## What Second Derivatives Are And What They Can Do For You

Previous supplemental reading for Why Stuff Can Orbit:

This is another supplemental piece because it’s too much to include in the next bit of Why Stuff Can Orbit. I need some more stuff about how a mathematical physicist would look at something.

This is also a story about approximations. A lot of mathematics is really about approximations. I don’t mean numerical computing. We all know that when we compute we’re making approximations. We use 0.333333 instead of one-third and we use 3.141592 instead of π. But a lot of precise mathematics, what we call analysis, is also about approximations. We do this by a logical structure that works something like this: take something we want to prove. Now for every positive number ε we can find something — a point, a function, a curve — that’s no more than ε away from the thing we’re really interested in, and which is easier to work with. Then we prove whatever we want to with the easier-to-work-with thing. And since ε can be as tiny a positive number as we want, we can suppose ε is a tinier difference than we can hope to measure. And so the difference between the thing we’re interested in and the thing we’ve proved something interesting about is zero. (This is the part that feels like we’re pulling a scam. We’re not, but this is where it’s worth stopping and thinking about what we mean by “a difference between two things”. When you feel confident this isn’t a scam, continue.) So we proved whatever we proved about the thing we’re interested in. Take an analysis course and you will see this all the time.

When we get into mathematical physics we do a lot of approximating functions with polynomials. Why polynomials? Yes, because everything is polynomials. But also because polynomials make so much mathematical physics easy. Polynomials are easy to calculate, if you need numbers. Polynomials are easy to integrate and differentiate, if you need analysis. Here that’s the calculus that tells you about patterns of behavior. If you want to approximate a continuous function you can always do it with a polynomial. The polynomial might have to be infinitely long to approximate the entire function. That’s all right. You can chop it off after finitely many terms. This finite polynomial is still a good approximation. It’s just good for a smaller region than the infinitely long polynomial would have been.

Necessary qualifiers: pages 65 through 82 of any book on real analysis.

So. Let me get to functions. I’m going to use a function named ‘f’ because I’m not wasting my energy coming up with good names. (When we get back to the main Why Stuff Can Orbit sequence this is going to be ‘U’ for potential energy or ‘E’ for energy.) It’s got a domain that’s the real numbers, and a range that’s the real numbers. To express this in symbols I can write $f: \Re \rightarrow \Re$. If I have some number called ‘x’ that’s in the domain then I can tell you what number in the domain is matched by the function ‘f’ to ‘x’: it’s the number ‘f(x)’. You were expecting maybe 3.5? I don’t know that about ‘f’, not yet anyway. The one thing I do know about ‘f’, because I insist on it as a condition for appearing, is that it’s continuous. It hasn’t got any jumps, any gaps, any regions where it’s not defined. You could draw a curve representing it with a single, if wriggly, stroke of the pen.

I mean to build an approximation to the function ‘f’. It’s going to be a polynomial expansion, a set of things to multiply and add together that’s easy to find. To make this polynomial expansion this I need to choose some point to build the approximation around. Mathematicians call this the “point of expansion” because we froze up in panic when someone asked what we were going to name it, okay? But how are we going to make an approximation to a function if we don’t have some particular point we’re approximating around?

(One answer we find in grad school when we pick up some stuff from linear algebra we hadn’t been thinking about. We’ll skip it for now.)

I need a name for the point of expansion. I’ll use ‘a’. Many mathematicians do. Another popular name for it is ‘x0‘. Or if you’re using some other variable name for stuff in the domain then whatever that variable is with subscript zero.

So my first approximation to the original function ‘f’ is … oh, shoot, I should have some new name for this. All right. I’m going to use ‘F0‘ as the name. This is because it’s one of a set of approximations, each of them a little better than the old. ‘F1‘ will be better than ‘F0‘, but ‘F2‘ will be even better, and ‘F2038‘ will be way better yet. I’ll also say something about what I mean by “better”, although you’ve got some sense of that already.

I start off by calling the first approximation ‘F0‘ by the way because you’re going to think it’s too stupid to dignify with a number as big as ‘1’. Well, I have other reasons, but they’ll be easier to see in a bit. ‘F0‘, like all its sibling ‘Fn‘ functions, has a domain of the real numbers and a range of the real numbers. The rule defining how to go from a number ‘x’ in the domain to some real number in the range?

$F^0(x) = f(a)$

That is, this first approximation is simply whatever the original function’s value is at the point of expansion. Notice that’s an ‘x’ on the left side of the equals sign and an ‘a’ on the right. This seems to challenge the idea of what an “approximation” even is. But it’s legit. Supposing something to be constant is often a decent working assumption. If you failed to check what the weather for today will be like, supposing that it’ll be about like yesterday will usually serve you well enough. If you aren’t sure where your pet is, you look first wherever you last saw the animal. (Or, yes, where your pet most loves to be. A particular spot, though.)

We can make this rigorous. A mathematician thinks this is rigorous: you pick any margin of error you like. Then I can find a region near enough to the point of expansion. The value for ‘f’ for every point inside that region is ‘f(a)’ plus or minus your margin of error. It might be a small region, yes. Doesn’t matter. It exists, no matter how tiny your margin of error was.

But yeah, that expansion still seems too cheap to work. My next approximation, ‘F1‘, will be a little better. I mean that we can expect it will be closer than ‘F0‘ was to the original ‘f’. Or it’ll be as close for a bigger region around the point of expansion ‘a’. What it’ll represent is a line. Yeah, ‘F0‘ was a line too. But ‘F0‘ is a horizontal line. ‘F1‘ might be a line at some completely other angle. If that works better. The second approximation will look like this:

$F^1(x) = f(a) + m\cdot\left(x - a\right)$

Here ‘m’ serves its traditional yet poorly-explained role as the slope of a line. What the slope of that line should be we learn from the derivative of the original ‘f’. The derivative of a function is itself a new function, with the same domain and the same range. There’s a couple ways to denote this. Each way has its strengths and weaknesses about clarifying what we’re doing versus how much we’re writing down. And trying to write down almost anything can inspire confusion in analysis later on. There’s a part of analysis when you have to shift from thinking of particular problems to how problems work then.

