My All 2020 Mathematics A to Z: Tiling

Mr Wu, author of the Singapore Maths Tuition blog, had an interesting suggestion for the letter T: Talent. As in mathematical talent. It’s a fine topic but, in the end, too far beyond my skills. I could share some of the legends about mathematical talent I’ve received. But what that says about the culture of mathematicians is a deeper and more important question.

So I picked my own topic for the week. I do have topics for next week — U — and the week after — V — chosen. But the letters W and X? I’m still open to suggestions. I’m open to creative or wild-card interpretations of the letters. Especially for X and (soon) Z. Thanks for sharing any thoughts you care to.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.


Think of a floor. Imagine you are bored. What do you notice?

What I hope you notice is that it is covered. Perhaps by carpet, or concrete, or something homogeneous like that. Let’s ignore that. My floor is covered in small pieces, repeated. My dining room floor is slats of wood, about three and a half feet long and two inches wide. The slats are offset from the neighbors so there’s a pleasant strong line in one direction and stippled lines in the other. The kitchen is squares, one foot on each side. This is a grid we could plot high school algebra functions on. The bathroom is more elaborate. It has white rectangles about two inches long, tan rectangles about two inches long, and black squares. Each rectangle is perpendicular to ones of the other color, and arranged to bisect those. The black squares fill the gaps where no rectangle would fit.

Move from my house to pure mathematics. It’s easy to turn the floor of a room into abstract mathematics. We start with something to tile. Usually this is the infinite, two-dimensional plane. The thing you get if you have a house and forget the walls. Sometimes we look to tile the hyperbolic plane, a different geometry that we of course represent with a finite circle. (Setting particular rules about how to measure distance makes this equivalent to a funny-shaped plane.) Or the surface of a sphere, or of a torus, or something like that. But if we don’t say otherwise, it’s the plane.

What to cover it with? … Smaller shapes. We have a mathematical tiling if we have a collection of not-overlapping open sets. And if those open sets, plus their boundaries, cover the whole plane. “Cover” here means what “cover” means in English, only using more technical words. These sets — these tiles — can be any shape. We can have as many or as few of them as we like. We can even add markings to the tiles, give them colors or patterns or such, to add variety to the puzzles.

(And if we want, we can do this in other dimensions. There are good “tiling” questions to ask about how to fill a three-dimensional space, or a four-dimensional one, or more.)

Having an unlimited collection of tiles is nice. But mathematicians learn to look for how little we need to do something. Here, we look for the smallest number of distinct shapes. As with tiling an actual floor, we can get all the tiles we need. We can rotate them, too, to any angle. We can flip them over and put the “top” side “down”, something kitchen tiles won’t let us do. Can we reflect them? Use the shape we’d get looking at the mirror image of one? That’s up to whoever’s writing this paper.

What shapes will work? Well, squares, for one. We can prove that by looking at a sheet of graph paper. Rectangles would work too. We can see that by drawing boxes around the squares on our graph paper. Two-by-one blocks, three-by-two blocks, 40-by-1 blocks, these all still cover the paper and we can imagine covering the plane. If we like, we can draw two-by-two squares. Squares made up of smaller squares. Or repeat this: draw two-by-one rectangles, and then group two of these rectangles together to make two-by-two squares.

We can take it on faith that, oh, rectangles π long by e wide would cover the plane too. These can all line up in rows and columns, the way our squares would. Or we can stagger them, like bricks or my dining room’s wood slats are.

How about parallelograms? Those, it turns out, tile exactly as well as rectangles or squares do. Grids or staggered, too. Ah, but how about trapezoids? Surely they won’t tile anything. Not generally, anyway. The slanted sides will, most of the time, only fit in weird winding circle-like paths.

Unless … take two of these trapezoid tiles. We’ll set them down so the parallel sides run horizontally in front of you. Rotate one of them, though, 180 degrees. And try setting them — let’s say so the longer slanted line of both trapezoids meet, edge to edge. These two trapezoids come together. They make a parallelogram, although one with a slash through it. And we can tile parallelograms, whether or not they have a slash.

OK, but if you draw some weird quadrilateral shape, and it’s not anything that has a more specific name than “quadrilateral”? That won’t tile the plane, will it?

It will! In one of those turns that surprises and impresses me every time I run across it again, any quadrilateral can tile the plane. It opens up so many home decorating options, if you get in good with a tile maker.

That’s some good news for quadrilateral tiles. How about other shapes? Triangles, for example? Well, that’s good news too. Take two of any identical triangle you like. Turn one of them around and match sides of the same length. The two triangles, bundled together like that, are a quadrilateral. And we can use any quadrilateral to tile the plane, so we’re done.

How about pentagons? … With pentagons, the easy times stop. It turns out not every pentagon will tile the plane. The pentagon has to be of the right kind to make it fit. If the pentagon is in one of these kinds, it can tile the plane. If not, not. There are fifteen families of tiling known. The most recent family was discovered in 2015. It’s thought that there are no other convex pentagon tilings. I don’t know whether the proof of that is generally accepted in tiling circles. And we can do more tilings if the pentagon doesn’t need to be convex. For example, we can cut any parallelogram into two identical pentagons. So we can make as many pentagons as we want to cover the plane. But we can’t assume any pentagon we like will do it.

Hexagons look promising. First, a regular hexagon tiles the plane, as strategy games know. There are also at least three families of irregular hexagons that we know can tile the plane.

And there the good times end. There are no convex heptagons or octagons or any other shape with more sides that tile the plane.

Not by themselves, anyway. If we have more than one tile shape we can start doing fine things again. Octagons assisted by squares, for example, will tile the plane. I’ve lived places with that tiling. Or something that looks like it. It’s easier to install if you have square tiles with an octagon pattern making up the center, and triangle corners a different color. These squares come together to look like octagons and squares.

And this leads to a fun avenue of tiling. Hao Wang, in the early 60s, proposed a sort of domino-like tiling. You may have seen these in mathematics puzzles, or in toys. Each of these Wang Tiles, or Wang Dominoes, is a square. But the square is cut along the diagonals, into four quadrants. Each quadrant is a right triangle. Each quadrant, each triangle, is one of a finite set of colors. Adjacent triangles can have the same color. You can place down tiles, subject only to the rule that the tile edge has to have the same color on both sides. So a tile with a blue right-quadrant has to have on its right a tile with a blue left-quadrant. A tile with a white upper-quadrant on its top has, above it, a tile with a white lower-quadrant.

