The Summer 2017 Mathematics A To Z: Arithmetic


And now as summer (United States edition) reaches its closing months I plunge into the fourth of my A To Z mathematics-glossary sequences. I hope I know what I’m doing! Today’s request is one of several from Gaurish, who’s got to be my top requester for mathematical terms and whom I thank for it. It’s a lot easier writing these things when I don’t have to think up topics. Gaurish hosts a fine blog, For the love of Mathematics, which you might consider reading.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Arithmetic.

Arithmetic is what people who aren’t mathematicians figure mathematicians do all day. I remember in my childhood a Berenstain Bears book about people’s jobs. Its mathematician was an adorable little bear adding up sums on the chalkboard, in an observatory, on the Moon. I liked every part of this. I wouldn’t say it’s the whole reason I became a mathematician but it did made the prospect look good early on.

People who aren’t mathematicians are right. At least, the bulk of what mathematics people do is arithmetic. If we work by volume. Arithmetic is about the calculations we do to evaluate or solve polynomials. And polynomials are everything that humans find interesting. Arithmetic is adding and subtracting, of multiplication and division, of taking powers and taking roots. Arithmetic is changing the units of a thing, and of breaking something into several smaller units, or of merging several smaller units into one big one. Arithmetic’s role in commerce and in finance must overwhelm the higher mathematics. Higher mathematics offers cohomologies and Ricci tensors. Arithmetic offers a budget.

This is old mathematics. There’s evidence of humans twenty thousands of years ago recording their arithmetic computations. My understanding is the evidence is ambiguous and interpretations vary. This seems fair. I assume that humans did such arithmetic then, granting that I do not know how to interpret archeological evidence. The thing is that arithmetic is older than humans. Animals are able to count, to do addition and subtraction, perhaps to do harder computations. (I crib this from The Number Sense:
How the Mind Creates Mathematics
, by Stanislas Daehaene.) We learn it first, refining our rough instinctively developed sense to something rigorous. At least we learn it at the same time we learn geometry, the other branch of mathematics that must predate human existence.

The primality of arithmetic governs how it becomes an adjective. We will have, for example, the “arithmetic progression” of terms in a sequence. This is a sequence of numbers such as 1, 3, 5, 7, 9, and so on. Or 4, 9, 14, 19, 24, 29, and so on. The difference between one term and its successor is the same as the difference between the predecessor and this term. Or we speak of the “arithmetic mean”. This is the one found by adding together all the numbers of a sample and dividing by the number of terms in the sample. These are important concepts, useful concepts. They are among the first concepts we have when we think of a thing. Their familiarity makes them easy tools to overlook.

Consider the Fundamental Theorem of Arithmetic. There are many Fundamental Theorems; that of Algebra guarantees us the number of roots of a polynomial equation. That of Calculus guarantees us that derivatives and integrals are joined concepts. The Fundamental Theorem of Arithmetic tells us that every whole number greater than one is equal to one and only one product of prime numbers. If a number is equal to (say) two times two times thirteen times nineteen, it cannot also be equal to (say) five times eleven times seventeen. This may seem uncontroversial. The budding mathematician will convince herself it’s so by trying to work out all the ways to write 60 as the product of prime numbers. It’s hard to imagine mathematics for which it isn’t true.

But it needn’t be true. As we study why arithmetic works we discover many strange things. This mathematics that we know even without learning is sophisticated. To build a logical justification for it requires a theory of sets and hundreds of pages of tight reasoning. Or a theory of categories and I don’t even know how much reasoning. The thing that is obvious from putting a couple objects on a table and then a couple more is hard to prove.

As we continue studying arithmetic we start to ponder things like Goldbach’s Conjecture, about even numbers (other than two) being the sum of exactly two prime numbers. This brings us into number theory, a land of fascinating problems. Many of them are so accessible you could pose them to a person while waiting in a fast-food line. This befits a field that grows out of such simple stuff. Many of those are so hard to answer that no person knows whether they are true, or are false, or are even answerable.

And it splits off other ideas. Arithmetic starts, at least, with the counting numbers. It moves into the whole numbers and soon all the integers. With division we soon get rational numbers. With roots we soon get certain irrational numbers. A close study of this implies there must be irrational numbers that must exist, at least as much as “four” exists. Yet they can’t be reached by studying polynomials. Not polynomials that don’t already use these exotic irrational numbers. These are transcendental numbers. If we were to say the transcendental numbers were the only real numbers we would be making only a very slight mistake. We learn they exist by thinking long enough and deep enough about arithmetic to realize there must be more there than we realized.

Thought compounds thought. The integers and the rational numbers and the real numbers have a structure. They interact in certain ways. We can look for things that are not numbers, but which follow rules like that for addition and for multiplication. Sometimes even for powers and for roots. Some of these can be strange: polynomials themselves, for example, follow rules like those of arithmetic. Matrices, which we can represent as grids of numbers, can have powers and even something like roots. Arithmetic is inspiration to finding mathematical structures that look little like our arithmetic. We can find things that follow mathematical operations but which don’t have a Fundamental Theorem of Arithmetic.

And there are more related ideas. These are often very useful. There’s modular arithmetic, in which we adjust the rules of addition and multiplication so that we can work with a finite set of numbers. There’s floating point arithmetic, in which we set machines to do our calculations. These calculations are no longer precise. But they are fast, and reliable, and that is often what we need.

So arithmetic is what people who aren’t mathematicians figure mathematicians do all day. And they are mistaken, but not by much. Arithmetic gives us an idea of what mathematics we can hope to understand. So it structures the way we think about mathematics.

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48 Altered States


I saw this intriguing map produced by Brian Brettschneider.

He made it on and for Twitter, as best I can determine. I found it from a stray post in Usenet newsgroup soc.history.what-if, dedicated to ways history could have gone otherwise. It also covers ways that it could not possibly have gone otherwise but would be interesting to see happen. Very different United States state boundaries are part of the latter set of things.

The location of these boundaries is described in English and so comes out a little confusing. It’s hard to make concise. Every point in, say, this alternate Missouri is closer to Missouri’s capital of … uhm … Missouri City than it is to any other state’s capital. And the same for all the other states. All you kind readers who made it through my recent A To Z know a technical term for this. This is a Voronoi Diagram. It uses as its basis points the capitals of the (contiguous) United States.

It’s an amusing map. I mean amusing to people who can attach concepts like amusement to maps. It’d probably be a good one to use if someone needed to make a Risk-style grand strategy game map and didn’t want to be to beholden to the actual map.

No state comes out unchanged, although a few don’t come out too bad. Maine is nearly unchanged. Michigan isn’t changed beyond recognition. Florida gets a little weirder but if you showed someone this alternate shape they’d recognize the original. No such luck with alternate Tennessee or alternate Wyoming.

The connectivity between states changes a little. California and Arizona lose their border. Washington and Montana gain one; similarly, Vermont and Maine suddenly become neighbors. The “Four Corners” spot where Utah, Colorado, New Mexico, and Arizona converge is gone. Two new ones look like they appear, between New Hampshire, Massachusetts, Rhode Island, and Connecticut; and between Pennsylvania, Maryland, Virginia, and West Virginia. I would be stunned if that weren’t just because we can’t zoom far enough in on the map to see they’re actually a pair of nearby three-way junctions.

I’m impressed by the number of borders that are nearly intact, like those of Missouri or Washington. After all, many actual state boundaries are geographic features like rivers that a Voronoi Diagram doesn’t notice. How could Ohio come out looking anything like Ohio?

The reason comes to historical subtleties. At least once you get past the original 13 states, basically the east coast of the United States. The boundaries of those states were set by colonial charters, with boundaries set based on little or ambiguous information about what the local terrain was actually like, and drawn to reward or punish court factions and favorites. Never mind the original thirteen (plus Maine and Vermont, which we might as well consider part of the original thirteen).

After that, though, the United States started drawing state boundaries and had some method to it all. Generally a chunk of territory would be split into territories and later states that would be roughly rectangular, so far as practical, and roughly similar in size to the other states carved of the same area. So for example Missouri and Alabama are roughly similar to Georgia in size and even shape. Louisiana, Arkansas, and Missouri are about equal in north-south span and loosely similar east-to-west. Kansas, Nebraska, South Dakota, and North Dakota aren’t too different in their north-to-south or east-to-west spans.

There’s exceptions, for reasons tied to the complexities of history. California and Texas get peculiar shapes because they could. Michigan has an upper peninsula for quirky reasons that some friend of mine on Twitter discovers every three weeks or so. But the rough guide is that states look a lot more similar to one another than you’d think from a quick look. Mark Stein’s How The States Got Their Shapes is an endlessly fascinating text explaining this all.

If there is a loose logic to state boundaries, though, what about state capitals? Those are more quirky. One starts to see the patterns when considering questions like “why put California’s capital in Sacramento instead of, like, San Francisco?” or “Why Saint Joseph instead Saint Louis or Kansas City?” There is no universal guide, but there are some trends. Generally states end up putting their capitals in a city that’s relatively central, at least to the major population centers around the time of statehood. And, generally, not in one of the state’s big commercial or industrial centers. The desire to be geographically central is easy to understand. No fair making citizens trudge that far if they have business in the capital. Avoiding the (pardon) first tier of cities has subtler politics to it; it’s an attempt to get the government somewhere at least a little inconvenient to the money powers.

There’s exceptions, of course. Boston is the obviously important city in Massachusetts, Salt Lake City the place of interest for Utah, Denver the equivalent for Colorado. Capitals relocated; Atlanta is Georgia’s eighth(?) I think since statehood. Sometimes they were weirder. Until 1854 Rhode Island rotated between five cities, to the surprise of people trying to name a third city in Rhode Island. New Jersey settled on Trenton as compromise between the East and West Jersey capitals of Perth Amboy and Burlington. But if you look for a city that’s fairly central but not the biggest in the state you get to the capital pretty often.

So these are historical and cultural factors which combine to make a Voronoi Diagram map of the United States strange, but not impossibly strange, compared to what has really happened. Things are rarely so arbitrary as they seem at first.

The End 2016 Mathematics A To Z: Yang Hui’s Triangle


Today’s is another request from gaurish and another I’m glad to have as it let me learn things too. That’s a particularly fun kind of essay to have here.

Yang Hui’s Triangle.

It’s a triangle. Not because we’re interested in triangles, but because it’s a particularly good way to organize what we’re doing and show why we do that. We’re making an arrangement of numbers. First we need cells to put the numbers in.

Start with a single cell in what’ll be the top middle of the triangle. It spreads out in rows beneath that. The rows are staggered. The second row has two cells, each one-half width to the side of the starting one. The third row has three cells, each one-half width to the sides of the row above, so that its center cell is directly under the original one. The fourth row has four cells, two of which are exactly underneath the cells of the second row. The fifth row has five cells, three of them directly underneath the third row’s cells. And so on. You know the pattern. It’s the one that pins in a plinko board take. Just trimmed down to a triangle. Make as many rows as you find interesting. You can always add more later.

In the top cell goes the number ‘1’. There’s also a ‘1’ in the leftmost cell of each row, and a ‘1’ in the rightmost cell of each row.

What of interior cells? The number for those we work out by looking to the row above. Take the cells to the immediate left and right of it. Add the values of those together. So for example the center cell in the third row will be ‘1’ plus ‘1’, commonly regarded as ‘2’. In the third row the leftmost cell is ‘1’; it always is. The next cell over will be ‘1’ plus ‘2’, from the row above. That’s ‘3’. The cell next to that will be ‘2’ plus ‘1’, a subtly different ‘3’. And the last cell in the row is ‘1’ because it always is. In the fourth row we get, starting from the left, ‘1’, ‘4’, ‘6’, ‘4’, and ‘1’. And so on.