So I will define a new function, spoken of as f-prime, this way:

$f'(x) = \frac{df}{dx}\left(x\right)$

If you look closely you realize there’s two different meanings of ‘x’ here. One is the ‘x’ that appears in parentheses. It’s the value in the domain of f and of f’ where we want to evaluate the function. The other ‘x’ is the one in the lower side of the derivative, in that $\frac{df}{dx}$. That’s my sloppiness, but it’s not uniquely mine. Mathematicians keep this straight by using the symbols $\frac{df}{dx}$ so much they don’t even see the ‘x’ down there anymore so have no idea there’s anything to find confusing. Students keep this straight by guessing helplessly about what their instructors want and clinging to anything that doesn’t get marked down. Sorry. But what this means is to “take the derivative of the function ‘f’ with respect to its variable, and then, evaluate what that expression is for the value of ‘x’ that’s in parentheses on the left-hand side”. We can do some things that avoid the confusion in symbols there. They all require adding some more variables and some more notation in, and it looks like overkill for a measly definition like this.

Anyway. We really just want the deriviate evaluated at one point, the point of expansion. That is:

$m = f'(a) = \frac{df}{dx}\left(a\right)$

which by the way avoids that overloaded meaning of ‘x’ there. Put this together and we have what we call the tangent line approximation to the original ‘f’ at the point of expansion:

$F^1(x) = f(a) + f'(a)\cdot\left(x - a\right)$

This is also called the tangent line, because it’s a line that’s tangent to the original function. A plot of ‘F1‘ and the original function ‘f’ are guaranteed to touch one another only at the point of expansion. They might happen to touch again, but that’s luck. The tangent line will be close to the original function near the point of expansion. It might happen to be close again later on, but that’s luck, not design. Most stuff you might want to do with the original function you can do with the tangent line, but the tangent line will be easier to work with. It exactly matches the original function at the point of expansion, and its first derivative exactly matches the original function’s first derivative at the point of expansion.

We can do better. We can find a parabola, a second-order polynomial that approximates the original function. This will be a function ‘F2(x)’ that looks something like:

$F^2(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 m_2 \left(x - a\right)^2$

What we’re doing is adding a parabola to the approximation. This is that curve that looks kind of like a loosely-drawn U. The ‘m2‘ there measures how spread out the U is. It’s not quite the slope, but it’s kind of like that, which is why I’m using the letter ‘m’ for it. Its value we get from the second derivative of the original ‘f’:

$m_2 = f''(a) = \frac{d^2f}{dx^2}\left(a\right)$

We find the second derivative of a function ‘f’ by evaluating the first derivative, and then, taking the derivative of that. We can denote it with two ‘ marks after the ‘f’ as long as we aren’t stuck wrapping the function name in ‘ marks to set it out. And so we can describe the function this way:

$F^2(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 f''(a) \left(x - a\right)^2$

This will be a better approximation to the original function near the point of expansion. Or it’ll make larger the region where the approximation is good.

If the first derivative of a function at a point is zero that means the tangent line is horizontal. In physics stuff this is an equilibrium. The second derivative can tell us whether the equilibrium is stable or not. If the second derivative at the equilibrium is positive it’s a stable equilibrium. The function looks like a bowl open at the top. If the second derivative at the equilibrium is negative then it’s an unstable equilibrium.

We can make better approximations yet, by using even more derivatives of the original function ‘f’ at the point of expansion:

$F^3(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 f''(a) \left(x - a\right)^2 + \frac{1}{3\cdot 2} f'''(a) \left(x - a\right)^3$

There’s better approximations yet. You can probably guess what the next, fourth-degree, polynomial would be. Or you can after I tell you the fraction in front of the new term will be $\frac{1}{4\cdot 3\cdot 2}$. The only big difference is that after about the third derivative we give up on adding ‘ marks after the function name ‘f’. It’s just too many little dots. We start writing, like, ‘f(iv)‘ instead. Or if the Roman numerals are too much then ‘f(2038)‘ instead. Or if we don’t want to pin things down to a specific value ‘f(j)‘ with the understanding that ‘j’ is some whole number.

We don’t need all of them. In physics problems we get equilibriums from the first derivative. We get stability from the second derivative. And we get springs in the second derivative too. And that’s what I hope to pick up on in the next installment of the main series.

• #### elkement (Elke Stangl) 4:20 pm on Wednesday, 7 June, 2017 Permalink | Reply

This is great – I’ve just written a very short version of that (a much too succinct one) … as an half-hearted attempt to explain Taylor expansions that I need in an upcoming post. But now I won’t feel bad anymore about its incomprehensibility and simply link to this post of yours :-)

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• #### Joseph Nebus 12:17 am on Friday, 9 June, 2017 Permalink | Reply

Aw, goodness, thank you so. That’s so kind of you.

Liked by 1 person

## Reading the Comics, May 31, 2017: Feast Week Edition

You know we’re getting near the end of the (United States) school year when Comic Strip Master Command orders everyone to clear out their mathematics jokes. I’m assuming that’s what happened here. Or else a lot of cartoonists had word problems on their minds eight weeks ago. Also eight weeks ago plus whenever they originally drew the comics, for those that are deep in reruns. It was busy enough to split this week’s load into two pieces and might have been worth splitting into three, if I thought I had publishing dates free for all that.

Larry Wright’s Motley Classics for the 28th of May, a rerun from 1989, is a joke about using algebra. Occasionally mathematicians try to use the the ability of people to catch things in midair as evidence of the sorts of differential equations solution that we all can do, if imperfectly, in our heads. But I’m not aware of evidence that anyone does anything that sophisticated. I would be stunned if we didn’t really work by a process of making a guess of where the thing should be and refining it as time allows, with experience helping us make better guesses. There’s good stuff to learn in modeling how to catch stuff, though.