In 1961 Wang conjectured that if a finite set of these tiles will tile the plane, then there must be a periodic tiling. That is, if you picked up the plane and slid it a set horizontal and vertical distance, it would all look the same again. This sort of translation is common. All my floors do that. If we ignore things like the bounds of their rooms, or the flaws in their manufacture or installation or where a tile broke in some mishap.

This is not to say you couldn’t arrange them aperiodically. You don’t even need Wang Tiles for that. Get two colors of square tile, a white and a black, and lay them down based on whether the next decimal digit of π is odd or even. No; Wang’s conjecture was that if you had tiles that you could lay down aperiodically, then you could also arrange them to set down periodically. With the black and white squares, lay down alternate colors. That’s easy.

In 1964, Robert Berger proved Wang’s conjecture was false. He found a collection of Wang Tiles that could only tile the plane aperiodically. In 1966 he published this in the Memoirs of the American Mathematical Society. The 1964 proof was for his thesis. 1966 was its general publication. I mention this because while doing research I got irritated at how different sources dated this to 1964, 1966, or sometimes 1961. I want to have this straightened out. It appears Berger had the proof in 1964 and the publication in 1966.

I would like to share details of Berger’s proof, but haven’t got access to the paper. What fascinates me about this is that Berger’s proof used a set of 20,426 different tiles. I assume he did not work this all out with shards of construction paper, but then, how to get 20,426 of anything? With computer time as expensive as it was in 1964? The mystery of how he got all these tiles is worth an essay of its own and regret I can’t write it.

Berger conjectured that a smaller set might do. Quite so. He himself reduced the set to 104 tiles. Donald Knuth in 1968 modified the set down to 92 tiles. In 2015 Emmanuel Jeandel and Michael Rao published a set of 11 tiles, using four colors. And showed by computer search that a smaller set of tiles, or fewer colors, would not force some aperiodic tiling to exist. I do not know whether there might be other sets of 11, four-colored, tiles that work. You can see the set at the top of Wikipedia’s page on Wang Tiles.

These Wang Tiles, all squares, inspired variant questions. Could there be other shapes that only aperiodically tile the plane? What if they don’t have to be squares? Raphael Robinson, in 1971, came up with a tiling using six shapes. The shapes have patterns on them too, usually represented as colored lines. Tiles can be put down only in ways that fit and that make the lines match up.

Among my readers are people who have been waiting, for 1800 words now, for Roger Penrose. It’s now that time. In 1974 Penrose published an aperiodic tiling, one based on pentagons and using a set of six tiles. You’ve never heard of that either, because soon after he found a different set, based on a quadrilateral cut into two shapes. The shapes, as with Wang Tiles or Robinson’s tiling, have rules about what edges may be put against each other. Penrose — and independently Robert Ammann — also developed another set, this based on a pair of rhombuses. These have rules about what edges may tough one another, and have patterns on them which must line up.

The Penrose tiling became, and stayed famous. (Ammann, an amateur, never had much to do with the mathematics community. He died in 1994.) Martin Gardner publicized it, and it leapt out of mathematicians’ hands into the popular culture. At least a bit. That it could give you nice-looking floors must have helped.

To show that the rhombus-based Penrose tiling is aperiodic takes some arguing. But it uses tools already used in this essay. Remember drawing rectangles around several squares? And then drawing squares around several of these rectangles? We can do that with these Penrose-Ammann rhombuses. From the rhombus tiling we can draw bigger rhombuses. Ones which, it turns out, follow the same edge rules that the originals do. So that we can go again, grouping these bigger rhombuses into even-bigger rhombuses. And into even-even-bigger rhombuses. And so on.

What this gets us is this: suppose the rhombus tiling is periodic. Then there’s some finite-distance horizontal-and-vertical move that leaves the pattern unchanged. So, the same finite-distance move has to leave the bigger-rhombus pattern unchanged. And this same finite-distance move has to leave the even-bigger-rhombus pattern unchanged. Also the even-even-bigger pattern unchanged.

Keep bundling rhombuses together. You get eventually-big-enough-rhombuses. Now, think of how far you have to move the tiles to get a repeat pattern. Especially, think how many eventually-big-enough-rhombuses it is. This distance, the move you have to make, is less than one eventually-big-enough rhombus. (If it’s not you aren’t eventually-big-enough yet. Bundle them together again.) And that doesn’t work. Moving one tile over without changing the pattern makes sense. Moving one-half a tile over? That doesn’t. So the eventually-big-enough pattern can’t be periodic, and so, the original pattern can’t be either. This is explained in graphic detail a nice Powerpoint slide set from Professor Alexander F Ritter, A Tour Of Tilings In Thirty Minutes.

It’s possible to do better. In 2010 Joshua E S Socolar and Joan M Taylor published a single tile that can force an aperiodic tiling. As with the Wang Tiles, and Robinson shapes, and the Penrose-Ammann rhombuses, markings are part of it. They have to line up so that the markings — in two colors, in the renditions I’ve seen — make sense. With the Penrose tilings, you can get away from the pattern rules for the edges by replacing them with little notches. The Socolar-Taylor shape can make a similar trade. Here the rules are complex enough that it would need to be a three-dimensional shape, one that looks like the dilithium housing of the warp core. You can see the tile — in colored, marked form, and also in three-dimensional tile shape — at the PDF here. It’s likely not coming to the flooring store soon.

It’s all wonderful, but is it useful? I could go on a few hundred words about, particularly, crystals and quasicrystals. These are important for materials science. Especially these days as we have harnessed slightly-imperfect crystals to be our computers. I don’t care. These are lovely to look at. If you see nothing appealing in a great heap of colors and polygons spread over the floor there are things we cannot communicate about. Tiling is a delight; what more do you need?

Thanks for your attention. This and all of my 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. See you next week, I hope.