It’s a neat little arithmetic project. It has useful application beyond the joy of making something neat. Many neat little arithmetic projects don’t have that. But the numbers in each row give us binomial coefficients, which we often want to know. That is, if we wanted to work out (a + b) to, say, the third power, we would know what it looks like from looking at the fourth row of Yanghui’s Triangle. It will be 1\cdot a^4 + 4\cdot a^3 \cdot b^1 + 6\cdot a^2\cdot b^2 + 4\cdot a^1\cdot b^3 + 1\cdot b^4 . This turns up in polynomials all the time.

Look at diagonals. By diagonal here I mean a line parallel to the line of ‘1’s. Left side or right side; it doesn’t matter. Yang Hui’s triangle is bilaterally symmetric around its center. The first diagonal under the edges is a bit boring but familiar enough: 1-2-3-4-5-6-7-et cetera. The second diagonal is more curious: 1-3-6-10-15-21-28 and so on. You’ve seen those numbers before. They’re called the triangular numbers. They’re the number of dots you need to make a uniformly spaced, staggered-row triangle. Doodle a bit and you’ll see. Or play with coins or pool balls.

The third diagonal looks more arbitrary yet: 1-4-10-20-35-56-84 and on. But these are something too. They’re the tetrahedronal numbers. They’re the number of things you need to make a tetrahedron. Try it out with a couple of balls. Oranges if you’re bored at the grocer’s. Four, ten, twenty, these make a nice stack. The fourth diagonal is a bunch of numbers I never paid attention to before. 1-5-15-35-70-126-210 and so on. This is — well. We just did tetrahedrons, the triangular arrangement of three-dimensional balls. Before that we did triangles, the triangular arrangement of two-dimensional discs. Do you want to put in a guess what these “pentatope numbers” are about? Sure, but you hardly need to. If we’ve got a bunch of four-dimensional hyperspheres and want to stack them in a neat triangular pile we need one, or five, or fifteen, or so on to make the pile come out neat. You can guess what might be in the fifth diagonal. I don’t want to think too hard about making triangular heaps of five-dimensional hyperspheres.

There’s more stuff lurking in here, waiting to be decoded. Add the numbers of, say, row four up and you get two raised to the third power. Add the numbers of row ten up and you get two raised to the ninth power. You see the pattern. Add everything in, say, the top five rows together and you get the fifth Mersenne number, two raised to the fifth power (32) minus one (31, when we’re done). Add everything in the top ten rows together and you get the tenth Mersenne number, two raised to the tenth power (1024) minus one (1023).

Or add together things on “shallow diagonals”. Start from a ‘1’ on the outer edge. I’m going to suppose you started on the left edge, but remember symmetry; it’ll be fine if you go from the right instead. Add to that ‘1’ the number you get by moving one cell to the right and going up-and-right. And then again, go one cell to the right and then one cell up-and-right. And again and again, until you run out of cells. You get the Fibonacci sequence, 1-1-2-3-5-8-13-21-and so on.

We can even make an astounding picture from this. Take the cells of Yang Hui’s triangle. Color them in. One shade if the cell has an odd number, another if the cell has an even number. It will create a pattern we know as the Sierpiński Triangle. (Wacław Sierpiński is proving to be the surprise special guest star in many of this A To Z sequence’s essays.) That’s the fractal of a triangle subdivided into four triangles with the center one knocked out, and the remaining triangles them subdivided into four triangles with the center knocked out, and on and on.

By now I imagine even my most skeptical readers agree this is an interesting, useful mathematical construct. Also that they’re wondering why I haven’t said the name “Blaise Pascal”. The Western mathematical tradition knows of this from Pascal’s work, particularly his 1653 Traité du triangle arithmétique. But mathematicians like to say their work is universal, and independent of the mere human beings who find it. Constructions like this triangle give support to this. Yang lived in China, in the 12th century. I imagine it possible Pascal had hard of his work or been influenced by it, by some chain, but I know of no evidence that he did.

And even if he had, there are other apparently independent inventions. The Avanti Indian astronomer-mathematician-astrologer Varāhamihira described the addition rule which makes the triangle work in commentaries written around the year 500. Omar Khayyám, who keeps appearing in the history of science and mathematics, wrote about the triangle in his 1070 Treatise on Demonstration of Problems of Algebra. Again so far as I am aware there’s not a direct link between any of these discoveries. They are things different people in different traditions found because the tools — arithmetic and aesthetically-pleasing orders of things — were ready for them.

Yang Hui wrote about his triangle in the 1261 book Xiangjie Jiuzhang Suanfa. In it he credits the use of the triangle (for finding roots) was invented around 1100 by mathematician Jia Xian. This reminds us that it is not merely mathematical discoveries that are found by many peoples at many times and places. So is Boyer’s Law, discovered by Hubert Kennedy.

Theorem Thursday: The Five-Color Map Theorem


People think mathematics is mostly counting and arithmetic. It’s what we get at when we say “do the math[s]”. It’s why the mathematician in the group is the one called on to work out what the tip should be. Heck, I attribute part of my love for mathematics to a Berenstain Bears book which implied being a mathematician was mostly about adding up sums in a base on the Moon, which is an irresistible prospect. In fact, usually counting and arithmetic are, at least, minor influences on real mathematics. There are legends of how catastrophically bad at figuring mathematical genius can be. But usually isn’t always, and this week I’d like to show off a case where counting things and adding things up lets us prove something interesting.

The Five-Color Map Theorem.

No, not four. I imagine anyone interested enough to read a mathematics blog knows the four-color map theorem. It says that you only need four colors to color a map. That’s true, given some qualifiers. No discontiguous chunks that need the same color. Two regions with the same color can touch at a point, they just can’t share a line or curve. The map is on a plane or the surface of a sphere. Probably some other requirements. I’m not going to prove that. Nobody has time for that. The best proofs we’ve figured out for it amount to working out how every map fits into one of a huge number of cases, and trying out each case. It’s possible to color each of those cases with only four colors, so, we’re done. Nice but unenlightening and way too long to deal with.

The five-color map theorem is a lot like the four-color map theorem, with this difference: it says that you only need five colors to color a map. Same qualifiers as before. Yes, it’s true because the four-color map theorem is true and because five is more than four. We can do better than that. We can prove five colors are enough even without knowing whether four colors will do. And it’s easy. The ease of the five-color map theorem gave people reason to think four colors would be maybe harder but still manageable.

The proof I want to show uses one of mathematicians’ common tricks. It employs the same principle which Hercules used to slay the Hydra, although it has less cauterizing lake-monster flesh with flaming torches, as that’s considered beneath the dignity of the Academy anymore except when grading finals for general-requirements classes. The part of the idea we do use is to take a problem which we might not be able to do and cut it down to one we can do. Properly speaking this is a kind of induction proof. In those we start from problems we can do and show that if we can do those, we can do all the complicated problems. But we come at it by cutting down complicated problems and making them simple ones.

So suppose we start with a map that’s got some huge number of territories to color. I’m going to start with the United States states which were part of the Dominion of New England. As I’m sure I don’t need to remind any readers, American or otherwise, this was a 17th century attempt by the English to reorganize their many North American colonies into something with fewer administrative irregularities. It lasted almost long enough for the colonists to hear about it. At that point the Glorious Revolution happened (not involving the colonists) and everybody went back to what they were doing before.

Please enjoy my little map of the place. It gives all the states a single color because I don’t really know how to use QGIS and it would probably make my day job easier if I did. (Well, QGIS is open-source software, so its interface is a disaster and its tutorials gibberish. The only way to do something with it is to take flaming torches to it.)

Map showing New York, New Jersey, and New England (Connecticut, Rhode Island, Massachusetts, Vermont, New Hampshire, and Maine) in a vast white space.
States which, in their 17th-century English colonial form, were part of the Dominion of New England (1685-1689). More or less. If I’ve messed up don’t tell me as it doesn’t really matter for this problem.

There’s eight regions here, eight states, so it’s not like we’re at the point we can’t figure how to color this with five different colors. That’s all right. I’m using this for a demonstration. Pretend the Dominion of New England is so complicated we can’t tell whether five colors are enough. Oh, and a spot of lingo: if five colors are enough to color the map we say the map is “colorable”. We say it’s “5-colorable” if we want to emphasize five is enough colors.

So imagine that we erase the border between Maine and New Hampshire. Combine them into a single state over the loud protests of the many proud, scary Mainers. But if this simplified New England is colorable, so is the real thing. There’s at least one color not used for Greater New Hampshire, Vermont, or Massachusetts. We give that color to a restored Maine. If the simplified map can be 5-colored, so can the original.

Maybe we can’t tell. Suppose the simplified map is still too complicated to make it obvious. OK, then. Cut out another border. How about we offend Roger Williams partisans and merge Rhode Island into Massachusetts? Massachusetts started out touching five other states, which makes it a good candidate for a state that needed a sixth color. With Rhode Island reduced to being a couple counties of the Bay State, Greater Massachusetts only touches four other states. It can’t need a sixth color. There’s at least one of our original five that’s free.

OK, but, how does that help us find a color for Rhode Island? Maine it’s easy to see why there’s a free color. But Rhode Island?

Well, it’ll have to be the same color as either Greater New Hampshire or Vermont or New York. At least one of them has to be available. Rhode Island doesn’t touch them. Connecticut’s color is out because Rhode Island shares a border with it. Same with Greater Massachusetts’s color. But we’ve got three colors for the taking.

But is our reduced map 5-colorable? Even with Maine part of New Hampshire and Rhode Island part of Massachusetts it might still be too hard to tell. There’s six territories in it, after all. We can simplify things a little. Let’s reverse the treason of 1777 and put Vermont back into New York, dismissing New Hampshire’s claim on the territory as obvious absurdity. I am never going to be allowed back into New England. This Greater New York needs one color for itself, yes. And it touches four other states. But these neighboring states don’t touch each other. A restored Vermont could use the same color as New Jersey or Connecticut. Greater Massachusetts and Greater New Hampshire are unavailable, but there’s still two choices left.

And now look at the map we have remaining. There’s five states in it: Greater New Hampshire, Greater Massachusetts, Greater New York, Regular Old Connecticut, and Regular old New Jersey. We have five colors. Obviously we can give the five territories different colors.

This is one case, one example map. That’s all we need. A proper proof makes things more abstract, but uses the same pattern. Any map of a bunch of territories is going to have at least one territory that’s got at most five neighbors. Maybe it will have several. Look for one of them. If you find a territory with just one neighbor, such as Maine had, remove that border. You’ve got a simpler map and there must be a color free for the restored territory.

If you find a territory with just two neighbors, such as Rhode Island, take your pick. Merge it with either neighbor. You’ll still have at least one color free for the restored territory. With three neighbors, such as Vermont or Connecticut, again you have your choice. Merge it with any of the three neighbors. You’ll have a simpler map and there’ll be at least one free color.

If you have four neighbors, the way New York has, again pick a border you like and eliminate that. There is a catch. You can imagine one of the neighboring territories reaching out and wrapping around to touch the original state on more than one side. Imagine if Massachusetts ran far out to sea, looped back through Canada, and came back to touch New Jersey, Vermont from the north, and New York from the west. That’s more of a Connecticut stunt to pull, I admit. But that’s still all right. Most of the colonies tried this sort of stunt. And even if Massachusetts did that, we would have colors available. It would be impossible for Vermont and New Jersey to touch. We’ve got a theorem that proves it.

Yes, it’s the Jordan Curve Theorem, here to save us right when we might get stuck. Just like I promised last week. In this case some part of the border of New York and Really Big Massachusetts serves as our curve. Either Vermont or New Jersey is going to be inside that curve, and the other state is outside. They can’t touch. Thank you.

