touch with the common person as ever. In spite of this urgent need, I hope they kept a few in reserve, they were discontinued. Now looking at inflation, one pound in 1984 could buy you the same as two pounds 63 today. So that would mean that a 1/2p, when it was discontinued, would have the same value as 1.3p today, more than a penny. Interesting. Let's think about the other coin that was discontinued, the lovely farthing, discontinued in 1960. One pound in 1960 would be worth 16 pounds and 18p today. That little asterisk is to remind me that the inflation calculator goes to the end of the last full calendar year. So I'm talking now in October, 2022, this is up to the end of 2021. So things will only have got worse since then. Let's say it's worth about 40 pounds today. So a farthing, back in 1960, that's equivalent to 1.7p today. So actually, not only should we be getting rid of the penny, we might have to get rid of the 2p as well, and just, you know, get rid of those coppers altogether. We'd miss them, but maybe we would like our purses being a bit lighter. We could manage without them. You might think but how are we going to pay for things that have, you know, whose values are 99p or something. That's okay. I mean, Canada actually got rid of their one cent coin a long time ago in 2013. Their smallest coin now is five cents. And what happens is you can still have things priced at any amount you want. You can still deal electronically in whole numbers of pennies or cents, but when you get change in a shop, it's just rounded to the nearest five. And that's, you know, no one worries about that. So we might, it might be time to say goodbye to the penny. But, you know, one day we may not use coins at all. I think that day is happily quite far off. Coins are so interesting and fascinating, and they're bits of history and mathematics, as we've seen, in our pockets. So I hope you've enjoyed this mathematical guide to coins. In my next lecture in the Maths and Money series, we're looking at the mathematics of game theory, which is how maths can help us to find the best strategies when we're buying, selling, bargaining, bartering and auctioning. So thank you very much and I hope to see you then. (audience applauding) - When you were talking earlier on about the people cheating with various fake coinage, there was a question that came in about how about an alloy of gold, silver, and a denser metal in such a proportion that the overall density is the same as that of gold. - Yes. So, and that's, yeah, that's a great question. So you could imagine potentially being able to put in, manage to mix things, maybe do a bit of allegation, but for bad and nefarious purposes, would allow you to get something that matches the density and therefore the mass. So depending on what metals you use, potentially you might be able to detect in the color, the luster, what it actually looks like, how soft it is. So lead for example, is potentially quite soft. If you're really worried about a batch of coins, then you can do an assay, you can test them chemically. If the testing of the densities and the volumes doesn't do it for you. So there are ways still to detect those things. It might just be if someone was super clever like that, it might be a bit harder, yeah. - Thank you so much, Professor Hart. I'm very sorry to say we are out of time now for any further questions, but thank you and please, I hope we will see you all again on Tuesday the 22nd of November, for Professor Hart's Maths of Game Theory. Thank you so much. (audience applauding)