How April 2015 Treated My Mathematics Blog

(I apologize if the formatting is messed up. For some reason preview is not working, and I will not be trying the new page for entering posts if I can at all help it. I will fix when I can, if it needs fixing.)

As it’s the start of the month I want to try understanding the readership of my blogs, as WordPress gives me statistics. It’s been a more confusing month than usual, though. One thing is easy to say: the number of pages read was 1,047, an all-time high around these parts for a single month. It’s up from 1,022 in March, and 859 in February. And it’s the second month in a row there’ve been more than a thousand readers. That part’s easy.

The number of visitors has dropped. It was down to 389 in April, from a record 468 in March and still-higher 407 in April. This is, if WordPress doesn’t lead me awry, my fifth-highest number of viewers. This does mean the number of views per visitor was my highest since June of 2013. The blog had 2.69 views per visitor, compared to 2.18 in March and 2.11 in February. It’s one of my highest views-per-visitor on record anyway. Perhaps people quite like what they see and are archive-binging. I approve of this. I’m curious why the number of readers dropped so, though, particularly when I look at my humor blog statistics (to be posted later).

I’m confident the readers are there, though. The number of likes on my mathematics blog was 297, up from March’s 265 and February’s 179. It’s the highest on record far as WordPress will tell me. So readers are more engaged, or else they’re clicking like from the WordPress Reader or an RSS feed. Neither gets counted as a page view or a visitor. That’s another easy part. The number of comments is down to 64, from March’s record 93, but March seems to have been an exceptional month. February had 56 comments so I’m not particularly baffled by April’s drop.

May starts out with 23,884 total views, and 472 people following specifically through WordPress.

It’s a truism that my most popular posts are the trapezoids one and the Reading The Comics posts, but for April that was incredibly true. Most popular the past thirty days were:

  1. How Many Trapezoids I Can Draw.
  2. Reading The Comics, April 10, 2015: Getting Into The Story Problem Edition.
  3. Reading The Comics, April 15, 2015: Tax Day Edition.
  4. Reading The Comics, April 20, 2015: History Of Mathematics Edition.
  5. Reading The Comics, March 31, 2015: Closing Out March Edition.

I am relieved that I started giving all these Comics posts their own individual “Edition” titles. Otherwise there’d be no way to tell them apart.

The nations sending me the most readers were, as ever, the United States (662), Canada (82), and the United Kingdom (47), with Slovenia once again strikingly high (36). Hong Kong came in with 24 readers, Italy 23, and Austria a mere 18. Elke Stangl’s had a busy month, I know.

This month’s single-reader countries were Czech Republic, Morocco, the Netherlands, Puerto Rico, Romania, Taiwan, and Vietnam. Romania’s the only one that sent me a single reader last month. India bounced back from five readers to six.

Among the search terms bringing people to me were no poems. Among the interesting phrases were:

  • what point is driving the area difference between two triangles (A good question!)
  • how do you say 1,898,600,000,000,000,000,000,000,000 (I almost never do.)
  • is julie larson still drawing the dinette set (Yes, to the best of my knowledge.)
  • jpe fast is earth spinning? (About once per day, although the answer can be surprisingly difficult to say! But also figure about 465 times the cosine of your latitude meters per second, roughly.)
  • origin is the gateway to your entire gaming universe. (Again, I don’t know what this means, and I’m a little scared to find out.)
  • i hate maths 2015 photos (Well, that just hurts.)
  • getting old teacher jokes (Again, that hurts, even if it’s not near my birthday.)
  • two trapezoids make a (This could be a poem, actually.)
  • how to draw 2 trapezoids (I’d never thought about that one. Shall have to consider writing it.)

I don’t know quite what it all means, other than that I need to write about comic strips and trapezoids more somehow.

My Mathematics Blog, As March 2015 Would Have It

And now for my monthly review of publication statistics. This is a good month to do it with, since it was a record month: I had 1,022 pages viewed around these parts, the first time (according to WordPress) that I’ve had more than a thousand in a month. In January I’d had 944, and in February a mere 859, which I was willing to blame on the shortness of that month. March’s is a clean record, though, more views per day than either of those months.

The total number of visitors was up, too, to 468. That’s compared to 438 in January and 407 in short February, although it happens it’s not a record; that’s still held by January 2013 and its 473 visitors. The number of views per visitor keeps holding about steady: from 2.16 in January to 2.11 in February to 2.18 in March. It appears that I’m getting a little better at finding people who like to read what I like to write, but haven’t caught that thrilling transition from linear to exponential growth.

The new WordPress statistics tell me I had a record 265 likes in March, up from January’s 196 and February’s 179. The number of comments rose from January’s 51 and February’s 56 to a full 93 for March. I take all this as supporting evidence that I’m better at reaching people lately. (Although I do wonder if it counts backlinks from one of my articles to another as a comment.)

The mathematics blog starts the month at 22,837 total views, and with 454 WordPress followers.

The most popular articles in March, though, were the set you might have guessed without actually reading things around here:

I admit I thought the “how interesting is a basketball tournament?” thing would be more popular, but it’s hampered by having started out in the middle of the month. I might want to start looking at the most popular articles of the past 30 days in the middle of the month too.

The countries sending me the greatest number of readers were the usual set: the United States at 658 in first place, and Canada in second at 66. The United Kingdom was a strong third at 57, and Austria in fourth place at 30.

Sending me a single reader each were Belgium, Ecuador, Israel, Japan, Lebanon, Mexico, Nepal, Norway, Portugal, Romania, Samoa, Saudi Arabia, Slovakia, Thailand, the United Arab Emirates, Uruguay, and Venezuela. The repeats from February were Japan, Mexico, Romania, and Venezuela. Japan is on a three-month streak, while Mexico has sent me a solitary reader four months in a row. India’s declined slightly in reading me, from 6 to 5. Ah well.

Among the interesting search terms were:

  • right trapezoid 5 (I loved this anime as a kid)
  • a short comic strip on reminding people on how to order decimals correctly (I hope they found what they were looking for)
  • are there other ways to draw a trapezoid (try with food dye on the back of your pet rabbit!)
  • motto of ideal gas (veni vidi v = nRT/P ?)
  • rectangular states (the majority of United States states are pretty rectangular, when you get down to it)
  • what is the definition of rerun (I don’t think this has come up before)
  • what are the chances of consecutive friday the 13th’s in a year (I make it out at 3/28, or a touch under 11 percent; anyone have another opinion?)