If you have five neighbors, the way Massachusetts has, well, maybe you’re lucky. We are here. None of its neighboring states touches more than two others. We can cut out a border easily and have colors to spare. But we could be in trouble. We could have a map in which all the bordering states touch three or four neighbors and that seems like it would run out of colors. Let me show a picture of that.

The map shows a pentagonal region A which borders five regions, B, C, D, E, and F. Each of those regions borders three or four others. B is entirely enclosed by regions A, C, and D, although from B's perspective they're all enclosed by it.
A hypothetical map with five regions named by an uninspired committee.

So this map looks dire even when you ignore that line that looks like it isn’t connected where C and D come together. Flood fill didn’t run past it, so it must be connected. It just doesn’t look right. Everybody has four neighbors except the province of B, which has three. The province of A has got five. What can we do?

Call on the Jordan Curve Theorem again. At least one of the provinces has to be landlocked, relative to the others. In this case, the borders of provinces A, D, and C come together to make a curve that keeps B in the inside and E on the outside. So we’re free to give B and E the same color. We treat this in the proof by doing a double merger. Erase the boundary between provinces A and B, and also that between provinces A and E. (Or you might merge B, A, and F together. It doesn’t matter. The Jordan Curve Theorem promises us there’ll be at least one choice and that’s all we need.)

So there we have it. As long as we have a map that has some provinces with up to five neighbors, we can reduce the map. And reduce it again, if need be, and again and again. Eventually we’ll get to a map with only five provinces and that has to be 5-colorable.

Just … now … one little nagging thing. We’re relying on there always being some province with at most five neighbors. Why can’t there be some horrible map where every province has six or more neighbors?

Counting will tell us. Arithmetic will finish the job. But we have to get there by way of polygons.

That is, the easiest way to prove this depends on a map with boundaries that are all polygons. That’s all right. Polygons are almost the polynomials of geometry. You can make a polygon that looks so much like the original shape the eye can’t tell the difference. Look at my Dominion of New England map. That’s computer-rendered, so it’s all polygons, and yet all those shore and river boundaries look natural.

But what makes up a polygon? Well, it’s a bunch of straight lines. We call those ‘edges’. Each edge starts and ends at a corner. We call those ‘vertices’. These edges come around and close together to make a ‘face’, a territory like we’ve been talking about. We’re going to count all the regions that have a certain number of neighboring other regions.

Specifically, F2 will represent however many faces there are that have two sides. F3 will represent however many faces there are that have three sides. F4 will represent however many faces there are that have four sides. F10 … yeah, you got this.

One thing you didn’t get. The outside counts as a face. We need this to make the count come out right, so we can use some solid-geometry results. In my map that’s the vast white space that represents the Atlantic Ocean, the other United States, the other parts of Canada, the Great Lakes, all the rest of the world. So Maine, for example, belongs to F2 because it touches New Hampshire and the great unknown void of the rest of the universe. Rhode Island belongs to F3 similarly. New Hampshire’s in F4.

Any map has to have at least one thing that’s in F2, F3, F4, or F5. They touch at most two, three, four or five neighbors. (If they touched more, they’d represent a face that was a polygon of even more sides.)

How do we know? It comes from Euler’s Formula, which starts out describing the ways corners and edges and faces of a polyhedron fit together. Our map, with its polygon on the surface of the sphere, turns out to be just as good as a polyhedron. It looks a little less blocky, but that doesn’t show.

By Euler’s Formula, there’s this neat relationship between the number of vertices, the number of edges, and the number of faces in a polyhedron. (This is the same Leonhard Euler famous for … well, everything in mathematics, really. But in this case it’s for his work with shapes.) It holds for our map too. Call the number of vertices V. Call the number of edges E. Call the number of faces F. Then:

V - E + F = 2

Always true. Try drawing some maps yourself, using simple straight lines, and see if it works. For that matter, look at my Really Really Simplified map and see if it doesn’t hold true still.

One of those blocky diagrams of New York, New Jersey, and New England, done in that way transit maps look, only worse because I'm not so good at this.
A very simplified blocky diagram of my Dominion of New England, with the vertices and edges highlighted so they’re easy to count if you want to do that.

Here’s one of those insights that’s so obvious it’s hard to believe. Every edge ends in two vertices. Three edges meet at every vertex. (We don’t have more than three territories come together at a point. If that were to happen, we’d change the map a little to find our coloring and then put it back afterwards. Pick one of the territories and give it a disc of area from the four or five or more corners. The troublesome corner is gone. Once we’ve done with our proof, shrink the disc back down to nothing. Coloring done!) And therefore 2E = 3V .

A polygon has the same number of edges as vertices, and if you don’t believe that then draw some and count. Every edge touches exactly two regions. Every vertex touches exactly three edges. So we can rework Euler’s formula. Multiply it by six and we get 6V - 6E + 6F = 12 . And from doubling the equation about edges and vertices equation in the last paragraph, 4E = 6V . So if we break up that 6E into 4E and 2E we can rewrite that Euler’s formula again. It becomes 6V - 4E - 2E + 6F = 12. 6V – 4E is zero, so, -2E + 6F = 12 .

Do we know anything about F itself?

Well, yeah. F = F_2 + F_3 + F_4 + F_5 + F_6 + \cdots . The number of faces has to equal the sum of the number of faces of two edges, and of three edges, and of four edges, and of five edges, and of six edges, and on and on. Counting!

Do we know anything about how E and F relate?

Well, yeah. A polygon in F2 has two edges. A polygon in F3 has three edges. A polygon in F4 has four edges. And each edge runs up against two faces. So therefore 2E = 2F_2 + 3F_3 + 4F_4 + 5F_5 + 6F_6 + \cdots . This goes on forever but that’s all right. We don’t need all these terms.

Because here’s what we do have. We know that -2E + 6F = 12 . And we know how to write both E and F in terms of F2, F3, F4, and so on. We’re going to show at least one of these low-subscript Fsomethings has to be positive, that is, there has to be at least one of them.

Start by just shoving our long sum expressions into the modified Euler’s Formula we had. That gives us this:

-(2F_2 + 3F_3 + 4F_4 + 5F_5 + 6F_6 + \cdots) + 6(F_2 + F_3 + F_4 + F_5 + F_6 + \cdots) = 12

Doesn’t look like we’ve got anywhere, does it? That’s all right. Multiply that -1 and that 6 into their parentheses. And then move the terms around, so that we group all the terms with F2 together, and all the terms with F3 together, and all the terms with F4 together, and so on. This gets us to:

(-2 + 6) F_2 + (-3 + 6) F_3 + (-4 + 6) F_4 + (-5 + 6) F_5  + (-6 + 6) F_6 + (-7 + 6) F_7 + (-8 + 6) F_8 + \cdots = 12

I know, that’s a lot of parentheses. And it adds negative numbers to positive which I guess we’re allowed to do but who wants to do that? Simplify things a little more:

4 F_2 + 3 F_3 + 2 F_4 + 1 F_5 + 0 F_6 - 1 F_7 - 2 F_8 - \cdots = 12

And now look at that. Each Fsubscript has to be zero or a positive number. You can’t have a negative number of shapes. If you can I don’t want to hear about it. Most of those Fsubscript‘s get multiplied by a negative number before they’re added up. But the sum has to be a positive number.

There’s only one way that this sum can be a positive number. At least one of F2, F3, F4, or F5 has to be a positive number. So there must be at least one region with at most five neighbors. And that’s true without knowing anything about our map. So it’s true about the original map, and it’s true about a simplified map, and about a simplified-more map, and on and on.

And that is why this hydra-style attack method always works. We can always simplify a map until it obviously can be colored with five colors. And we can go from that simplified map back to the original map, and color it in just fine. Formally, this is an existence proof: it shows there must be a way to color a map with five colors. But it does so the devious way, by showing a way to color the map. We don’t get enough existence proofs like that. And, at its critical point, we know the proof is true because we can count the number of regions and the number of edges and the number of corners they have. And we can add and subtract those numbers in the right way. Just like people imagine mathematicians do all day.

Properly this works only on the surface of a sphere. Euler’s Formula, which we use for the proof, depends on that. We get away with it on a piece of paper because we can pretend this is just a part of the globe so small we don’t see how flat it is. The vast white edge we suppose wraps around the whole world. And that’s fine since we mostly care about maps on flat surfaces or on globes. If we had a map that needed three dimensions, like one that looked at mining and water and overflight and land-use rights, things wouldn’t be so easy. Nor would they work at all if the map turned out to be on an exotic shape like a torus, a doughnut shape.

But this does have a staggering thought. Suppose we drew boundary lines. And suppose we found an arrangement of them so that we needed more than five colors. This would tell us that we have to be living on a surface such as a torus, the doughnut shape. We could learn something about the way space is curved by way of an experiment that never looks at more than where two regions come together. That we can find information about the whole of space, global information, by looking only at local stuff amazes me. I hope it at least surprises you.

From fiddling with this you probably figure the four-color map theorem should follow right away. Maybe involve a little more arithmetic but nothing too crazy. I agree, it so should. It doesn’t. Sorry.

How Interesting Is A Baseball Score? Some Partial Results


Meanwhile I have the slight ongoing quest to work out the information-theory content of sports scores. For college basketball scores I made up some plausible-looking score distributions and used that. For professional (American) football I found a record of all the score outcomes that’ve happened, and how often. I could use experimental results. And I’ve wanted to do other sports. Soccer was asked for. I haven’t been able to find the scoring data I need for that. Baseball, maybe the supreme example of sports as a way to generate statistics … has been frustrating.

The raw data is available. Retrosheet.org has logs of pretty much every baseball game, going back to the forming of major leagues in the 1870s. What they don’t have, as best I can figure, is a list of all the times each possible baseball score has turned up. That I could probably work out, when I feel up to writing the scripts necessary, but “work”? Ugh.

Some people have done the work, although they haven’t shared all the results. I don’t blame them; the full results make for a boring sort of page. “The Most Popular Scores In Baseball History”, at ValueOverReplacementGrit.com, reports the top ten most common scores from 1871 through 2010. The essay also mentions that as of then there were 611 unique final scores. And that lets me give some partial results, if we trust that blogger post from people I never heard of before are accurate and true. I will make that assumption over and over here.

There’s, in principle, no limit to how many scores are possible. Baseball contains many implied infinities, and it’s not impossible that a game could end, say, 580 to 578. But it seems likely that after 139 seasons of play there can’t be all that many more scores practically achievable.

Suppose then there are 611 possible baseball score outcomes, and that each of them is equally likely. Then the information-theory content of a score’s outcome is negative one times the logarithm, base two, of 1/611. That’s a number a little bit over nine and a quarter. You could deduce the score for a given game by asking usually nine, sometimes ten, yes-or-no questions from a source that knew the outcome. That’s a little higher than the 8.7 I worked out for football. And it’s a bit less than the 10.8 I estimate for college basketball.

And there’s obvious rubbish there. In no way are all 611 possible outcomes equally likely. “The Most Popular Scores In Baseball History” says that right there in the essay title. The most common outcome was a score of 3-2, with 4-3 barely less popular. Meanwhile it seems only once, on the 28th of June, 1871, has a baseball game ended with a score of 49-33. Some scores are so rare we can ignore them as possibilities.