Also I want to say some very good words about Jantze’s graphical design. The mock textbook cover for the title panel on the left is so spot-on for a particular era in mathematics textbooks it’s uncanny. The all-caps Helvetica, the use of two slightly different tans, the minimalist cover art … I know shelves stuffed full in the university mathematics library where every book looks like that. Plus, “[Mathematics Thing] And Their Applications” is one of the roughly four standard approved mathematics book titles. He paid good attention to his references.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 28th deploys a big old whiteboard full of equations for the “secret” of the universe. This makes a neat change from finding the “meaning” of the universe, or of life. The equations themselves look mostly like gibberish to me, but Wise and Aldrich make good uses of their symbols. The symbol $\vec{B}$, a vector-valued quantity named B, turns up a lot. This symbol we often use to represent magnetic flux. The B without a little arrow above it would represent the intensity of the magnetic field. Similarly an $\vec{H}$ turns up. This we often use for magnetic field strength. While I didn’t spot a $\vec{E}$ — electric field — which would be the natural partner to all this, there are plenty of bare E symbols. Those would represent electric potential. And many of the other symbols are what would naturally turn up if you were trying to model how something is tossed around by a magnetic field. Q, for example, is often the electric charge. ω is a common symbol for how fast an electromagnetic wave oscillates. (It’s not the frequency, but it’s related to the frequency.) The uses of symbols is consistent enough, in fact, I wonder if Wise and Aldrich did use a legitimate sprawl of equations and I’m missing the referenced problem.

John Graziano’s Ripley’s Believe It Or Not for the 28th mentions how many symbols are needed to write out the numbers from 1 to 100. Is this properly mathematics? … Oh, who knows. It’s just neat to know.

Mark O’Hare’s Citizen Dog rerun for the 29th has the dog Fergus struggle against a word problem. Ordinary setup and everything, but I love the way O’Hare draws Fergus in that outfit and thinking hard.

The Eric the Circle rerun for the 29th by ACE10203040 is a mistimed Pi Day joke.

Bill Amend’s FoxTrot Classicfor the 31st, a rerun from the 7th of June, 2006, shows the conflation of “genius” and “good at mathematics” in everyday use. Amend has picked a quixotic but in-character thing for Jason Fox to try doing. Euclid’s Fifth Postulate is one of the classic obsessions of mathematicians throughout history. Euclid admitted the thing — a confusing-reading mess of propositions — as a postulate because … well, there’s interesting geometry you can’t do without it, and there doesn’t seem any way to prove it from the rest of his geometric postulates. So it must be assumed to be true.

There isn’t a way to prove it from the rest of the geometric postulates, but it took mathematicians over two thousand years of work at that to be convinced of the fact. But I know I went through a time of wanting to try finding a proof myself. It was a mercifully short-lived time that ended in my humbly understanding that as smart as I figured I was, I wasn’t that smart. We can suppose Euclid’s Fifth Postulate to be false and get interesting geometries out of that, particularly the geometries of the surface of the sphere, and the geometry of general relativity. Jason will surely sometime learn.

• #### goldenoj 9:08 pm on Sunday, 4 June, 2017 Permalink | Reply

Just found these recently. I really enjoy them and catching up is fun. Thanks!

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• #### Joseph Nebus 1:05 am on Wednesday, 7 June, 2017 Permalink | Reply

Thanks for finding the pieces. I hope you enjoy; they’re probably my most reliable feature around here.

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## How May 2017 Treated My Mathematics Blog

The big news is that in May my mathematics blog crept back above a thousand page views. It had been a whole month since it had reached this threshold of purely imaginary significance. For what was a slow writing month — only twelve posts — marred by my computer dying and a nasty cold the final week, the numbers aren’t bad.

In May there were 1,029 pages viewed here. That’s up from April’s 994 and March’s 1,026. The number of unique visitors is down for the third month running, though, down to 662 from April’s 696 and March’s 699. The happy implication: people reading more posts as they visit. You know, liking my writing more.

I still feel like trying to rig up some compensation for that bizarre event back in … September 2015, wasn’t it? … when suddenly everybody’s statistics everywhere dropped and we blamed it on them no longer counting mobile devices. But if that were so, surely they’ve put them back? There’s no way the non-mobile-device readership is growing fast enough that these numbers should be about stable.

I’d think, anyway. There were 78 posts liked in May, down from April’s 90 and March’s 85. Not to pout or anything but WordPress does tell me that in June 2015 there were 518 likes around here and I can’t think, gosh, what was different then? … Well, it was one of my A To Z months, with posts 28 days of the month, and that usually encourages cross-reading. The number of comments just cratered, though: there were only 8 all month, down from 16 in April and 15 in May. Clearly I’m failing to encourage conversation and I don’t know how to turn that around.

According to Insights the most popular day for reading stuff was Thursday, with 16 percent of page views then. In April Sunday was the busiest day again with 16 percent of page views; in March it was 18 percent, on Tuesdays. I may give up on tracking this; obviously, each day is about equally likely to be the most popular. The most popular reading time was the hour of 6 pm, with 11 percent of page views coming before 7 pm. In April the same hour got 11 percent of page views again. In March it got 12 percent. I might experiment with the designated posting hour to find a more popular time, but obviously most people are going to read right after the thing is published.

So what was popular writing around here in April? I don’t want to say I knew this would happen, but one of the top five posts was one for which I wrote eleven words, and which I predicted to myself would be among the motnh’s top posts.

1. How Many Grooves Are On A Record’s Side? People want simple answers to their questions.
2. Reading the Comics, May 27, 2017: Panels Edition and I’m surprised this took the lead in the month’s Reading the Comics races, given how little time it had to do it.
3. How Many Trapezoids I Can Draw as see above comment about people wanting answers
4. Theorem Thursday: The Jordan Curve Theorem which I was thinking about at the mall on Thursday. Something or other made me think of it and how much I liked my description of how you prove the theorem.
5. Dabbing and the Pythagorean Theorem which, really, I should do more like given how popular this kind of post is.

Now the roster of the 52 countries that sent me readers in May, and how many each of them did. Spoiler: the United States tops the list.

Country Views
United States 658
United Kingdom 38
Australia 28
Italy 23
India 19
Singapore 15
Slovenia 13
Turkey 13
Spain 12
South Africa 11
Austria 10
Switzerland 10
Denmark 7
Mexico 7
New Zealand 7
Puerto Rico 7
Philippines 6
Brazil 5
Oman 5
Russia 5
Sweden 5
Germany 4
Chile 3
France 3
Netherlands 3
European Union 2
Indonesia 2
Pakistan 2
Peru 2
Argentina 1 (*)
Bahamas 1
Belgium 1
Colombia 1
Czech Republic 1
Finland 1 (**)
Iceland 1
Israel 1
Japan 1
Nigeria 1
Poland 1
Portugal 1 (**)
Saudi Arabia 1
Slovakia 1
Sri Lanka 1
St. Kitts & Nevis 1
Taiwan 1
US Virgin Islands 1
Ukraine 1
Uruguay 1
Venezuela 1

There had been 45 countries sending readers in April and 56 in March. European Union makes its big return.