Well, with luck, I should have a fresh comic strips post soon and some more writing in the curious mix between information theory and college basketball.

How To Numerically Integrate Like A Mathematician

I have a guest post that I mean to put up shortly which is a spinoff of the talk last month about calculating logarithms. There are several ways to define a logarithm but one of the most popular is to define it as an integral. That has the advantages of allowing the logarithm to be studied using a lot of powerful analytic tools already built up for calculus, and allow it to be calculated numerically because there are a lot of methods for calculating logarithms out there. I wanted to precede that post with a discussion of a couple of the ways to do these numerical integrations.

A great way to interpret integrating a function is to imagine drawing a plot of function; the integral is the net area between the x-axis and the plot of that function. That may be pretty hard to do, though, so we fall back on a standard mathematician’s trick that they never tell you about in grade school, probably for good reason: don’t bother doing the problem you actually have, and instead do a problem that looks kind of like it but that you are able to do.

Normally, for what’s called a definite integral, we’re interested in the area underneath a curve and across an “interval”, that is, between some minimum and some maximum value on the x-axis. Definite integrals are the kind we can approximate numerically. An indefinite integral gives a function that would tell us what the definite integral on any interval would be, but that takes symbolic mathematics to work out and that’s way beyond this article’s scope.

A squiggly function between two vertical green lines that show the interval to be integrated over.
The red curve here represents some function. Any function will do, really. This is just one of them.

While we may have no idea what the area underneath a complicated squiggle on some interval is, we do know what the area inside a rectangle is. So if we pretend we’re interested in the area of the rectangle instead of the original area, good. Take my little drawing of a generic function here, the wavey red curve. The integral of it from wherever that left vertical green line is to the right is the area between the x-axis, the horizontal black line, and the red curve.

A generic function, with rectangles approximating the function's value along the y-axis based on the function's value at the left and at the right of the area we're integrating over.
For the Rectangle Rule, find the area of a rectangle that approximates the function whose integral you’re actually interested in. Shown here are the left-endpoint (yellow line) and the right-endpoint (blue line) approximations, but any rule can be used to find the approximation.

If we use the “Rectangle Rule”, we draw a horizontal line based on the value of the function somewhere from the left line to the right. The yellow line up top is based on the value at the left endpoint. The blue line is based on the value the function has at the right endpoint. We can use any point, although the most popular ones are the left endpoint, the right endpoint, and the midpoint, because those are nice, easy picks to make. (And if we’re trying to integrate a function whose definition we don’t know, for example because it’s the data we got from an experiment, these will often be the only data points we have.) The area under the curve is going to be something like the area of the rectangle bounded by the green lines, the horizontal black line, and the blue horizontal line or the yellow horizontal line.

Drawn this way you might complain this approximation is rubbish: the area of the blue-topped rectangle is obviously way too low, and that of the yellow-topped rectangle is way too high. The mathematician’s answer to this is: oh, hush. We were looking for easy, not good. The area is the width of the interval times the value of the function at the chosen point; how much easier can you get?

(It also happens that the blue rectangle obviously gives too low an area, while the yellow gives too high an area. This is a coincidence, caused by my not thinking to make my function wiggle up and down quite enough. Generally speaking neither the left- nor the right-endpoints are maximum or minimum values for the function. It can be useful analytically to select the points that are “where the function is its highest” and “where the function is its lowest” — this lets you find the upper and lower bounds for the area — but that’s generally too hard to use computationally.)

A generic function divided along the x-axis, with rectangles approximating the function's value along the y-axis.
For the Composite Rectangle Rule, slice the area you want to integrate into a bunch of small strips — not necessarily all the same width — and find the areas of the rectangles that approximate the function in each of those smaller strips.

But we can turn into a good approximation. What makes the blue or the yellow lines lousy approximations is that the function changes a lot in the distance between the green lines. If we were to chop up the strip into a bunch of smaller ones, and use the rectangle rule on each of those pieces, the function would change less in each of those smaller pieces, and so we’d get an area total that’s closer to the actual area. We find the distance between a pair of adjacent vertical green lines, multiply that by the height of the function at the chosen point, and add that to the running total. This is properly called the “Composite Rectangle Rule”, although it’s really only textbooks introducing the idea that make a fuss about including the “composite”. It just makes so much sense to break the interval up that we do that all the time and forget to explicitly say that except in the class where we introduce this.

(And, notice, in my drawings that in some of the regions behind vertical green lines the left-endpoint and the right-endpoint are not where the function gets its highest, or lowest, value. They can just be points.)

There’s nothing special about the Rectangle Rule that makes it uniquely suited for composition. It’s just easier to draw that way. Any numerical integration rule lets you do the same trick. Also, it’s very common to make all the smaller rectangles — called the subintervals — the same width, but that’s not because the method needs that to work. It’s easier to calculate if all the subintervals are the same width, because then you don’t have to remember how wide each different subinterval is.

A generic function, with a trapezoid approximating the function's value. The trapezoid matches the original function at the left and the right ends of the interval we integrate over.
For the Trapezoid Rule, approximate the area under the function by finding the area of a right trapezoid (or trapezium) which is as wide as the interval you want to integrate over and which matches the original function at the left- and the right-endpoints.

Rectangles are probably the easiest shape of all to deal with, but they’re not the only easy shapes. Trapezoids, or trapeziums if you prefer, are hardly a challenge to find the area for. This gives me the next really popular integration rule, the “Trapezoid Rule” or “Trapezium Rule” as your dialect favors. We take the function and approximate its area by working out the area of the trapezoid formed by the left green edge, the bottom black edge, the right green edge, and the sloping blue line that goes from where the red function touches the left end to where the function touches the right end. This is a little harder than the Rectangle Rule: we have to multiply the width of the interval between the green lines by the arithmetic mean of the function’s value at the left and at the right endpoints. That means, evaluate the function at the left endpoint and at the right endpoint, add those two values together, and divide by two. Not much harder and it’s pleasantly more accurate than the Rectangle Rule.