(You may wonder how incompetent baseball players of the 1870s were that a game could get to 49-33. Not so bad as you imagine. But the equipment and conditions they were playing with were unspeakably bad by modern standards. Notably, the playing field couldn’t be counted on to be flat and level and well-mowed. There would be unexpected divots or irregularities. This makes even simple ground balls hard to field. The baseball, instead of being replaced with every batter, would stay in the game. It would get beaten until it was a little smashed shell of unpredictable dynamics and barely any structural integrity. People were playing without gloves. If a game ran long enough, they would play at dusk, without lights, with a muddy ball on a dusty field. And sometimes you just have four innings that get out of control.)

What’s needed is a guide to what are the common scores and what are the rare scores. And I haven’t found that, nor worked up the energy to make the list myself. But I found some promising partial results. In a September 2008 post on Baseball-Fever.com, user weskelton listed the 24 most common scores and their frequency. This was for games from 1993 to 2008. One might gripe that the list only covers fifteen years. True enough, but if the years are representative that’s fine. And the top scores for the fifteen-year survey look to be pretty much the same as the 139-year tally. The 24 most common scores add up to just over sixty percent of all baseball games, which leaves a lot of scores unaccounted for. I am amazed that about three in five games will have a score that’s one of these 24 choices though.

But that’s something. We can calculate the information content for the 25 outcomes, one each of the 24 particular scores and one for “other”. This will under-estimate the information content. That’s because “other” is any of 587 possible outcomes that we’re not distinguishing. But if we have a lower bound and an upper bound, then we’ve learned something about what the number we want can actually be. The upper bound is that 9.25, above.

The information content, the entropy, we calculate from the probability of each outcome. We don’t know what that is. Not really. But we can suppose that the frequency of each outcome is close to its probability. If there’ve been a lot of games played, then the frequency of a score and the probability of a score should be close. At least they’ll be close if games are independent, if the score of one game doesn’t affect another’s. I think that’s close to true. (Some games at the end of pennant races might affect each other: why try so hard to score if you’re already out for the year? But there’s few of them.)

The entropy then we find by calculating, for each outcome, a product. It’s minus one times the probability of that outcome times the base-two logarithm of the probability of that outcome. Then add up all those products. There’s good reasons for doing it this way and in the college-basketball link above I give some rough explanations of what the reasons are. Or you can just trust that I’m not lying or getting things wrong on purpose.

So let’s suppose I have calculated this right, using the 24 distinct outcomes and the one “other” outcome. That makes out the information content of a baseball score’s outcome to be a little over 3.76 bits.

As said, that’s a low estimate. Lumping about two-fifths of all games into the single category “other” drags the entropy down.

But that gives me a range, at least. A baseball game’s score seems to be somewhere between about 3.76 and 9.25 bits of information. I expect that it’s closer to nine bits than it is to four bits, but will have to do a little more work to make the case for it.

A Timeline Of Mathematics Education


https://twitter.com/dannytybrown/status/670174694390239232

As Danny Brown’s tweet above promises, this is an interesting timeline. It’s a “work in progress” presentation by one David Allen that tries to summarize the major changes in the teaching of mathematics in the United States.

It’s a presentation made on Prezi, and it appears to require Flash (and at one point it breaks, at least on my computer, and I have to move around rather than use the forward/backward buttons). And the compilation is cryptic. It reads better as a series of things for further research than anything else. Still, it’s got fascinating data points, such as when algebra became a prerequisite for college, and when it and geometry moved from being college-level mathematics to high school-level mathematics.

Reading the Comics, November 1, 2015: Uncertainty and TV Schedules Edition


Brian Fies’s Mom’s Cancer is a heartbreaking story. It’s compelling reading, but people who are emotionally raw from lost love ones, or who know they’re particularly sensitive to such stories, should consider before reading that the comic is about exactly what the title says.

But it belongs here because in the October 29th and the November 2nd installments are about a curiosity of area, and volume, and hypervolume, and more. That is that our perception of how big a thing is tends to be governed by one dimension, the length or the diameter of the thing. But its area is the square of that, its volume the cube of that, its hypervolume some higher power yet of that. So very slight changes in the diameter produce great changes in the volume. Conversely, though, great changes in volume will look like only slight changes. This can hurt.

Tom Toles’s Randolph Itch, 2 am from the 29th of October is a Roman numerals joke. I include it as comic relief. The clock face in the strip does depict 4 as IV. That’s eccentric but not unknown for clock faces; IIII seems to be more common. There’s not a clear reason why this should be. The explanation I find most nearly convincing is an aesthetic one. Roman numerals are flexible things, and can be arranged for artistic virtue in ways that Arabic numerals make impossible.

The aesthetic argument is that the four-character symbol IIII takes up nearly as much horizontal space as the VIII opposite it. The two-character IV would look distractingly skinny. Now, none of the symbols takes up exactly the same space as their counterpart. X is shorter than II, VII longer than V. But IV-versus-VIII does seem like the biggest discrepancy. Still, Toles’s art shows it wouldn’t look all that weird. And he had to conserve line strokes, so that the clock would read cleanly in newsprint. I imagine he also wanted to avoid using different representations of “4” so close together.

Jon Rosenberg’s Scenes From A Multiverse for the 29th of October is a riff on both quantum mechanics — Schödinger’s Cat in a box — and the uncertainty principle. The uncertainty principle can be expressed as a fascinating mathematical construct. It starts with Ψ, a probability function that has spacetime as its domain, and the complex-valued numbers as its range. By applying a function to this function we can derive yet another function. This function-of-a-function we call an operator, because we’re saying “function” so much it’s starting to sound funny. But this new function, the one we get by applying an operator to Ψ, tells us the probability that the thing described is in this place versus that place. Or that it has this speed rather than that speed. Or this angular momentum — the tendency to keep spinning — versus that angular momentum. And so on.

If we apply an operator — let me call it A — to the function Ψ, we get a new function. What happens if we apply another operator — let me call it B — to this new function? Well, we get a second new function. It’s much the way if we take a number, and multiply it by another number, and then multiply it again by yet another number. Of course we get a new number out of it. What would you expect? This operators-on-functions things looks and acts in many ways like multiplication. We even use symbols that look like multiplication: AΨ is operator A applied to function Ψ, and BAΨ is operator B applied to the function AΨ.

Now here is the thing we don’t expect. What if we applied operator B to Ψ first, and then operator A to the product? That is, what if we worked out ABΨ? If this was ordinary multiplication, then, nothing all that interesting. Changing the order of the real numbers we multiply together doesn’t change what the product is.

Operators are stranger creatures than real numbers are. It can be that BAΨ is not the same function as ABΨ. We say this means the operators A and B do not commute. But it can be that BAΨ is exactly the same function as ABΨ. When this happens we say that A and B do commute.

Whether they do or they don’t commute depends on the operators. When we know what the operators are we can say whether they commute. We don’t have to try them out on some functions and see what happens, although that sometimes is the easiest way to double-check your work. And here is where we get the uncertainty principle from.

The operator that lets us learn the probability of particles’ positions does not commute with the operator that lets us learn the probability of particles’ momentums. We get different answers if we measure a particle’s position and then its velocity than we do if we measure its velocity and then its position. (Velocity is not the same thing as momentum. But they are related. There’s nothing you can say about momentum in this context that you can’t say about velocity.)

The uncertainty principle is a great source for humor, and for science fiction. It seems to allow for all kinds of magic. Its reality is no less amazing, though. For example, it implies that it is impossible for an electron to spiral down into the nucleus of an atom, collapsing atoms the way satellites eventually fall to Earth. Matter can exist, in ways that let us have solid objects and chemistry and biology. This is at least as good as a cat being perhaps boxed.

Jan Eliot’s Stone Soup Classics for the 29th of October is a rerun from 1995. (The strip itself has gone to Sunday-only publication.) It’s a joke about how arithmetic is easy when you have the proper motivation. In 1995 that would include catching TV shows at a particular time. You see, in 1995 it was possible to record and watch TV shows when you wanted, but it required coordinating multiple pieces of electronics. It would often be easier to just watch when the show actually aired. Today we have it much better. You can watch anything you want anytime you want, using any piece of consumer electronics you have within reach, including several current models of microwave ovens and programmable thermostats. This does, sadly, remove one motivation for doing arithmetic. Also, I’m not certain the kids’ TV schedule is actually consistent with what was on TV in 1995.

Oh, heck, why not. Obviously we’re 14 minutes before the hour. Let me move onto the hour for convenience. It’s 744 minutes to the morning cartoons; that’s 12.4 hours. Taking the morning cartoons to start at 8 am, that means it’s currently 14 minutes before 24 minutes before 8 pm. I suspect a rounding error. Let me say they’re coming up on 8 pm. 194 minutes to Jeopardy implies the game show is on at 11 pm. 254 minutes to The Simpsons puts that on at midnight, which is probably true today, though I don’t think it was so in 1995 just yet. 284 minutes to Grace puts that on at 12:30 am.

I suspect that Eliot wanted it to be 978 minutes to the morning cartoons, which would bump Oprah to 4:00, Jeopardy to 7:00, Simpsons and Grace to 8:00 and 8:30, and still let the cartoons begin at 8 am. Or perhaps the kids aren’t that great at arithmetic yet.

Stephen Beals’s Adult Children for the 30th of October tries to build a “math error” out of repeated use of the phrase “I couldn’t care less”. The argument is that the thing one cares least about is unique. But why can’t there be two equally least-cared-about things?

We can consider caring about things as an optimization problem. Optimization problems are about finding the most of something given some constraints. If you want the least of something, multiply the thing you have by minus one and look for the most of that. You may giggle at this. But it’s the sensible thing to do. And many things can be equally high, or low. Take a bundt cake pan, and drizzle a little water in it. The water separates into many small, elliptic puddles. If the cake pan were perfectly formed, and set on a perfectly level counter, then the bottom of each puddle would be at the same minimum height. I grant a real cake pan is not perfect; neither is any counter. But you can imagine such.

Just because you can imagine it, though, must it exist? Think of the “smallest positive number”. The idea is simple. Positive numbers are a set of numbers. Surely there’s some smallest number. Yet there isn’t; name any positive number and we can name a smaller number. Divide it by two, for example. Zero is smaller than any positive number, but it’s not itself a positive number. A minimum might not exist, at least not within the confines where we are to look. It could be there is not something one could not care less about.

So a minimum might or might not exist, and it might or might not be unique. This is why optimization problems are exciting, challenging things.

A bedbug declares that 'according to our quantum mechanical computations, our entire observable universe is almost certainly Fred Wardle's bed.'
Niklas Eriksson’s Carpe Diem for the 1st of November, 2015. I’m not sure how accurately the art depicts bedbugs, although I’m also not sure how accurately Eriksson should.

Niklas Eriksson’s Carpe Diem for the 1st of November is about understanding the universe by way of observation and calculation. We do rely on mathematics to tell us things about the universe. Immanuel Kant has a bit of reputation in mathematical physics circles for this observation. (I admit I’ve never seen the original text where Kant observed this, so I may be passing on an urban legend. My love has several thousands of pages of Kant’s writing, but I do not know if any of them touch on natural philosophy.) If all we knew about space was that gravitation falls off as the square of the distance between two things, though, we could infer that space must have three dimensions. Otherwise that relationship would not make geometric sense.

Jeff Harris’s kids-information feature Shortcuts for the 1st of November was about the Harvard Computers. By this we mean the people who did the hard work of numerical computation, back in the days before this could be done by electrical and then electronic computer. Mathematicians relied on people who could do arithmetic in those days. There is the folkloric belief that mathematicians are inherently terrible at arithmetic. (I suspect the truth is people assume mathematicians must be better at arithmetic than they really are.) But here, there’s the mathematics of thinking what needs to be calculated, and there’s the mathematics of doing the calculations.