There were 21 single-reader countries in May, way up from April’s 10 but still down from March’s 26. Argentina was a single-reader country in April also. Finland and Portugal have been single-reader countries for three months.

The month starts with 49,247 page views from some 22,212 logged distinct visitors since WordPress started telling us about those. WordPress tells me also there are 662 followers on WordPress, people who’ve gone and clicked the ‘Follow On WordPress’ button at the top right of the page in the hopes that I’ll follow back and increase their readership count. We all know how the game works.

And then what are popular search terms bringing folks here? What you’d expect given the most popular posts.

• comics conversation
• how many grooves are on typical record or cd ? how they are arranged?
• origin is the gateway to your entire gaming universe.
• peacetips football prediction
• only yestetday dividing fractions
• animated rolling dice 7

Plus some 146 unknown search terms. I’d be interested to know what those are too.

Well, thanks all of you for being around for this. I hope it’s a good month ahead.

You know, the arrangement of CDs is probably an interesting subject. I love that sort of technical-detail stuff too. It’s probably only slightly mathematics but I bet I can find a pretext to include it here. If someone’s interested.

## Something Cute I Never Noticed Before About Infinite Sums

This is a trifle, for which I apologize. I’ve been sick. But I ran across this while reading Carl B Boyer’s The History of the Calculus and its Conceptual Development. This is from the chapter “A Century Of Anticipation”, developments leading up to Newton and Leibniz and The Calculus As We Know It. In particular, while working out the indefinite integrals for simple powers — x raised to a whole number — John Wallis, whom you’ll remember from such things as the first use of the ∞ symbol and beating up Thomas Hobbes for his lunch money, noted this:

$\frac{0 + 1}{1 + 1} = \frac{1}{2}$

Which is fine enough. But then Wallis also noted that

$\frac{0 + 1 + 2}{2 + 2 + 2} = \frac{1}{2}$

And furthermore that

$\frac{0 + 1 + 2 + 3}{3 + 3 + 3 + 3} = \frac{1}{2}$

$\frac{0 + 1 + 2 + 3 + 4}{4 + 4 + 4 + 4 + 4} = \frac{1}{2}$

$\frac{0 + 1 + 2 + 3 + 4 + 5}{5 + 5 + 5 + 5 + 5 + 5} = \frac{1}{2}$

And isn’t that neat? Wallis goes on to conclude that this is true not just for finitely many terms in the numerator and denominator, but also if you carry on infinitely far. This seems like a dangerous leap to make, but they treated infinities and infinitesimals dangerously in those days.

What makes this work is — well, it’s just true; explaining how that can be is kind of like explaining how it is circles have a center point. All right. But we can prove that this has to be true at least for finite terms. A sum like 0 + 1 + 2 + 3 is an arithmetic progression. It’s the sum of a finite number of terms, each of them an equal difference from the one before or the one after (or both).

Its sum will be equal to the number of terms times the arithmetic mean of the first and last. That is, it’ll be the number of terms times the sum of the first and the last terms and divided that by two. So that takes care of the numerator. If we have the sum 0 + 1 + 2 + 3 + up to whatever number you like which we’ll call ‘N’, we know its value has to be (N + 1) times N divided by 2. That takes care of the numerator.

The denominator, well, that’s (N + 1) cases of the number N being added together. Its value has to be (N + 1) times N. So the fraction is (N + 1) times N divided by 2, itself divided by (N + 1) times N. That’s got to be one-half except when N is zero. And if N were zero, well, that fraction would be 0 over 0 and we know what kind of trouble that is.

It’s a tiny bit, although you can use it to make an argument about what to expect from $\int{x^n dx}$, as Wallis did. And it delighted me to see and to understand why it should be so.

• #### elkement (Elke Stangl) 4:38 pm on Monday, 5 June, 2017 Permalink | Reply

It reminds me of the famous story about young Gauss – when he baffled his teacher with this somewhat related ‘trick’ of adding up numbers between 1 to 100 very quickly (by actually calculating 101*50).

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• #### Joseph Nebus 1:09 am on Wednesday, 7 June, 2017 Permalink | Reply

That’s exactly what crossed my mind, especially as I realized I was doing the sum of 1 through 100 at least implicitly. It feels so playful to have something like that turn up.

Liked by 1 person

## Reading the Comics, May 27, 2017: Panels Edition

Can’t say this was too fast or too slow a week for mathematically-themed comic strips. A bunch of the strips were panel comics, so that’ll do for my theme.

Norm Feuti’s Retail for the 21st mentions every (not that) algebra teacher’s favorite vague introduction to group theory, the Rubik’s Cube. Well, the ways you can rotate the various sides of the cube do form a group, which is something that acts like arithmetic without necessarily being numbers. And it gets into value judgements. There exist algorithms to solve Rubik’s cubes. Is it a show of intelligence that someone can learn an algorithm and solve any cube? — But then, how is solving a Rubik’s cube, with or without the help of an algorithm, a show of intelligence? At least of any intelligence more than the bit of spatial recognition that’s good for rotating cubes around?

Norm Feuti’s Retail for the 21st of May, 2017. A few weeks ago I ran across a book about the world of competitive Rubik’s Cube solving. I haven’t had the chance to read it, but am interested by the ways people form rules for what would seem like a naturally shapeless feature such as solving Rubik’s Cubes. Not featured: the early 80s Saturday morning cartoon that totally existed because somehow that made sense back then.

I don’t see that learning an algorithm for a problem is a lack of intelligence. No more than using a photo reference shows a lack of drawing skill. It’s still something you need to learn, and to apply, and to adapt to the cube as you have it to deal with. Anyway, I never learned any techniques for solving it either. Would just play for the joy of it. Here’s a page with one approach to solving the cube, if you’d like to give it a try yourself. Good luck.