If that’s not good enough for you, you can break the interval up into a bunch of subintervals, just as with the Composite Rectangle Rule, and find the areas of all the trapezoids created there. This is properly called the “Composite Trapezoid Rule”, but again, after your Numerical Methods I class you won’t see the word “composite” prefixed to the name again.

And yet we can do better still. We’ll remember this when we pause a moment and think about what we’re really trying to do. When we do a numerical integration like this we want to find, instead of the area underneath our original curve, the area underneath a curve that looks like it but that’s easier to deal with. (Yes, we’re calling the straight lines of the Rectangle and Trapezoid Rules “curves”. Hush.) We can use any curve that we know how to deal with. Parabolas — the curving arc that you see if, say, you shoot the water from a garden hose into the air — may not seem terribly easy to deal with, but it turns out it’s not hard to figure out the area underneath a slice of one of them. This gives us the integration technique called “Simpson’s Rule”.

The Simpson here is Thomas Simpson, 1710 – 1761, who in accord with Mathematics Law did not discover or invent the rule named for him. Johannes Kepler knew the rule a century before Simpson got into the game, at minimum, and both Galileo’s student Bonaventura Cavalieri (who introduced logarithms to Italy, and was one of those people creeping up on infinitesimal calculus ahead of Newton) and the English mathematician/physicist James Gregory (who discovered diffraction grating, and seems to be the first person to have published a proof of the Fundamental Theorem of Calculus) were in on it too. But Simpson wrote a number of long-lived textbooks about calculus, which probably is why his name got tied to this method.

A generic function, with a parabola approximating the function's value. The parabola matches the original function at the left, middle, and right of the region we're integrating over.
For Simpson’s Rule, approximate the function you’re interested in with a parabola that exactly matches the original function at, at minimum, the left endpoint, the midpoint, and the right endpoint.

In Simpson’s Rule, you need the value of the function at the left endpoint, the midpoint, and the right endpoint of the interval. You can draw the parabola which connects those points — it’s the blue curve in my drawing — and find the area underneath that parabola. The formula may sound a little funny but it isn’t hard: the area underneath the parabola is one-third the width of the interval times the sum of the value at the left endpoint, the value at the right endpoint, and four times the value at the midpoint. It’s a bit more work but it’s a lot more accurate than the Trapezoid Rule.

There are literally infinitely many more rules you could use, with such names as “Simpson’s Three-Eighths Rule” (also called “Simpson’s Second Rule”) or “Boole’s Rule”[1], but they’re based on similar tricks of making a function that looks like the one you’re interested in but whose area you know how to calculate exactly. For the Simpson’s Three-Eighth Rule, for example, you make a cubic polynomial instead of a parabola. If you’re good at finding the areas underneath wedges of circles or underneath hyperbolas or underneath sinusoidal functions, go ahead and use those. You can find the balance of ease of use and accuracy of result that you like.

[1]: Boole’s Rule is also known as Bode’s Rule, because of a typo in the 1972 edition of the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, or as everyone ever has always referred to this definitive mathematical reference, Abromowitz and Stegun. (Milton Abromowitz and Irene Stegun were the reference’s editors.)

My Math Blog Statistics, September 2014

Since it’s the start of a new month it’s time to review statistics for the previous month, which gives me the chance to list a bunch of countries, which is strangely popular with readers. I don’t pretend to understand this, I just accept the inevitable.

In total views I haven’t seen much change the last several months: September 2014 looks to be closing out with about 558 pages viewed, not a substantial change from August’s 561, and triflingly fewer than July’s 589. The number of unique visitors has been growing steadily, though: 286 visitors in September, compared to 255 the month before, and 231 the month before that. One can choose to read this as the views per visitor dropping to 1.95, its lowest figure since March, but I’ll take it as more people finding things that interest them, at least.

As to what those things are — well, mostly it’s comic strip posts, which I suppose makes sense given that they’re quite accessible and often contain jokes people understand. The most popular articles for September 2014 were:

As usual the country sending me the greatest number of readers was the United States (347), with Canada (29), Austria (27), the United Kingdom (26), and Puerto Rico and Turkey (20 each) coming up close behind. My single-reader countries for September were Bahrain, Brazil, Costa Rica, Czech Republic, Estonia, Finland, Germany, Iceland, Jamaica, Kazakhstan, Malaysia, the Netherlands, Pakistan, Saudi Arabia, Slovenia, and Sweden. Finland, Germany, and Sweden were single-reader countries in August, too, but at least none of them were single-reader countries in July as well.

Among the search terms bringing people here the past month have been:

I got to my 17,882nd reader this month, a little short of that tolerably nice and round 18,000 readers. If I don’t come down with sudden-onset boringness, though, I’ll reach that in the next week or so, especially if I have a couple more days of twenty or thirty readers.

The Math Blog Statistics, May 2014

And on to the tracking of how my little mathematics blog is doing. As readership goes, things are looking good — my highest number of page views since January 2013, and third-highest ever, and also my highest number of unique viewers since January 2013 (unique viewer counts aren’t provided for before December 2012, so who knows what happened before that). The total number of page views rose from 565 in April to 751, and the number of unique visitors rose from 238 to 315. This is a remarkably steady number of views per visitor, though — 2.37 rising to 2.38, as if that were a significant difference. I passed visitor number 15,000 somewhere around the 5th of May, and at number 15,682 right now that puts me on track to hit 16,000 somewhere around the 13th.

As with April, the blog’s felt pretty good to me. I think I’m hitting a pretty good mixture of writing about stuff that interest me and finding readers who’re interested to read it. I’m hoping I can keep that up another month.

The most popular articles of the month — well, I suspect someone was archive-binging on the mathematics comics ones because, here goes:

  1. How Many Trapezoids I Can Draw, which will be my memorial
  2. Reading the Comics, May 13, 2014: Good Class Problems Edition, which was a tiny bit more popular than …
  3. Reading the Comics, May 26, 2014: Definitions Edition, the last big entry in the Math Comics of May sequence.
  4. Some Things About Joseph Nebus is just my little biographic page and I have no idea why anyone’s even looking at that.
  5. Reading the Comics, May 18, 2014: Pop Math of the 80s Edition is back on the mathematics comics, as these things should be,
  6. Reading the Comics, May 4, 2014: Summing the Series Edition and what the heck, let’s just mention this one too.
  7. The ideal gas equation is my headsup to a good writer’s writings.
  8. Where Does A Plane Touch A Sphere? is a nicely popular bit motivated by the realization that a tangent point is an important calculus concept and nevertheless a subtler thing than one might realize.