Their existence tends to be mentioned as a rare bit of human interest in numerical mathematics books, usually in the preface in which the author speaks with amazement of how people who did computing were once called computers. I wonder if books about font and graphic design mention how people who typed used to be called typewriters in their prefaces.

Who Was Jonas Moore?


I imagine I’m not the only person to have not realized the anniversary of Jonas Moore’s death was upon us again. Granted he’s not in anyone’s short list of figures from mathematical history. The easiest thing to say about him is that he appears to have coined common shorthands for the trigonometric functions: cot for cotangent, that sort of thing. Perhaps nothing exciting, but it’s something that had to be done.

Moore’s more interesting than that. The Renaissance Mathematicus has a biographic essay. Particularly of interest is that Moore oversaw the building of the Royal Observatory in Greenwich, and paid for the first instruments put into it. And, with Samuel Pepys, he founded the Royal Mathematical School at Christ’s Hospital, to train men in scientific navigation. As such he’s got a place in the story of longitude, and time-keeping, and our understanding of how to measure things.

That won’t put him onto your short list of important figures in the history of mathematics and science. But it’s interesting anyway.

Reading the Comics, June 20, 2015: Blatantly Padded Edition, Part 1


I confess. I’m padding my post count with the end-of-the-week roundup of mathematically-themed comic strips. While what I’ve got is a little long for a single post it’s not outrageously long. But I realized that if I split this into two pieces then, given how busy last week was around here, and how I have an A To Z post ready for Monday already, I could put together a string of eight days of posting. And that would look so wonderful in the “fireworks display” of posts that WordPress puts together for its annual statistics report. Please don’t think worse of me for it.

John Graziano’s Ripley’s Believe It or Not (June 17) presents the trivia point that Harvard University is older than calculus. That’s fair enough to say, although I don’t think it merits Graziano’s exclamation point. A proper historical discussion of when calculus was invented has to be qualified. It’s a big, fascinating invention; such things don’t have unambiguous origin dates. You can see what are in retrospect obviously the essential ideas of calculus in historical threads weaving through thousands of years and every mathematically-advanced culture. But calculus as we know it, the set of things that you will see in an Introduction To Calculus textbook, got organized into a coherent set of ides that we call that, now, in the late 17th century. Most of its notation took shape by the mid-18th century, especially as Leonhard Euler promoted many of the symbols and much of the notation that we still use today.

John Graziano’s Ripley’s Believe It or Not is still a weird attribution even if I can’t think of a better one.

Hector D Cantu and Carlos Castellanos’s Baldo (June 18) reminds us that all you really need to do mathematics well is have a problem which you’re interested in. But what isn’t that true of?

J C Duffy’s The Fusco Brothers (June 18) is about the confusion between what positive and negative mean in test screenings. I’ve written about this before. The use of positive for what is typically bad news, and negative for what is typically good news, seems to trace to statistical studies. The test amounts to an experiment. We measure something in a complicated system, like a body. Is that measurement consistent with what we might normally expect, or is it so far away from normal that it’s implausible that it might be just chance? The “positive” then reflects finding that whatever is measured is unlikely to be that far from normal just by chance.

Larry Wright’s Motley (June 18, rerun from June 18, 1987) uses a bit of science and mathematics as a signifier of intelligence. In the context of a game show, though, “23686 π” is an implausible answer. Unless the question was “what’s the area of a circle with radius 23683?” there’s just no way 2368 would even come up. I suspect “hydromononucleatic acid” isn’t at thing either.

Mark Parisi’s Off The Mark (June 18) is this week’s anthropomorphic numerals joke.

Arnold is sparring with a kangaroo. Told to outsmart him, Arnold recites the Pythagorean Theorem. He gets clobbered.
Bud Grace’s The Piranha Club for the 19th of June, 2015.

Bud Grace’s The Piranha Club (June 19) is another strip to use mathematics as a signifier of intelligence. And, hey, guy punched by a kangaroo, what’s not to like? (In the June 20 strip the kangaroo’s joey emerges from a pouch and punches him too, so I suppose the kangaroo’s female, never mind what the 19th says.)

Reading the Comics, April 20, 2015: History of Mathematics Edition


This is a bit of a broad claim, but it seems Comic Strip Master Command was thinking of all mathematics one lead-time ago. There’s a comic about the original invention of mathematics, and another showing off 20th century physics equations. This seems as much of the history of mathematics as one could reasonably expect from the comics page.

Mark Anderson’s Andertoons gets its traditional appearance around here with the April 17th strip. It features a bit of arithmetic that is indeed lovely but wrong.

Continue reading “Reading the Comics, April 20, 2015: History of Mathematics Edition”

Roller Coaster Immortality Update!


Several years ago I had the chance to go to Lakemont Park, in Altoona, Pennsylvania. It’s a lovely and very old amusement park, featuring the oldest operating roller coaster, Leap The Dips. As roller coasters go it’s not very large and not very fast, but it’s a great ride. It does literally and without exaggeration leap off the track, though not far enough to be dangerous. I recommend the park and the ride to people who have cause to be in the middle of Pennsylvania.

I wondered whether any boards in it might date from the original construction in 1902 by the E Joy Morris company. If we make some assumptions we can turn this into a probability problem. It’s a problem of a type that always seems to be answered 1/e. (The problem is “what is the probability that any particular piece of wood has lasted 100 years, if a piece of wood has a one percent chance of needing replacement every year?”) That’s a probability of about 37 percent. But I doubted this answer meant anything. My skepticism came from wondering why every piece of wood should be equally likely to survive every year. Different pieces serve different structural roles, and will be exposed to the elements differently. How can I be sure that the probability one piece needs replacement is independent of the probability some other piece needs replacement? But if they’re not independent then my calculation doesn’t give a relevant answer.

The Leap-The-Dips roller coaster at Lakemont Park, Altoona, Pennsylvania.
The Leap-The-Dips roller coaster at Lakemont Park, Altoona, Pennsylvania.

A recent post on the Usenet roller coaster enthusiast newsgroup rec.roller-coaster, in a discussion titled “Age a coaster should be preserved”, suggests I was right in my skepticism. Derek Gee writes:

According to the video documentary the park produced around
1999, all of the original upright lumber was found to be in excellent shape.
The E. Joy Morris company had waterproofed it by sealing it in ten coats of
paint and it was old-growth hardwood. All the horizontal lumber was
replaced as I recall.

I am aware this is not an academically rigorous answer to the question of how much of the roller coaster’s original construction is still in place. But it is a lead. It suggests that quite a bit of the antique ride is as antique as could be.

A bit more about Thomas Hobbes


You might remember a post from last April, Thomas Hobbes and the Doing of Important Mathematics, timed to the renowned philosopher’s birthday. I talked about him because a good bit of his intellectual life was spent trying to achieve mathematical greatness, which he never did.

Recently I’ve had the chance to read Douglas M Jesseph’s Squaring The Circle: The War Between Hobbes And Wallis, about Hobbes’s attempts to re-build mathematics on an intellectual foundation he found more satisfying, and the conflict this put him in with mainstream mathematicians, particularly John Wallis (algebra and calculus pioneer, and popularizer of the ∞ symbol). The situation of Hobbes’s mathematical ambitions is more complicated than I realized, although the one thing history teaches us is that the situation is always more complicated than we realized, and I wanted to at least make my writings about Hobbes a bit less incomplete. Jesseph’s book can’t be fairly reduced to a blog post, of course, and I’d recommend it to people who want to really understand what the fuss was all about. It’s a very good idea to have some background in philosophy and in 17th century English history going in, though, because it turns out a lot of the struggle — and particularly the bitterness with which Hobbes and Wallis fought, for decades — ties into the religious and political struggles of England of the 1600s.

Hobbes’s project, I better understand now, was not merely the squaring of the circle or the solving of other ancient geometric problems like the doubling of the cube or the trisecting of an arbitrary angle, although he did claim to have various proofs or approximate proofs of them. He seems to have been interested in building a geometry on more materialist grounds, more directly as models of the real world, instead of the pure abstractions that held sway then (and, for that matter, now). This is not by itself a ridiculous thing to do: we are almost always better off for having multiple independent ways to construct something, because the differences in those ways teaches us not just about the thing, but about the methods we use to discover things. And purely abstract constructions have problems also: for example, if a line can be decomposed into nothing but an enormous number of points, and absolutely none of those points has any length, then how can the line have length? You can answer that, but it’s going to require a pretty long running start.

Trying to re-build the logical foundations of mathematics is an enormously difficult thing to do, and it’s not surprising that someone might fail to do so perfectly. Whole schools of mathematicians might be needed just to achieve mixed success. And Hobbes wasn’t able to attract whole schools of mathematicians, in good part because of who he was.

Hobbes achieved immortality as an important philosopher with the publication of Leviathan. What I had not appreciated and Jesseph made clear was that in the context of England of the 1650s, Hobbes’s views on the natures of God, King, Society, Law, and Authority managed to offend — in the “I do not know how I can continue to speak with a person who holds views like that” — pretty much everybody in England who had any strong opinion about anything in politics, philosophy, or religion. I do not know for a fact that Hobbes then went around kicking the pet dogs of any English folk who didn’t have strong opinions about politics, philosophy, or religion, but I can’t rule it out. At least part of the relentlessness and bitterness with which Wallis (and his supporters) attacked Hobbes, and with which Hobbes (and his supporters) attacked back, can be viewed as a spinoff of the great struggle between the Crown and Parliament that produced the Civil War, the Commonwealth, and the Restoration, and in that context it’s easier to understand why all parties carried on, often quibbling about extremely minor points, well past the point that their friends were advising them that the quibbling was making themselves look bad. Hobbes was a difficult person to side with, even when he was right, and a lot of his mathematics just wasn’t right. Some of it I’m not sure ever could be made right, however many ingenious people you had working to avoid flaws.

An amusing little point that Jesseph quotes is a bit in which Hobbes, making an argument about the rights that authority has, asserts that if the King decreed that Euclid’s Fifth Postulate should be taught as false, then false it would be in the kingdom. The Fifth Postulate, also known as the Parallel Postulate, is one of the axioms on which classical Greek geometry was built and it was always the piece that people didn’t like. The other postulates are all nice, simple, uncontroversial, common-sense things like “all right angles are equal”, the kinds of things so obvious they just have to be axioms. The Fifth Postulate is this complicated-sounding thing about how, if a line is crossed by two non-parallel lines, you can determine on which side of the first line the non-parallel lines will meet.

It wouldn’t be really understood or accepted for another two centuries, but, you can suppose the Fifth Postulate to be false. This gives you things named “non-Euclidean geometries”, and the modern understanding of the universe’s geometry is non-Euclidean. In picking out an example of something a King might decree and the people would have to follow regardless of what was really true, Hobbes picked out an example of something that could be decreed false, and that people could follow profitably.

That’s not mere ironical luck, probably. A streak of mathematicians spent a long time trying to prove the Fifth Postulate was unnecessary, at least, by showing it followed from the remaining and non-controversial postulates, or at least that it could be replaced with something that felt more axiomatic. Of course, in principle you can use any set of axioms you like to work, but some sets produce more interesting results than others. I don’t know of any interesting geometry which results from supposing “not all right angles are equal”; supposing that the Fifth Postule is untrue gives us general relativity, which is quite nice to have.

Again I have to warn that Jesseph’s book is not always easy reading. I had to struggle particularly over some of the philosophical points being made, because I’ve got only a lay understanding of the history of philosophy, and I was able to call on my love (a professional philosopher) for help at points. I imagine someone well-versed in philosophy but inexperienced with mathematics would have a similar problem (although — don’t let the secret out — you’re allowed to just skim over the diagrams and proofs and go on to the explanatory text afterwards). But for people who want to understand the scope and meaning of the fighting better, or who just want to read long excerpts of the wonderful academic insulting that was current in the era, I do recommend it. Check your local college or university library.