Bob Weber Jr and Jay Stephens’s Oh, Brother! for the 22nd is a word-problem avoidance joke. It’s a slight thing to include, but the artwork is nice.

Brian and Ron Boychuk’s Chuckle Brothers for the 23rd is a very slight thing to include, but it’s looking like a slow week. I need something here. If you don’t see it then things picked up. They similarly tried sprucing things up the 27th, with another joke for taping onto the door.

Nate Fakes’s Break of Day for the 24th features the traditional whiteboard full of mathematics scrawls as a sign of intelligence. The scrawl on the whiteboard looks almost meaningful. The integral, particularly, looks like it might have been copied from a legitimate problem in polar or cylindrical coordinates. I say “almost” because while I think that some of the r symbols there are r’ I’m not positive those aren’t just stray marks. If they are r’ symbols, it’s the sort of integral that comes up when you look at surfaces of spheres. It would be the electric field of a conductive metal ball given some charge, or the gravitational field of a shell. These are tedious integrals to solve, but fortunately after you do them in a couple of introductory physics-for-majors classes you can just look up the answers instead.

Samson’s Dark Side of the Horse for the 26th is the Roman numerals joke for this installment. I feel like it ought to be a pie chart joke too, but I can’t find a way to make it one.

Izzy Ehnes’s The Best Medicine Cartoon for the 27th is the anthropomorphic numerals joke for this paragraph.

## Getting Into Shapes

This is, in part, a post for myself. They all are, but this is moreso. My day job includes some Geographic Information Services stuff, which is how we say “maps” when we want to be taken seriously as Information Technology professionals. When we make maps, what we really do is have a computer draw polygons, and then put dots on them. A common need is to put a dot in the middle of a polygon. Yes, this sounds silly, but describe your job this abstractly and see how it comes out.

The trouble is polygons can be complicated stuff. Can be, not are. If the polygon is, like, the border of your building’s property it’s probably not too crazy. It’s probably a rectangle, or at least a trapezoid. Maybe there’s a curved boundary. If you need a dot, such as to place the street address or a description of the property, you can make a good guess about where to put it so it’s inside the property and not too close to an edge.

But you can’t always. The polygons can be complicated. Especially if you’re representing stuff that reflects government or scientific or commercial interest. There’s good reasons to be interested in the boundaries between the low-low tide and the high-high tide lines of a beach, but that’s not going to look like anything simple for any realistic property. Finding a representative spot to fix labels or other business gets tricky.

So this crossed my Twitter feed and I’ll probably want to refer back to it at some point. It’s an algorithm, published last August by Vladimir Agafonkin at Mapbox, which uses some computation tricks to find a reasonable center.

The approach is, broadly, of a kind with many numerical methods. It tries to find an answer by taking a guess and then seeing if any obvious variations will make it a little better. If you can, then, repeat these variations. Eventually, usually, you’ll get to a pretty good answer. It may not be the exact best possible answer, but that’s all right. We accept that we’ll have a merely approximate answer, but we’ll get it more quickly than we otherwise would have. Often this is fine. Nobody will be upset that the label on a map would be “better” moved one pixel to the right if they get the map ten seconds faster. Optimization is often like that.

I have not tried putting this code into mine yet; I’ve just now read it and I have some higher-priority tasks at work. But I’m hoping to remember that this exists and to see whether I can use it.

## Dabbing and the Pythagorean Theorem

The picture explains itself nicely. Just a thought on an average day.

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I enjoyed this article from Fox Sports. Apparently, a French Precalculus textbook created a homework problem asking if football (soccer) superstar Paul Pogba is doing the perfect dab by creating two right triangles.

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## Reading the Comics, May 20, 2017: Major Computer Malfunction Week Edition

I was hit by a massive computer malfunction this week, the kind that forced me to buy a new computer and spend half a week copying stuff over from a limping hard drive and hoping it would maybe work if I held things just right. Mercifully, Comic Strip Master Command gave me a relatively easy week. No huge rush of mathematically-themed comic strips and none that are going to take a thousand words of writing to describe. Let’s go.

Sam Hepburn’s Questionable Quotebook for the 14th includes this week’s anthropomorphic geometry sketch underneath its big text block.

Eric the Circle for the 15th, this one by “Claire the Square”, is the rare Eric the Circle to show off properties of circles. So maybe that’s the second anthropomorphic geometry sketch for the week. If the week hadn’t been dominated by my computer woes that might have formed the title for this edition.

Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 15th puts a mathematician in mortal peril and leaves him there to die. As is traditional for this sort of puzzle the mathematician left a dying clue. (Mathematicians were similarly kind to their investigators on the 4th of July, 2016 and the 9th of July, 2012. I was expecting the answer to be someone with a four-letter and an eight-letter name, none of which anybody here had. Doesn’t matter. It’ll never stand up in court.

John Graziano’s Ripley’s Believe It Or Not for the 17th features one of those astounding claims that grows out of number theory. Graziano asserts that there are an astounding 50,613,244,155,051,856 ways to score exactly 100 points in (ten-pin) bowling. I won’t deny that this seems high to me. But partitioning a number — that is, taking a (positive) whole number and writing down the different ways one can add up (positive) whole numbers to get that sum — often turns up a lot of possibilities. That there should be many ways to get a score of 100 by adding between ten and twenty numbers that could be between zero and ten each, plus the possibility of adding pairs of the numbers (for spares) or trios of numbers (for strikes) makes this less astonishing.

Wikipedia led me to this page, from Balmoral Software, about all the different ways there are to score different numbers. The most surprising thing it reveals to me is that 100 isn’t even the score with the greatest number of possible scores. 77 is. There are 172,542,309,343,731,946 ways to score exactly 77 points. I agree this ought to make me feel better about my game. It doesn’t. It turns out there are, altogether, something like 5,726,805,883,325,784,576 possible different outcomes for a bowling game. And how we can tell that, given there’s no practical way to go and list all of them, is described at the end of the page.

The technique is called “divide and conquer”. There’s no way to list all the outcomes of ten frames of bowling, but there’s certainly a way to list all the outcomes of one. Or two. Or three. So, work out how many possible scores there would be in few enough frames you can handle that. Then combine these shortened games into one that’s the full ten frames. There’s some trouble in matching up the ends of the short games. A spare or a strike in the last frame of a shortened game means one has to account for the first or first two frames of the next one. But this is still an easier problem than the one we started with.