I think without actually checking this is the first month I’ve noticed with seven countries sending me twenty or more visitors each — the United States (438), Canada (39), Australia (38), Sweden (31), Denmark (21), and Singapore and the United Kingdom (20 each). Austria came in at 19, too. Sixteen countries sent me one visitor each: Antigua and Barbuda, Colombia, Guernsey, Hong Kong, Ireland, Italy, Jamaica, Japan, Kuwait, Lebanon, Mexico, Morocco, Norway, Peru, Poland, Swaziland, and Switzerland. Morocco’s the only one to have been there last month.

And while I lack for search term poetry, some of the interesting searches that brought people here include:

  • working mathematically comics
  • [ They’ve come to the right place, then. ]
  • how do you say 1898600000000000000000000000 in words [ I never do. ]
  • two trapezoids make a [ This is kind of beautiful as it is. ]
  • when you take a trapeziod apart how many trangles will you have?
  • -7/11,5/-8which is greter rational number and why
  • origin is the gateway to your entire gaming universe. [ This again is rather beautiful. ]
  • venn diagram on cartoons and amusement parks [ Beats me. ]

The Math Blog Statistics, April 2014

Another month’s gone by and the statistics about viewership were pretty gratifying for April 2014. But I’m feeling awfully good about the place, because I’ve felt more gratified by the mathematics blog lately. It’s felt to me like there’ve been more comments and more interaction the past couple weeks, and it’s felt like it’s getting closer to supporting a community, which is thrilling, if not exactly measurable given what WordPress shares with me.

In March 2014, according to last month’s statistics survey, there were 453 views from 257 distinct viewers. That jumped pretty noticeably this month to 565 views, albeit from 235 distinct viewers, a views-per-visitor jump from 1.76 to 2.40. I suspect there’s some archive-bingers, and I’m happy to give anyone that thrill. It’s my greatest viewer count since June 2013, and the fourth-highest since December 2012 when WordPress started sharing statistics on unique visitors. I also noted at the start of April that while I’d reached 14,000 visitors in March I’d need a stroke of luck to reach 15,000 in April. I came close: the month topped out with my 14,931st view.

The most popular articles of the past thirty days were:

  1. How Dirac Made Every Number, the answer to that puzzle of how to construct any counting number using precisely four 2’s and ordinary operations (it’s a forehead slapper once you’ve seen it)
  2. Reading the Comics, April 27, 2014: The Poetry of Calculus Edition, as everyone wants to see some calculus poetry
  3. Can You Be As Clever As Dirac For A Little Bit in which that Dirac puzzle was laid out and the rules given
  4. How Many Trapezoids I Can Draw, always with the trapezoids
  5. Reading the Comics, April 21, 2014: Bill Amend In Name Only Edition, which includes a bundle of a lot of mathematics comics from Bill Amend’s FoxTrot in case you need some

The countries sending me the most viewers were the United States (294), Canada (65), Denmark (29), Austria (27), and the United kingdom (26), and I count nine countries sending me at least ten views each, which I think is a record but I haven’t been keeping track of that number. Sending me a single viewer each were Belgium, Brazil, Ecuador, Finland, Greece, Hungary, Malaysia, Morocco, Oman, Sweden, and Venezuela. Belgium, Brazil, Hungary, and Sweden were single-country viewers last month, and Hungary’s got a three-month single-viewer streak going. So, ah, hi, whoever that is in Hungary. Apparently nobody has ever visited me from Honduras.

Once again there’s a shortage of search term poetry, but there were some fair queries the past month, including:

The Math Blog Statistics, March 2014

It’s the start of a fresh month, so let me carry on my blog statistics reporting. In February 2014, apparently, there were a mere 423 pages viewed around here, with 209 unique visitors. That’s increased a bit, to 453 views from 257 visitors, my second-highest number of views since last June and second-highest number of visitors since last April. I can make that depressing, though: it means views per visitor dropped from 2.02 to 1.76, but then, they were at 1.76 in January anyway. And I reached my 14,000th page view, which is fun, but I’d need an extraordinary bit of luck to get to 15,000 this month.

March’s most popular articles were a mix of the evergreens — trapezoids and comics — with a bit of talk about March Madness serving as obviously successful clickbait:

  1. How Many Trapezoids I Can Draw, and again, nobody’s found one I overlooked.
  2. Calculating March Madness, and the tricky problem of figuring out the chance of getting a perfect bracket.
  3. Reading The Comics, March 1, 2014: Isn’t It One-Half X Squared Plus C? Edition, showing how well an alleged joke will make comic strips popular.
  4. Reading The Comics, March 26, 2014: Kitchen Science Department, showing that maybe it’s just naming the comics installments that matters.
  5. What Are The Chances Of An Upset, which introduces some of the interesting quirks of the bracket and seed system of playoffs, such as the apparent advantage an eleventh seed has over an eighth seed.

There’s a familiar set of countries sending me the most readers: as ever the United States up top (277), with Denmark in second (26) and Canada in third (17). That’s almost a tie, though, as the United Kingdom (16), Austria (15), and the Philippines (13) could have taken third easily. I don’t want to explicitly encourage international rivalries to drive up my page count here, I’m just pointing it out. Singapore is in range too. The single-visitor countries this past month were the Bahamas, Belgium, Brazil, Colombia, Hungary, Mexico, Peru, Rwanda, Saudi Arabia, Spain, Sri Lanka, Sweden, Syria, and Taiwan. Hungary, Peru, and Saudi Arabia are the only repeat visitors from February, and nobody’s got a three-month streak going.

There wasn’t any good search-term poetry this month; mostly it was questions about trapezoids, but there were a couple interesting ones:

So, that’s where things stand: I need to get back to writing about trapezoids and comic strips.