Reading the Comics, November 20, 2014: Ancient Events Edition


I’ve got enough mathematics comics for another roundup, and this time, the subjects give me reason to dip into ancient days: one to the most famous, among mathematicians and astronomers anyway, of Greek shipwrecks, and another to some point in the midst of winter nearly seven thousand years ago.

Eric the Circle (November 15) returns “Griffinetsabine” to the writer’s role and gives another “Shape Single’s Bar” scene. I’m amused by Eric appearing with his ex: x is practically the icon denoting “this is an algebraic expression”, while geometry … well, circles are good for denoting that, although I suspect that triangles or maybe parallelograms are the ways to denote “this is a geometric expression”. Maybe it’s the little symbol for a right angle.

Jim Meddick’s Monty (November 17) presents Monty trying to work out just how many days there are to Christmas. This is a problem fraught with difficulties, starting with the obvious: does “today” count as a shopping day until Christmas? That is, if it were the 24th, would you say there are zero or one shopping days left? Also, is there even a difference between a “shopping day” and a “day” anymore now that nobody shops downtown so it’s only the stores nobody cares about that close on Sundays? Sort all that out and there’s the perpetual problem in working out intervals between dates on the Gregorian calendar, which is that you have to be daft to try working out intervals between dates on the Gregorian calendar. The only worse thing is trying to work out the intervals between Easters on it. My own habit for this kind of problem is to use the United States Navy’s Julian Date conversion page. The Julian date is a straight serial number, counting the number of days that have elapsed since noon Universal Time at what’s called the 1st of January, 4713 BCE, on the proleptic Julian calendar (“proleptic” because nobody around at the time was using, or even imagined, the calendar, but we can project back to what date that would have been), a year picked because it’s the start of several astronomical cycles, and it’s way before any specific recordable dates in human history, so any day you might have to particularly deal with has a positive number. Of course, to do this, we’re transforming the problem of “counting the number of days between two dates” to “counting the number of days between a date and January 1, 4713 BCE, twice”, but the advantage of that is, the United States Navy (and other people) have worked out how to do that and we can use their work.

Bill Hind’s kids-sports comic Cleats (November 19, rerun) presents Michael offering basketball advice that verges into logic and set theory problems: making the ball not go to a place outside the net is equivalent to making the ball go inside the net (if we decide that the edge of the net counts as either inside or outside the net, at least), and depending on the problem we want to solve, it might be more convenient to think about putting the ball into the net, or not putting the ball outside the net. We see this, in logic, in a set of relations called De Morgan’s Laws (named for Augustus De Morgan, who put these ideas in modern mathematical form), which describe what kinds of descriptions — “something is outside both sets A and B at one” or “something is not inside set A or set B”, or so on — represent the same relationship between the thing and the sets.

Tom Thaves’s Frank and Ernest (November 19) is set in the classic caveman era, with prehistoric Frank and Ernest and someone else discovering mathematics and working out whether a negative number times a negative number might be positive. It’s not obvious right away that they should, as you realize when you try teaching someone the multiplication rules including negative numbers, and it’s worth pointing out, a negative times a negative equals a positive because that’s the way we, the users of mathematics, have chosen to define negative numbers and multiplication. We could, in principle, have decided that a negative times a negative should give us a negative number. This would be a different “multiplication” (or a different “negative”) than we use, but as long as we had logically self-consistent rules we could do that. We don’t, because it turns out negative-times-negative-is-positive is convenient for problems we like to do. Mathematics may be universal — something following the same rules we do has to get the same results we do — but it’s also something of a construct, and the multiplication of negative numbers is a signal of that.

Goofy sees the message 'buried treasure in back yard' in his alphabet soup; what are the odds of that?
The Mickey Mouse comic rerun the 20th of November, 2014.

Mickey Mouse (November 20, rerun) — I don’t know who wrote or draw this, but Walt Disney’s name was plastered onto it — sees messages appearing in alphabet soup. In one sense, such messages are inevitable: jumble and swirl letters around and eventually, surely, any message there are enough letters for will appear. This is very similar to the problem of infinite monkeys at typewriters, although with the special constraint that if, say, the bowl has only two letters “L”, it’s impossible to get the word “parallel”, unless one of the I’s is doing an impersonation. Here, Goofy has the message “buried treasure in back yard” appear in his soup; assuming those are all the letters in his soup then there’s something like 44,881,973,505,008,615,424 different arrangements of letters that could come up. There are several legitimate messages you could make out of that (“treasure buried in back yard”, “in back yard buried treasure”), not to mention shorter messages that don’t use all those letters (“run back”), but I think it’s safe to say the number of possible sentences that make sense are pretty few and it’s remarkable to get something like that. Maybe the cook was trying to tell Goofy something after all.

Mark Anderson’s Andertoons (November 20) is a cute gag about the dangers of having too many axes on your plot.

Gary Delainey and Gerry Rasmussen’s Betty (November 20) mentions the Antikythera Mechanism, one of the most famous analog computers out there, and that’s close enough to pure mathematics for me to feel comfortable including it here. The machine was found in April 1900, in ancient shipwreck, and at first seemed to be just a strange lump of bronze and wood. By 1902 the archeologist Valerios Stais noticed a gear in the mechanism, but since it was believed the wreck far, far predated any gear mechanisms, the machine languished in that strange obscurity that a thing which can’t be explained sometimes suffers. The mechanism appears to be designed to be an astronomical computer, tracking the positions of the Sun and the Moon — tracking the actual moon rather than an approximate mean lunar motion — the rising and etting of some constellations, solar eclipses, several astronomical cycles, and even the Olympic Games. It’s an astounding mechanism, it’s mysterious: who made it? How? Are there others? What happened to them? How was the mechanical engineering needed for this developed, and what other projects did the people who created this also do? Any answers to these questions, if we ever know them, seem sure to be at least as amazing as the questions are.

Some Stuff About Edmond Halley


When I saw the Maths History tweet about Edmond Halley’s birthday I wondered if the November 8th date given was the relevant one since, after all, in 1656 England was still on the Julian calendar. The MacTutor biography of him makes clear that the 8th of November is his Gregorian-date birthday, and he was born on the 29th of October by the calendar his parents were using, although it’s apparently not clear he was actually born in 1656. Halley claimed it was 1656, at least, and he probably heard from people who knew.

Halley is famous for working out the orbit of the comet that’s gotten his name attached, and correctly so: working out the orbits of comets was one of the first great accomplishments of Newtonian mechanics, and Halley’s work took into account how Jupiter’s gravitation distorts the orbit of a comet. It’s great work. And he’s also famous within mathematical and physics circles because it’s fair to wonder whether, without his nagging and his financial support, Isaac Newton would have published his Principia Mathematica. Astronomers note him as the first Western European astronomer to set up shop in the southern hemisphere and produce a map of that part of the sky, as well.

That hardly exhausts what’s interesting about him: for example, he joined in the late-17th-century fad for diving bell companies (for a while, you couldn’t lose money excavating wrecked ships, until finally everyone did) and even explored the bed of the English Channel in a diving bell of his own design. This is to me the most terrifying thing he did, and that’s even with my awareness he led two scientific sailing expeditions, one of which was cut short after among other things irreconcilable differences with the ship’s other commissioned officer, Lieutenant Edward Harrison (who blamed Halley for the oblivion which Harrison’s book on longitude received), and the second of which included a pause in Recife when Halley was put under guard by a man claiming to be the English consul, and who was actually an agent of the Royal African Company considering whether to seize Halley’s ship[1] as a prize.

After his second expedition Halley published charts showing the magnetic declination, how far a magnetic compass’s “north” is from true north, and introduced one of those great conceptual breakthroughs that charts can give us: he connected the lines showing the points where the declination was equal. These isolines are a magnificent way to diagram three-dimensional information on a two-dimensional chart; we see them in topographic maps, as the contour curves showing where a hill rises or a valley sinks. We see them in weather maps, the lines where the temperature is 70 or 80 Fahrenheit (or 20 or 25 Celsius, if you rather) or where the wind speed is some sufficiently alarming figure. We see them (in three-dimensional form) in medical imaging, where a region of constant density gets the same color and this is used to understand a complicated shape within. Not all these uses derive directly from Halley; as with all really good, widely usable concepts many people discovered the concept, but Halley was among the first to put them to obvious, prominent use.

And something that might serve as comfort to anyone who’s taking a birthday hard: at age 65, Halley began a study of the moon’s saros, the cycle patterns of different relative positions the Sun and Moon have in the sky which describe when eclipses happen. One cycle takes a bit over eighteen years to complete. Halley lived long enough to complete this work.

[1] The Paramore, which — I note because this is just the kind of world it was back then — was constructed in 1694 at the Royal Dockyard at Deptford on the River Thames for a scientific circumnavigation of the globe, and first sailed in April 1698 under Tsar Peter the Great, then busy travelling western Europe under ineffective cover to learn things which might modernize Russia. Halley had hoped to sail in 1696, but he was waylaid by his appointment to the Mint at Chester, courtesy of Newton.

Reading the Comics, October 7, 2014: Repeated Comics Edition


Since my last roundup of mathematics-themed comic strips there’s been a modest drizzle of new ones, and I’m not sure that I can find any particular themes to them, except that Zach Weinersmith and the artistic collective behind Eric the Circle apparently like my attention. Well, what the heck; that’s easy enough to give.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 29) hopes to be that guy who appears somewhere around the fourth comment of every news article ever that mentions a correlation being found between two quantities. A lot of what’s valuable about science is finding causal links between things, but it’s only in rare and, often, rather artificial circumstances that such links are easy to show. What’s more often necessary is showing that as one quantity changes so does another, which allows one to suspect a link. Then, typically, one would look for a plausible reason they might have anything to do with one another, and look for ways to experiment and prove whether there is or is not.

But just because there is a correlation doesn’t by itself mean that one thing necessarily has anything to do with another. They could be coincidence, for example, or they could be influenced by some other confounding factor. To be worth mention in a decent journal, a correlation is probably going to be strong enough that it’s hard to believe it’s just coincidence, but there could yet be some confounding factor. And even if there is a causal link, in the complicated mess that is reality it can be difficult to discern which way the link flows. This is summarized in deductive logic by saying that correlation does not imply causation, but that uses deductive logic’s definition of “imply”.

In deductive logic to say “this implies that” means it is impossible for “this” to be true and “that” false simultaneously. It is perfectly permissible for both “this” and “that” to be true, and permissible for “this” to be false and “that” false, and — this is the point where Intro to Logic students typically crash — permissible for “this” to be false and “that” true. Colloquially, though, “imply” has a different connotation, something more along the lines of “this” and “that” have to both be false or both be true together. Don’t make that mistake on your logic test.

When a logician says that correlation does not imply causation, she is saying that it is imaginable for the correlation to be true while the causation is false. She is not saying the causation is false; she is just saying that the case is not proved from the fact of a correlation being true. And that’s so; if we just knew two things were correlated we would have to experiment to find whether there is a causal link. But finding a correlation one of the ways to start finding casual links; it’d be obviously daft not to use them as the start of one’s search. Anyway, that guy in about the fourth comment of every news report about a correlation just wants you to know it’s very important he tell you he’s smarter than journalists.

Saturday Morning Breakfast Cereal pops back up again (October 1) with an easier-to-describe joke about August Ferdinand Möbius and his rather famous strip, here applied to the old gag about walking to school uphill both ways. One hates to be a spoilsport, but Möbius was educated at home until 13, so this comic is not reliable as a shorthand biography of the renowned mathematician.