Bill Amend’s FoxTrot Classics for the 18th (rerun from the 25th of May, 2006) is your standard percentages and infinities joke. Really would have expected Paige’s mother to be wise to this game by now, but this sort of thing happens.

## Everything Interesting There Is To Say About Springs

I need another supplemental essay to get to the next part in Why Stuff Can Orbit. (Here’s the last part.) You probably guessed it’s about springs. They’re useful to know about. Why? That one killer Mystery Science Theater 3000 short, yes. But also because they turn up everywhere.

Not because there are literally springs in everything. Not with the rise in anti-spring political forces. But what makes a spring is a force that pushes something back where it came from. It pushes with a force that grows just as fast as the distance from where it came grows. Most anything that’s stable, that has some normal state which it tends to look like, acts like this. A small nudging away from the normal state gets met with some resistance. A bigger nudge meets bigger resistance. And most stuff that we see is stable. If it weren’t stable it would have broken before we got there.

(There are exceptions. Stable is, sometimes, about perspective. It can be that something is unstable but it takes so long to break that we don’t have to worry about it. Uranium, for example, is dying, turning slowly into stable elements like lead and helium. There will come a day there’s none left in the Earth. But it takes so long to break down that, barring surprises, the Earth will have broken down into something else first. And it may be that something is unstable, but it’s created by something that’s always going on. Oxygen in the atmosphere is always busy combining with other chemicals. But oxygen stays in the atmosphere because life keeps breaking it out of other chemicals.)

Now I need to put in some terms. Start with your thing. It’s on a spring, literally or metaphorically. Don’t care. If it isn’t being pushed in any direction then it’s at rest. Or it’s at an equilibrium. I don’t want to call this the ideal or natural state. That suggests some moral superiority to one way of existing over another, and how do I know what’s right for your thing? I can tell you what it acts like. It’s your business whether it should. Anyway, your thing has an equilibrium.

Next term is the displacement. It’s how far your thing is from the equilibrium. If it’s really a block of wood on a spring, like it is in high school physics, this displacement is how far the spring is stretched out. In equations I’ll represent this as ‘x’ because I’m not going to go looking deep for letters for something like this. What value ‘x’ has will change with time. This is what makes it a physics problem. If we want to make clear that ‘x’ does depend on time we might write ‘x(t)’. We might go all the way and start at the top of the page with ‘x = x(t)’, just in case.

If ‘x’ is a positive number it means your thing is displaced in one direction. If ‘x’ is a negative number it was displaced in the opposite direction. By ‘one direction’ I mean ‘to the right, or else up’. By ‘the opposite direction’ I mean ‘to the left, or else down’. Yes, you can pick any direction you like but why are you making life harder for everyone? Unless there’s something compelling about the setup of your thing that makes another choice make sense just go along with what everyone else is doing. Apply your creativity and iconoclasm where it’ll make your life better instead.

Also, we only have to worry about one direction. This might surprise you. If you’ve played much with springs you might have noticed how they’re three-dimensional objects. You can set stuff swinging back and forth in two directions at once. That’s all right. We can describe a two-dimensional displacement as a displacement in one direction plus a displacement perpendicular to that. And if there’s no such thing as friction, they won’t interact. We can pretend they’re two problems that happen to be running on the same spring at the same time. So here I declare: we can ignore friction and pretend it doesn’t matter. We don’t have to deal with more than one direction at a time.

(It’s not only friction. There’s problems about how energy gets transmitted between ways the thing can oscillate. This is what causes what starts out as a big whack in one direction to turn into a middling little circular wobbling. That’s a higher level physics than I want to do right now. So here I declare: we can ignore that and pretend it doesn’t matter.)

Whether your thing is displaced or not it’s got some potential energy. This can be as large or as small as you like, down to some minimum when your thing is at equilibrium. The potential energy we represent as a number named ‘U’ because of good reasons that somebody surely had. The potential energy of a spring depends on the square of the displacement. We can write its value as ‘U = ½ k x2‘. Here ‘k’ is a number known as the spring constant. It describes how strongly the spring reacts; the bigger ‘k’ is, the more any displacement’s met with a contrary force. It’ll be a positive number. ½ is that same old one-half that you know from ideas being half-baked or going-off being half-cocked.

Potential energy is great. If you can describe a physics problem with its energy you’re in good shape. It lets us bring physical intuition into understanding things. Imagine a bowl or a Habitrail-type ramp that’s got the cross-section of your potential energy. Drop a little marble into it. How the marble rolls? That’s what your thingy does in that potential energy.

Also we have mathematics. Calculus, particularly differential equations, lets us work out how the position of your thing will change. We need one more piece for this. That’s the momentum of your thing. Momentum is traditionally represented with the letter ‘p’. And now here’s how stuff moves when you know the potential energy ‘U’:

$\frac{dp}{dt} = - \frac{\partial U}{\partial x}$

Let me unpack that. $\frac{dp}{dt}$ — also known as $\frac{d}{dt}p$ if that looks better — is “the derivative of p with respect to t”. It means “how the value of the momentum changes as the time changes”. And that is equal to minus one times …

You might guess that $\frac{\partial U}{\partial x}$ — also written as $\frac{\partial}{\partial x} U$ — is some kind of derivative. The $\partial$ looks kind of like a cursive d, after all. It’s known as the partial derivative, because it means we look at how ‘U’ changes as ‘x’ and nothing else at all changes. With the normal, ‘d’ style full derivative, we have to track how all the variables change as the ‘t’ we’re interested in changes. In this particular problem the difference doesn’t matter. But there are problems where it does matter and that’s why I’m careful about the symbols.