Sometime on Thursday, the 6th of March, it appears I registered my 14,000th visitor to the math blog here. WordPress believes it to be someone from either the United States, France, Germany, Canada, or Australia, which at least covers a respectable number of possible time zones. The number’s a nice, big, round one, which I admit is about all I can think of that’s particularly interesting about it; even Wikipedia figures the most likely things you’re looking for if you look for 14,000 anything is either the ISO specification or an asteroid discovered in March of 1993 and apparently not even named yet. (It’s designated 1993 FZ55.) Well, at least asteroid 15,000 has a name.

Stare too hard at any one statistic, though, and you’ll start to wonder how reliable it is; I know for example that multiple of those 14,000 page views were me, testing neurotically to see whether the WordPress statistics counter was actually registering page views (particularly in the earliest days, when I was less self-confident and was using tags worse). Surely my just loading a page to see if it registers shouldn’t count as an actual page view, but, how can WordPress tell the difference?

Taking the WordPress statistics as if they meant what they purport to mean, though, indicates that apparently the most interesting thing I ever did was forget how to show why the area of a trapezoid was the trapezoid formula. My most-read article of all time is How Many Trapezoids I Can Draw, which is still standing at six by the way; some of the other articles which went into that (like Setting Out to Trap A Zoid and How Do You Make A Trapezoid Right) are also in the top five. All that’s based around working out how to work out a formula for the area of a trapezoid and convince yourself it’s right. For some reason the Reading the Comics post from September 11, 2012 also made the cut (I suppose the date is boosting it there). The early post How Many Numbers Have We Named is also fairly reliably popular.

Some of the most popular from the past year have included Something Neat About Triangles — since that post is only two months old I figure it’s got a good future ahead of it — and Solving The Price Is Right’s “Any Number” Game, which maybe solves the game less than explains why it’s usually a pretty good one to watch. Also popular is Counting From 52 to 11,108, and Inder J Taneja’s fascinating project in producing numbers using the digits one through nine in ascending or descending order.

January 2014’s Statistics

So how does the first month of 2014 compare to the last month of 2013, in terms of popularity? The raw numbers are looking up: I went from 176 unique visitors looking at 352 pages in December up to 283 unique visitors looking at 498 pages. If WordPress’s statistics are to be believed that’s my greatest number of page views since June of 2013, and the greatest number of visitors since February. This hurt the ratio of views per visitor a little, which dropped from 2.00 to 1.76, but we can’t have everything unless I write stuff that lots of people want to read and they figure they want to read a lot more based on that, which is just crazy talk. The most popular articles, though, were:

  1. Something Neat About Triangles, this delightful thing about forming an equilateral triangle starting from any old triangle.
  2. How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me).
  3. Factor Finding, linking over to IvaSallay’s quite interesting blog with a great recreational mathematics puzzle (or educational puzzle, depending on how you came into it) that drove me and a friend crazy with this week’s puzzles.
  4. What’s The Worst Way To Pack? in which I go looking for the least-efficient packing of spheres and show off these neat Mystery Science Theater 3000 foam balls I got.
  5. Reading The Comics, December 29, 2013, the old year’s last bunch of mathematics-themed comic strips.

The countries sending me readers the most often were the United States (281), Canada (52), the United Kingdom (25), and Austria (23). Sending me just a single reader each this past month were a pretty good list:
Bulgaria, France, Greece, Israel, Morocco, the Netherlands, Norway, Portugal, Romania, Russia, Serbia, Singapore, South Korea, Spain, and Viet Nam. Returning on that list from last month are Norway, Romania, Spain, and Viet Nam, and none of those were single-country viewers back in November 2013.

December 2013’s Statistics

There’s a hopeful trend in my readership statistics for December 2013 around these parts: according to WordPress, my number of readers grew from 308 in November to 352 and the number of unique visitors grew from 158 to 176. Even the number of views per visitor grew, from 1.95 to 2.00. None of these are records, but the fact of improvement is a good one.

I can’t figure exactly how to get the report on most popular articles for the exact month of December, and was too busy with other things to check the past-30-day report on New Year’s Eve, but at least the most popular articles for the 30 days ending today were:

The countries sending me the most readers were the United States, Canada, Denmark and Austria (tied, and hi again, Elke), and the United Kingdom. Sending me just one viewer each were a slew of nations: Bangladesh, Cambodia, India, Japan, Jordan, Malaysia, Norway, Romania, Slovenia, South Africa, Spain, Sweden, Turkey, and Viet Nam. On that list last month were Jordan and Slovenia, so I’m also marginally interesting to a different group of people this time around.

This has all caused me to realize that I failed to promote my string of articles inspired by Arthur Christmas and getting to the recurrence theorem and the existential dread of the universe’s end during the Christmas season. Maybe next year, then.

September 2013’s Statistics

And as it’s the start of the month I have a fresh round of reviewing the statistics for readership around here. I have seen a nice increase in both views — from 367 to about 466 total views — and in visitors — from 175 to 236 — which maybe reflects the resumption of the school year (in the United States, anyway) and some more reliable posting (of original articles and of links to other people’s) on my part. (Maybe. If I’m reading this rightly I actually only posted nine new things in September, which is the same as in August. I’m surprised that WordPress’s statistics page doesn’t seem to report how many new articles there were in the month, though.) My contrarian nature forces me to note this means my views-per-reader ratio has dropped to 1.97, down from 2.10. I suppose as long as the views-per-reader statistic stays above 1.00 I’m not doing too badly.