Eric the Circle has had a couple strips by “Griffinetsabine”, one on October 3, and another on the 7th of October, based on the Shape Singles Bar. Both strips are jokes about two points connecting by a line, suggesting that Griffinetsabine knew the premise was good for a couple of variants. I’d have spaced out the publication of them farther but perhaps this was the best that could be done.

Mikael Wulff and Anders Morgenthaler’s Truth Facts (September 30) — a panel strip that’s often engaging in showing comic charts — gives a guide to what the number of digits you’ve memorized says about you. (For what it’s worth, I peter out at “897932”.) I’m mildly delighted to find that their marker for Isaac Newton is more or less correct: Newton did work out pi to fifteen decimal places, by using his binomial theorem and a calculation of the area within a particular wedge of the circle. (As I make it out Wulff and Morgenthaler put Newton at fourteen decimal points, but they might have read references to Newton working out “fifteen decimal points” as meaning something different to what I do.) Newton’s was not the best calculation of pi in the 1660s when he worked it out — Christoph Grienberger, an Austrian Jesuit astronomer, had calculated 38 decimal places a generation earlier — but I can’t blame Wulff and Morgenthaler for supposing Newton to be a more recognizable name than Grienberger. I imagine if Einstein or Stephen Hawking had done any particularly unique work in calculating the digits of pi they’d have appeared on the chart too.

John Graziano’s Ripley’s Believe It or Not (October 1) — and don’t tell me that attribution doesn’t look weird — shares a story about the followers of the Ancient Greek mathematician, philosopher, and mystic Pythagoras, that they were forbidden to wear wool, eat beans, or pick up things they had dropped. I have heard the beans thing before and I think I’ve heard the wool prohibition before, but I don’t remember hearing about them not being able to pick up things before.

I’m not sure I can believe it, though: Pythagoras was a strange fellow, so far as the historical record is clear. It’s hard to be sure just what is true about him and his followers, though, and what is made up, either out of devoted followers building up the figure they admire or out of critics making fun of a strange fellow with his own little cult. Perhaps it’s so, perhaps it’s not. I would like to see a primary source, and I don’t think any exist.

The Little King skates a figure 8 that requires less tricky curving.
Otto Soglow’s The Little King for 29 February 1948.

Otto Soglow’s The Little King (October 5; originally run February 29, 1948) provides its normal gentle, genial humor in the Little King working his way around the problem of doing a figure 8.

Machines That Give You Logarithms


As I’ve laid out the tools that the Harvard IBM Automatic Sequence-Controlled Calculator would use to work out a common logarithm, now I can show how this computer of the 1940s and 1950s would do it. The goal, remember, is to compute logarithms to a desired accuracy, using computers that haven’t got abundant memory, and as quickly as possible. As quickly as possible means, roughly, avoiding multiplication (which takes time) and doing as few divisions as can possibly be done (divisions take forever).

As a reminder, the tools we have are:

  1. We can work out at least some logarithms ahead of time and look them up as needed.
  2. The natural logarithm of a number close to 1 is log_e\left(1 + h\right) = h - \frac12h^2 + \frac13h^3 - \frac14h^4 + \frac15h^5 - \cdots .
  3. If we know a number’s natural logarithm (base e), then we can get its common logarithm (base 10): multiply the natural logarithm by the common logarithm of e, which is about 0.43429.
  4. Whether the natural or the common logarithm (or any other logarithm you might like) \log\left(a\cdot b\cdot c \cdot d \cdots \right) = \log(a) + \log(b) + \log(c) + \log(d) + \cdots

Now we’ll put this to work. The first step is which logarithms to work out ahead of time. Since we’re dealing with common logarithms, we only need to be able to work out the logarithms for numbers between 1 and 10: the common logarithm of, say, 47.2286 is one plus the logarithm of 4.72286, and the common logarithm of 0.472286 is minus two plus the logarithm of 4.72286. So we’ll start by working out the logarithms of 1, 2, 3, 4, 5, 6, 7, 8, and 9, and storing them in what, in 1944, was still a pretty tiny block of memory. The original computer using this could store 72 numbers at a time, remember, though to 23 decimal digits.

So let’s say we want to know the logarithm of 47.2286. We have to divide this by 10 in order to get the number 4.72286, which is between 1 and 10, so we’ll need to add one to whatever we get for the logarithm of 4.72286 is. (And, yes, we want to avoid doing divisions, but dividing by 10 is a special case. The Automatic Sequence-Controlled Calculator stored numbers, if I am not grossly misunderstanding things, in base ten, and so dividing or multiplying by ten was as fast for it as moving the decimal point is for us. Modern computers, using binary arithmetic, find it as fast to divide or multiply by powers of two, even though division in general is a relatively sluggish thing.)

We haven’t worked out what the logarithm of 4.72286 is. And we don’t have a formula that’s good for that. But: 4.72286 is equal to 4 times 1.1807, and therefore the logarithm of 4.72286 is going to be the logarithm of 4 plus the logarithm of 1.1807. We worked out the logarithm of 4 ahead of time (it’s about 0.60206, if you’re curious).

We can use the infinite series formula to get the natural logarithm of 1.1807 to as many digits as we like. The natural logarithm of 1.1807 will be about 0.1807 - \frac12 0.1807^2 + \frac13 0.1807^3 - \frac14 0.1807^4 + \frac15 0.1807^5 - \cdots or 0.16613. Multiply this by the logarithm of e (about 0.43429) and we have a common logarithm of about 0.07214. (We have an error estimate, too: we’ve got the natural logarithm of 1.1807 within a margin of error of \frac16 0.1807^6 , or about 0.000 0058, which, multiplied by the logarithm of e, corresponds to a margin of error for the common logarithm of about 0.000 0025.

Therefore: the logarithm of 47.2286 is about 1 plus 0.60206 plus 0.07214, which is 1.6742. And it is, too; we’ve done very well at getting the number just right considering how little work we really did.

Although … that infinite series formula. That requires a fair number of multiplications, at least eight as I figure it, however you look at it, and those are sluggish. It also properly speaking requires divisions, although you could easily write your code so that instead of dividing by 4 (say) you multiply by 0.25 instead. For this particular example number of 47.2286 we didn’t need very many terms in the series to get four decimal digits of accuracy, but maybe we got lucky and some other number would have required dozens of multiplications. Can we make this process, on average, faster?

And here’s one way to do it. Besides working out the common logarithms for the whole numbers 1 through 9, also work out the common logarithms for 1.1, 1.2, 1.3, 1.4, et cetera up to 1.9. And then …

We started with 47.2286. Divide by 10 (a free bit of work) and we have 4.72286. Divide 4.72286 is 4 times 1.180715. And 1.180715 is equal to 1.1 — the whole number and the first digit past the decimal — times 1.07337. That is, 47.2286 is 10 times 4 times 1.1 times 1.07337. And so the logarithm of 47.2286 is the logarithm of 10 plus the logarithm of 4 plus the logarithm of 1.1 plus the logarithm of 1.07337. We are almost certainly going to need fewer terms in the infinite series to get the logarithm of 1.07337 than we need for 1.180715 and so, at the cost of one more division, we probably save a good number of multiplications.

The common logarithm of 1.1 is about 0.041393. So the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) is 1.6435, which falls a little short of the actual logarithm we’d wanted, about 1.6742, but two or three terms in the infinite series should be enough to make that up.

Or we could work out a few more common logarithms ahead of time: those for 1.01, 1.02, 1.03, and so on up to Our original 47.2286 divided by 10 is 4.72286. Divide that by the first number, 4, and you get 1.180715. Divide 1.180715 by 1.1, the first two digits, and you get 1.07337. Divide 1.07337 by 1.07, the first three digits, and you get 1.003156. So 47.2286 is 10 times 4 times 1.1 times 1.07 times 1.003156. So the common logarithm of 47.2286 is the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) plus the logarithm of 1.07 (about 0.02938) plus the logarithm of 1.003156 (to be determined). Even ignoring the to-be-determined part that adds up to 1.6728, which is a little short of the 1.6742 we want but is doing pretty good considering we’ve reduced the whole problem to three divisions, looking stuff up, and four additions.

If we go a tiny bit farther, and also have worked out ahead of time the logarithms for 1.001, 1.002, 1.003, and so on out to 1.009, and do the same process all over again, then we get some better accuracy and quite cheaply yet: 47.2286 divided by 10 is 4.72286. 4.72286 divided by 4 is 1.180715. 1.180715 divided by 1.1 is 1.07337. 1.07337 divided by 1.07 is 1.003156. 1.003156 divided by 1.003 is 1.0001558.

So the logarithm of 47.2286 is the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) plus the logarithm of 1.07 (0.029383) plus the logarithm of 1.003 (0.001301) plus the logarithm of 1.001558 (to be determined). Leaving aside the to-be-determined part, that adds up to 1.6741.

And the to-be-determined part is great: if we used just a single term in this series, the margin for error would be, at most, 0.000 000 0052, which is probably small enough for practical purposes. The first term in the to-be-determined part is awfully easy to calculate, too: it’s just 1.0001558 – 1, that is, 0.0001558. Add that and we have an approximate logarithm of 1.6742, which is dead on.

And I do mean dead on: work out more decimal places of the logarithm based on this summation and you get 1.674 205 077 226 78. That’s no more than five billionths away from the correct logarithm for the original 47.2286. And it required doing four divisions, one multiplication, and five additions. It’s difficult to picture getting such good precision with less work.

Of course, that’s done in part by having stockpiled a lot of hard work ahead of time: we need to know the logarithms of 1, 1.1, 1.01, 1.001, and then 2, 1.2, 1.02, 1.002, and so on. That’s 36 numbers altogether and there are many ways to work out logarithms. But people have already done that work, and we can use that work to make the problems we want to do considerably easier.

But there’s the process. Work out ahead of time logarithms for 1, 1.1, 1.01, 1.001, and so on, to whatever the limits of your patience. Then take the number whose logarithm you want and divide (or multiply) by ten until you get your working number into the range of 1 through 10. Divide out the first digit, which will be a whole number from 1 through 9. Divide out the first two digits, which will be something from 1.1 to 1.9. Divide out the first three digits, something from 1.01 to 1.09. Divide out the first four digits, something from 1.001 to 1.009. And so on. Then add up the logarithms of the power of ten you divided or multiplied by with the logarithm of the first divisor and the second divisor and third divisor and fourth divisor, until you run out of divisors. And then — if you haven’t already got the answer as accurately as you need — work out as many terms in the infinite series as you need; probably, it won’t be very many. Add that to your total. And you are, amazingly, done.

Machines That Think About Logarithms


I confess that I picked up Edmund Callis Berkeley’s Giant Brains: Or Machines That Think, originally published 1949, from the library shelf as a source of cheap ironic giggles. After all, what is funnier than an attempt to explain to a popular audience that, wild as it may be to contemplate, electrically-driven machines could “remember” information and follow “programs” of instructions based on different conditions satisfied by that information? There’s a certain amount of that, though not as much as I imagined, and a good amount of descriptions of how the hardware of different partly or fully electrical computing machines of the 1940s worked.

But a good part, and the most interesting part, of the book is about algorithms, the ways to solve complicated problems without demanding too much computing power. This is fun to read because it showcases the ingenuity and creativity required to do useful work. The need for ingenuity will never leave us — we will always want to compute things that are a little beyond our ability — but to see how it’s done for a simple problem is instructive, if for nothing else to learn the kinds of tricks you can do to get the most of your computing resources.