So now we fall back on how to take derivatives. This gives us the equation that describes how the physics of your thing on a spring works:

$\frac{dp}{dt} = - k x$

You’re maybe underwhelmed. This is because we haven’t got any idea how the momentum ‘p’ relates to the displacement ‘x’. Well, we do, because I know and if you’re still reading at this point you know full well what momentum is. But let me make it official. Momentum is, for this kind of thing, the mass ‘m’ of your thing times how its position is changing, which is $\frac{dx}{dt}$. The mass of your thing isn’t changing. If you’re going to let it change then we’re doing some screwy rocket problem and that’s a different article. So its easy to get the momentum out of that problem. We get instead the second derivative of the displacement with respect to time:

$m\frac{d^2 x}{dt^2} = - kx$

Fine, then. Does that tell us anything about what ‘x(t)’ is? Not yet, but I will now share with you one of the top secrets that only real mathematicians know. We will take a guess to what the answer probably is. Then we’ll see in what circumstances that answer could possibly be right. Does this seem ad hoc? Fine, so it’s ad hoc. Here is the secret of mathematicians:

It’s fine if you get your answer by any stupid method you like, including guessing and getting lucky, as long as you check that your answer is right.

Oh, sure, we’d rather you get an answer systematically, since a system might give us ideas how to find answers in new problems. But if all we want is an answer then, by definition, we don’t care where it came from. Anyway, we’re making a particular guess, one that’s very good for this sort of problem. Indeed, this guess is our system. A lot of guesses at solving differential equations use exactly this guess. Are you ready for my guess about what solves this? Because here it is.

We should expect that

$x(t) = C e^{r t}$

Here ‘C’ is some constant number, not yet known. And ‘r’ is some constant number, not yet known. ‘t’ is time. ‘e’ is that number 2.71828(etc) that always turns up in these problems. Why? Because its derivative is very easy to take, and if we have to take derivatives we want them to be easy to take. The first derivative of $Ce^{rt}$ with respect to ‘t’ is $r Ce^{rt}$. The second derivative with respect to ‘t’ is $r^2 Ce^{rt}$. so here’s what we have:

$m r^2 Ce^{rt} = - k Ce^{rt}$

What we’d like to find are the values for ‘C’ and ‘r’ that make this equation true. It’s got to be true for every value of ‘t’, yes. But this is actually an easy equation to solve. Why? Because the $C e^{rt}$ on the left side has to equal the $C e^{rt}$ on the right side. As long as they’re not equal to zero and hey, what do you know? $C e^{rt}$ can’t be zero unless ‘C’ is zero. So as long as ‘C’ is any number at all in the world except zero we can divide this ugly lump of symbols out of both sides. (If ‘C’ is zero, then this equation is 0 = 0 which is true enough, I guess.) What’s left?

$m r^2 = -k$

OK, so, we have no idea what ‘C’ is and we’re not going to have any. That’s all right. We’ll get it later. What we can get is ‘r’. You’ve probably got there already. There’s two possible answers:

$r = \pm\sqrt{-\frac{k}{m}}$

You might not like that. You remember that ‘k’ has to be positive, and if mass ‘m’ isn’t positive something’s screwed up. So what are we doing with the square root of a negative number? Yes, we’re getting imaginary numbers. Two imaginary numbers, in fact:

$r = \imath \sqrt{\frac{k}{m}}, r = - \imath \sqrt{\frac{k}{m}}$

Which is right? Both. In some combination, too. It’ll be a bit with that first ‘r’ plus a bit with that second ‘r’. In the differential equations trade this is called superposition. We’ll have information that tells us how much uses the first ‘r’ and how much uses the second.

You might still be upset. Hey, we’ve got these imaginary numbers here describing how a spring moves and while you might not be one of those high-price physicists you see all over the media you know springs aren’t imaginary. I’ve got a couple responses to that. Some are semantic. We only call these numbers “imaginary” because when we first noticed they were useful things we didn’t know what to make of them. The label is an arbitrary thing that doesn’t make any demands of the numbers. If we had called them, oh, “Cardanic numbers” instead would you be upset that you didn’t see any Cardanos in your springs?

My high-class semantic response is to ask in exactly what way is the “square root of minus one” any less imaginary than “three”? Can you give me a handful of three? No? Didn’t think so.

And then the practical response is: don’t worry. Exponentials raised to imaginary numbers do something amazing. They turn into sine waves. Well, sine and cosine waves. I’ll spare you just why. You can find it by looking at the first twelve or so posts of any pop mathematics blog and its article about how amazing Euler’s Formula is. Given that Euler published, like, 2,038 books and papers through his life and the fifty years after his death it took to clear the backlog you might think, “Euler had a lot of Formulas, right? Identities too?” Yes, he did, but you’ll know this one when you see it.

What’s important is that the displacement of your thing on a spring will be described by a function which looks like this:

$x(t) = C_1 e^{\sqrt{\frac{k}{m}} t} + C_2 e^{-\sqrt{\frac{k}{m}} t}$

for two constants, ‘C1‘ and ‘C2‘. These were the things we called ‘C’ back when we thought the answer might be $Ce^{rt}$; there’s two of them because there’s two r’s. I give you my word this is equivalent to a formula like this, but you can make me show my work if you must:

$x(t) = A cos\left(\sqrt{\frac{k}{m}} t\right) + B sin\left(\sqrt{\frac{k}{m}} t\right)$

for some (other) constants ‘A’ and ‘B’. Cosine and sine are the old things you remember from learning about cosine and sine.

OK, but what are ‘A’ and ‘B’?

Generically? We don’t care. Some numbers. Maybe zero. Maybe not. The pattern, how the displacement changes over time, will be the same whatever they are. It’ll be regular oscillation. At one time your thing will be as far from the equilibrium as it gets, and not moving toward or away from the center. At one time it’ll be back at the center and moving as fast as it can. At another time it’ll be as far away from the equilibrium as it gets, but on the other side. At another time it’ll be back at the equilibrium and moving as fast as it ever does, but the other way. How far is that maximum? What’s the fastest it travels?

The answer’s in how we started. If we start at the equilibrium without any kind of movement we’re never going to leave the equilibrium. We have to get nudged out of it. But what kind of nudge? There’s three ways you can do to nudge something out.

You can tug it out some and let it go from rest. This is the easiest: then ‘A’ is however big your tug was and ‘B’ is zero.

You can let it start from equilibrium but give it a good whack so it’s moving at some initial velocity. This is the next-easiest: ‘A’ is zero, and ‘B’ is … no, not the initial velocity. You need to look at what the velocity of your thing is at the start. That’s the first derivative:

$\frac{dx}{dt} = -\sqrt{\frac{k}{m}}A sin\left(\sqrt{\frac{k}{m}} t\right) + \sqrt{\frac{k}{m}} B sin\left(\sqrt{\frac{k}{m}} t\right)$

The start is when time is zero because we don’t need to be difficult. when ‘t’ is zero the above velocity is $\sqrt{\frac{k}{m}} B$. So that product has to be the initial velocity. That’s not much harder.