The most popular articles the past month were:

  1. From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics, which is really just pointing and slightly setting up ElKement’s start to a series on quantum field theory which you can too understand;
  2. How Many Trapezoids I Can Draw, which is a persistent favorite and makes me suspect that I’ve hit on something that teachers ask students about. If I could think of a couple other nice little how-many-of-these-things problems there are I’d post them gladly, although that might screw up some people’s homework assignments;
  3. Reading the Comics, September 11, 2012, which is another persistent favorite and I can’t imagine that it’s entirely about the date (although the similar Reading the Comics entry for September 11 of 2013 just missed being one of the top articles this month so perhaps the subject lines are just that effective a bit of click-baiting);
  4. What Is Calculus I Like?, about my own realization that I never took a Calculus I course in the conditions that most people who take it do. I’d like more answers to the question of what experiences in intro-to-calculus courses are like, since I’m assuming that I will someday teach it again and while I think I can empathize with students, I would surely do better at understanding what they don’t understand if I knew better what people in similar courses went through;
  5. Some Difficult Math Problems That You Understand, which is again pointing to another blog — here, Maths In A Minute — with a couple of mathematics problems that pretty much anyone can understand on their first reading. The problems are hard ones, each of which has challenged the mathematical community for generations, so you aren’t going to solve them; but, thinking about them and trying to solve them is probably a great exercise and likely to lead you to discovering something you didn’t know.

I got the greatest number of readers from the United States again (271), with Canada (31) once more in second place. The United Kingdom’s climbed back into the top three (21), while August’s number-three, Denmark, dropped out of the top ten and behind both Singapore and the Philippines. I got a mass of single-reader countries this time, too: Azerbaijan, Bangladesh, Belgium, Cambodia, the Czech Republic, Indonesia, Israel, Italy, Mexico, Norway, Poland, Qatar, Spain, Sri Lanka, Switzerland, and Thailand. Bangladesh and Sri Lanka are repeats from last month, but my Estonian readership seems to have fled entirely. At least India and New Zealand still like me.

My August 2013 Statistics

As promised I’m keeping and publicizing my statistics, as WordPress makes them out, the better I hope to understand what I do well and what the rest is. I’ve had a modest uptick in views from July — 341 to 367 — as well as in unique visitors — 156 to 175 — although this means my views-per-visitor count has dropped from 2.19 to 2.10. That’s still my third-highest views-per-visitor count since WordPress started revealing that data to us.

The most popular articles of the past month were:

  1. Reading the Comics, September 11, 2012 (which I suspect reflects the date turning up high in Google searches, though there are many comics mentioned in it so perhaps it just casts a wide net);
  2. Just Answer 1/e Whenever Anyone Asks This Kind Of Question (part of the thread on the chance of the 1902-built Leap-the-Dips roller coaster having any of its original boards remaining)
  3. Just How Far Is The End Of The World? (the start of a string based on seeing the Sleeping Bear Dunes)
  4. Why I Say 1/e About This Roller Coaster (back to the Leap-the-Dips question)
  5. Augustin-Louis Cauchy’s birthday (a little biographical post which drew some comments because Cauchy was at the center of a really interesting time in mathematics)
  6. Professor Ludwig von Drake Explains Numerical Mathematics (which isn’t one of the top-five pieces this month but is a comic strip, so enjoy)

The countries sending me the most readers were the United States (202), Canada (30), and Denmark (19). Sending me just one each were Argentina, Bangladesh, Estonia, Finland, the Netherlands, Portugal, Singapore, Sri Lanka, Taiwan, and Viet Nam. Argentina, Estonia, and the Netherlands did the same last month; clearly I’m holding steady. And my readership in Slovenia doubled from last month’s lone reader.

I also learn that the search terms bringing people to me have been, most often, “trapezoid”, “descartes and the fly”, “joseph nebus”, “nebus wordpress”, and “any number the price is right”. I’ve had more entries than I realized mention The Price Is Right and ought to try making them easier for search engines to locate. My trapezoid work meanwhile — on their area and the number of different kinds of trapezoids — remains popular.

My June 2013 Statistics

I don’t understand why, but an awful lot of the advice I see about blogging says that it’s important not just to keep track of how your blog is doing, but also to share it, so that … numbers will like you more? I don’t know. But I can give it a try, anyway.

For June 2013, according to WordPress, I had some 713 page views, out of 246 unique visitors. That’s the second-highest number of page views I’ve had in any month this year (January had 831 views), and the third-highest I’ve had for all time (there were 790 in March 2012). The number of unique visitors isn’t so impressive; since WordPress started giving me that information in December 2012, I’ve had more unique visitors … actually, in every month but May 2013. On the other hand, the pages-per-viewer count of 2.90 is the best I’ve had; the implication seems to be that I’m engaging my audience.

The most popular posts for the past month were Counting From 52 to 11,108, which I believe reflects it getting picked for a class assignment somehow; A Cedar Point Follow-Up, which hasn’t got much mathematics in it but has got pretty pictures of an amusement park, and Solving The Price Is Right’s “Any Number” Game, which has got some original mathematics but also a pretty picture.

My all-time most popular posts are from the series about Trapezoids — working out how to find their area, and how many kinds of trapezoids there are — with such catchy titles as How Many Trapezoids I Can Draw, or How Do You Make A Trapezoid Right?, or Setting Out To Trap A Zoid, which should be recognized as a Dave Barry reference.

My most frequent commenters, “recent”, whatever that means, are Chiaroscuro and BunnyHugger (virtually tied), with fluffy, elkelement, MJ Howard, and Geoffrey Brent rounding out the top six.

The most common source of page clicks the past month was from the United States (468), with Brazil (51) and Canada (23) taking silver and bronze. And WordPress recorded one click each from Portugal, Serbia, Hungary, Macedonia (the Former Yugoslav Republic), Indonesia, Argentina, Poland, Slovenia, and Viet Nam. I’ve been to just one of those countries.

Some Many Ways Of Flatness

[ We didn’t break 3,100 yet, and too bad that. But over the day I did get my first readers from Turkey and the second from the United Arab Emirates that I’ve noticed. Also while my many posts about trapezoids are drawing search engine results, “frazz sequins” comes up a lot. ]

I think I’ve managed, more or less, acceptance that a piecewise constant interpolation makes the simplest way to estimate the population of Charlotte, North Carolina, when all I had to work with was the population data from the 1970 and the 1980 censuses. In 1970 the city had 840,347 people; in 1980 it had 971,391, and therefore the easiest guess to the population in 1975 would be the 1970 value, of 840,347. We suppose that on the 1st of April, 1970 — that Census Day — the population was the lower value, and then sometime before the 1st of April, 1980, it leapt up at once by the 131,044-person difference. Only … how do I know the population jumped up sometime after 1975?

Continue reading “Some Many Ways Of Flatness”