The example that most struck me and which I want to share is from the chapter on the IBM Automatic Sequence-Controlled Calculator, built at Harvard at a cost of “somewhere near 3 or 4 hundred thousand dollars, if we leave out some of the cost of research and development, which would have been done whether or not this particular machine had ever been built”. It started working in April 1944, and wasn’t officially retired until 1959. It could store 72 numbers, each with 23 decimal digits. Like most computers (then and now) it could do addition and subtraction very quickly, in the then-blazing speed of about a third of a second; it could do multiplication tolerably quickly, in about six seconds; and division, rather slowly, in about fifteen seconds.

The process I want to describe is the taking of logarithms, and why logarithms should be interesting to compute takes a little bit of justification, although it’s implicitly there just in how fast calculations get done. Logarithms let one replace the multiplication of numbers with their addition, for a considerable savings in time; better, they let you replace the division of numbers with subtraction. They further let you turn exponentiation and roots into multiplication and division, which is almost always faster to do. Many human senses seem to work on a logarithmic scale, as well: we can tell that one weight is twice as heavy as the other much more reliably than we can tell that one weight is four pounds heavier than the other, or that one light is twice as bright as the other rather than is ten lumens brighter.

What the logarithm of a number is depends on some other, fixed, quantity, known as the base. In principle any positive number will do as base; in practice, these days people mostly only care about base e (which is a little over 2.718), the “natural” logarithm, because it has some nice analytic properties. Back in the day, which includes when this book was written, we also cared about base 10, the “common” logarithm, because we mostly work in base ten. I have heard of people who use base 2, but haven’t seen them myself and must regard them as an urban legend. The other bases are mostly used by people who are writing homework problems for the part of the class dealing with logarithms. To some extent it doesn’t matter what base you use. If you work out the logarithm in one base, you can convert that to the logarithm in another base by a multiplication.

The logarithm of some number in your base is the exponent you have to raise the base to to get your desired number. For example, the logarithm of 100, in base 10, is going to be 2 because 102 is 100, and the logarithm of e1/3 (a touch greater than 1.3956), in base e, is going to be 1/3. To dig deeper in my reserve of in-jokes, the logarithm of 2038, in base 10, is approximately 3.3092, because 103.3092 is just about 2038. The logarithm of e, in base 10, is about 0.4343, and the logarithm of 10, in base e, is about 2.303. Your calculator will verify all that.

All that talk about “approximately” should have given you some hint of the trouble with logarithms. They’re only really easy to compute if you’re looking for whole powers of whatever your base is, and that if your base is 10 or 2 or something else simple like that. If you’re clever and determined you can work out, say, that the logarithm of 2, base 10, has to be close to 0.3. It’s fun to do that, but it’ll involve such reasoning as “two to the tenth power is 1,024, which is very close to ten to the third power, which is 1,000, so therefore the logarithm of two to the tenth power must be about the same as the logarithm of ten to the third power”. That’s clever and fun, but it’s hardly systematic, and it doesn’t get you many digits of accuracy.

So when I pick up this thread I hope to explain one way to produce as many decimal digits of a logarithm as you could want, without asking for too much from your poor Automatic Sequence-Controlled Calculator.

The Geometry of Thermodynamics (Part 1)


I should mention that Peter Mander’s Carnot Cycle blog has a fine entry, “The Geometry of Thermodynamics (Part I)” which admittedly opens with a diagram that looks like the sort of thing you create when you want to present a horrifying science diagram. That’s a bit of flavor.

Mander writes about part of what made J Willard Gibbs probably the greatest theoretical physicist that the United States has yet produced: Gibbs put much of thermodynamics into a logically neat system, the kind we still basically use today, and all the better saw represent it and understand it as a matter of surface geometries. This is an abstract kind of surface — looking at the curve traced out by, say, mapping the energy of a gas against its volume, or its temperature versus its entropy — but if you can accept the idea that we can draw curves representing these quantities then you get to use your understanding how how solid objects (and Gibbs even got made solid objects — James Clerk Maxwell, of Maxwell’s Equations fame, even sculpted some) look and feel.

This is a reblogging of only part one, although as Mander’s on summer holiday you haven’t missed part two.

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Volume One of the Scientific Papers of J. Willard Gibbs, published posthumously in 1906, is devoted to Thermodynamics. Chief among its content is the hugely long and desperately difficult “On the equilibrium of heterogeneous substances (1876, 1878)”, with which Gibbs single-handedly laid the theoretical foundations of chemical thermodynamics.

In contrast to James Clerk Maxwell’s textbook Theory of Heat (1871), which uses no calculus at all and hardly any algebra, preferring geometry as the means of demonstrating relationships between quantities, Gibbs’ magnum opus is stuffed with differential equations. Turning the pages of this calculus-laden work, one could easily be drawn to the conclusion that the writer was not a visual thinker.

But in Gibbs’ case, this is far from the truth.

The first two papers on thermodynamics that Gibbs published, in 1873, were in fact visually-led. Paper I deals with indicator diagrams and their comparative properties, while Paper II

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Reading the Comics, May 26, 2014: Definitions Edition


The most recent bunch of mathematics-themed comics left me feeling stumped for a theme. There’s no reason they have to have one, of course; cartoonists, as far as I know, don’t actually take orders from Comic Strip Master Command regarding what to write about, but often they seem to. Some of them seem to touch on definitions, at least, including of such ideas as the value of a quantity and how long it is between two events. I’ll take that.

Jef Mallet’s Frazz (May 23) does the kid-resisting-the-question sort of joke (not a word problem, for a change of pace), although I admit I didn’t care for the joke. I needed too long to figure out how the meaning of “value” for a variable might be ambiguous. Caulfield kind of has a point about mathematics needing to use precise words, but the process of making a word precise is a great and neglected part of mathematical history. Consider, for example: contemporary (English-language, at least) mathematicians define a prime number to be a counting number (1, 2, 3, et cetera) with exactly two factors. Why exactly two factors, except to rule out 1 as a prime number? But then why rule that 1 can’t be a prime number? As an idea gets used and explored we get a better idea of what’s interesting about it, and what it’s useful for, and can start seeing whether some things should be ruled out as not fitting a concept we want to describe, or be accepted as fitting because the concept is too useful otherwise and there’s no clear way to divide what we want from what we don’t.

I still can’t buy Caulfield’s proposition there, though.

Steve Boreman’s Little Dog Lost (May 25) circles around a bunch of mathematical concepts without quite landing on any of them. The obvious thing is the counting ability of animals: the crow asserts that crows can only count as high as nine, for example, and the animals try to work out ways to deal with the very large number of 2,615. The vulture asserts he’s been waiting for 2,615 days for the Little Dog to cross the road, and wonders how many years that’s been. The first installment of the strip, from the 26th of March, 2007, did indeed feature Vulture waiting for Little Dog to cross the road, although as I make it out there’s 2,617 days between those events.

At a guess, either Boreman was not counting the first and the last days of the interval between March 26, 2007, and May 25, 2014, or maybe he forgot the leap days. Finding how long it is between dates is a couple of kinds of messes, first because it isn’t necessarily clear whether to include the end dates, and second because the Gregorian calendar is a mess of months of varying lengths plus the fun of leap years, which include an exception for century years and an exception to the exception, making it all the harder. My preferred route for finding intervals is to not even try working the time out by myself, and instead converting every date to the Julian date, a simple serial count of the number of dates since noon Universal Time on the 1st of January, 4713 BC, on the Julian calendar. Let the Navy deal with leap days. I have better things to worry about.

Samson’s Dark Side Of The Horse (May 26) sees Horace trying to count sheep to get himself to sleep; different ways of denoting numbers confound him. I’m not sure if it’s known why counting sheep, or any task like that, is useful in getting to sleep. My guess would be that it just falls into the sort of activity that can be done without a natural endpoint and without demanding too much attention to keep one awake, while demanding enough attention that one isn’t thinking about the bank account or the noise inside the walls or the way the car lurches two lanes to the right every time one taps the brake at highway speeds. That’s a guess, though.

Tom Horacek’s Foolish Mortals (May 26) uses the “on a scale of one to ten” standard for something that’s not usually described so vaguely, and I like the way it teases the idea of how to measure things. The “scale of one to ten” is logically flawed, since we have no idea what the units are, how little of something one represents or how much the ten does, or even whether it’s a linear scale — the difference between “two” and “three” is the same as that between “three” and “four”, the way lengths and weight work — or a logarithmic one — the ratio between “two” and “three” equals that between “three” and “four”, the way stellar magnitudes, decibel sound readings, and Richter scale earthquake intensity measure work — or, for that matter, what normal ought to be. And yet there’s something useful in making the assessment, surely because the first step towards usefully quantifying a thing is to make a clumsy and imprecise quantification of it.

Dave Blazek’s Loose Parts (May 26) kind of piles together a couple references so a character can identify himself as a double major in mathematics and theology. Of course, the generic biography for a European mathematician, between about the end of the Western Roman Empire and the Industrial Revolution, is that he (males most often had the chance to do original mathematics) studied mathematics alongside theology and philosophy, and possibly astronomy, although that reflects more how the subjects were seen as rather intertwined, and education wasn’t as specialized and differentiated as it’s now become. (The other generic mathematician would be the shopkeeper or the exchequer, but nobody tells jokes about their mathematics.)

And, finally, Doug Savage’s Savage Chickens (May 28) brings up the famous typing monkeys (here just the one of them), and what really has to be counted as a bit of success for the project.

George Berkeley’s 329th Birthday


The stream of mathematics-trivia tweets brought to my attention that the 12th of March, 1685 [1], was the birthday of George Berkeley, who’d become the Bishop of Cloyne and be an important philosopher, and who’s gotten a bit of mathematical immortality for complaining about calculus. Granted everyone who takes it complains about calculus, but Berkeley had the good sorts of complaints, the ones that force people to think harder and more clearly about what they’re doing.

Berkeley — whose name I’m told by people I consider reliable was pronounced “barkley” — particularly protested the “fluxions” of calculus as it was practiced in the day in his 1734 tract The Analyst: Or A Discourse Addressed To An Infidel Mathematician, which as far as I know nobody I went to grad school with ever read either, so maybe you shouldn’t bother reading what I have to say about them.

Fluxions were meant to represent infinitesimally small quantities, which could be added to or subtracted from a number without changing the number, but which could be divided by one another to produce a meaningful answer. That’s a hard set of properties to quite rationalize — if you can add something to a number without changing the number, you’re adding zero; and if you’re dividing zero by zero you’re not doing division anymore — and yet calculus was doing just that. For example, if you want to find the slope of a curve at a single point on the curve you’d take the x- and y-coordinates of that point, and add an infinitesimally small number to the x-coordinate, and see how much the y-coordinate has to change to still be on the curve, and then divide those changes, which are too small to even be numbers, and get something out of it.

It works, at least if you’re doing the calculations right, and Berkeley supposed that it was the result of multiple logical errors cancelling one another out that they did work; but he termed these fluxions with spectacularly good phrasing “ghosts of departed quantities”, and it would take better than a century to put all his criticisms quite to rest. The result we know as differential calculus.

I should point out that it’s not as if mathematicians playing with their shiny new calculus tools were being irresponsible in using differentials and integrals despite Berkeley’s criticisms. Mathematical concepts work a good deal like inventions, in that it’s not clear what is really good about them until they’re used, and it’s not clear what has to be made better until there’s a body of experience working with them and seeing where the flaws. And Berkeley was hardly being unreasonable for insisting on logical rigor in mathematics.

[1] Berkeley was born in Ireland. I have found it surprisingly hard to get a clear answer about when Ireland switched from the Julian to the Gregorian calendar, so I have no idea whether this birthdate is old style or new style, and for that matter whether the 1685 represents the civil year or the historical year. Perhaps it suffices to say that Berkeley was born sometime around this time of year, a long while ago.