The third case is when you start with some displacement and some velocity. A combination of the two. Then, ugh. You have to figure out ‘A’ and ‘B’ that make both the position and the velocity work out. That’s the simultaneous solutions of equations, and not even hard equations. It’s more work is all. I’m interested in other stuff anyway.

Because, yeah, the spring is going to wobble back and forth. What I’d like to know is how long it takes to get back where it started. How long does a cycle take? Look back at that position function, for example. That’s all we need.

$x(t) = A cos\left(\sqrt{\frac{k}{m}} t\right) + B sin\left(\sqrt{\frac{k}{m}} t\right)$

Sine and cosine functions are periodic. They have a period of 2π. This means if you take the thing inside the parentheses after a sine or a cosine and increase it — or decrease it — by 2π, you’ll get the same value out. What’s the first time that the displacement and the velocity will be the same as their starting values? If they started at t = 0, then, they’re going to be back there at a time ‘T’ which makes true the equation

$\sqrt{\frac{k}{m}} T = 2\pi$

And that’s going to be

$T = 2\pi\sqrt{\frac{m}{k}}$

Maybe surprising thing about this: the period doesn’t depend at all on how big the displacement is. That’s true for perfect springs, which don’t exist in the real world. You knew that. Imagine taking a Junior Slinky from the dollar store and sticking a block of something on one end. Imagine stretching it out to 500,000 times the distance between the Earth and Jupiter and letting go. Would it act like a spring or would it break? Yeah, we know. It’s sad. Think of the animated-cartoon joy a spring like that would produce.

But this period not depending on the displacement is true for small enough displacements, in the real world. Or for good enough springs. Or things that work enough like springs. By “true” I mean “close enough to true”. We can give that a precise mathematical definition, which turns out to be what you would mean by “close enough” in everyday English. The difference is it’ll have Greek letters included.

So to sum up: suppose we have something that acts like a spring. Then we know qualitatively how it behaves. It oscillates back and forth in a sine wave around the equilibrium. Suppose we know what the spring constant ‘k’ is. Suppose we also know ‘m’, which represents the inertia of the thing. If it’s a real thing on a real spring it’s mass. Then we know quantitatively how it moves. It has a period, based on this spring constant and this mass. And we can say how big the oscillations are based on how big the starting displacement and velocity are. That’s everything I care about in a spring. At least until I get into something wild like several springs wired together, which I am not doing now and might never do.

And, as we’ll see when we get back to orbits, a lot of things work close enough to springs.

• #### tkflor 8:13 pm on Saturday, 20 May, 2017 Permalink | Reply

“I don’t want to call this the ideal or natural state. That suggests some moral superiority to one way of existing over another, and how do I know what’s right for your thing? ”
Why don’t you call it a “ground state”?

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• #### Joseph Nebus 6:49 am on Friday, 26 May, 2017 Permalink | Reply

You’re right. This is a ground state.

For folks just joining in, the “ground state” is what the system looks like when it’s got the least possible energy. At least the least energy consistent with it being a system at all. For a spring problem that’s the one where the thing is at rest, at the center, not displaced at all.

In a more complicated system you can have an equilibrium that’s stable and that isn’t the ground state. That isn’t the case here, but I wonder if thinking about that didn’t make me avoid calling it a ground state.

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## Reading the Comics, May 13, 2017: Quiet Tuesday Through Saturday Edition

From the Sunday and Monday comics pages I was expecting another banner week. And then there was just nothing from Tuesday on, at least not among the comic strips I read. Maybe Comic Strip Master Command has ordered jokes saved up for the last weeks before summer vacation.

Tony Cochrane’s Agnes for the 7th is a mathematics anxiety strip. It’s well-expressed, since Cochrane writes this sort of hyperbole well. It also shows a common attitude that words and stories are these warm, friendly things, while mathematics and numbers are cold and austere. Perhaps Agnes is right to say some of the problem is familiarity. It’s surely impossible to go a day without words, if you interact with people or their legacies; to go without numbers … well, properly impossible. There’s too many things that have to be counted. Or places where arithmetic sneaks in, such as getting enough money to buy a thing. But those don’t seem to be the kinds of mathematics people get anxious about. Figuring out how much change, that’s different.

I suppose some of it is familiarity. It’s easier to dislike stuff you don’t do often. The unfamiliar is frightening, or at least annoying. And humans are story-oriented. Even nonfiction forms stories well. Mathematics … has stories, as do all human projects. But the mathematics itself? I don’t know. There’s just beautiful ingenuity and imagination in a lot of it. I’d just been thinking of the just beautiful scheme for calculating logarithms from a short table. But it takes time to get to that beauty.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 7th is a fractions joke. It might also be a joke about women concealing their ages. Or perhaps it’s about mathematicians expressing things in needlessly complicated ways. I think that’s less a mathematician’s trait than a common human trait. If you’re expert in a thing it’s hard to resist the puckish fun of showing that expertise off. Or just sowing confusion where one may.

Daniel Shelton’s Ben for the 8th is a kid-doing-arithmetic problem. Even I can’t squeeze some deeper subject meaning out of it, but it’s a slow week so I’ll include the strip anyway. Sorry.

Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 8th is the return of anthropomorphic-geometry joke after what feels like months without. I haven’t checked how long it’s been without but I’m assuming you’ll let me claim that. Thank you.

• #### Joshua K. 4:53 am on Thursday, 18 May, 2017 Permalink | Reply

Perhaps the father in the “Ben” strip, rather than snoring, was telling his son about the set of integers.

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## Excuses, But Classed Up Some

Afraid I’m behind on resuming Why Stuff Can Orbit, mostly as a result of a power outage yesterday. It wasn’t a major one, but it did reshuffle all the week’s chores to yesterday when we could be places that had power, and kept me from doing as much typing as I wanted. I’m going to be riding this excuse for weeks.

So instead, here, let me pass this on to you.

It links to a post about the Legendre Transform, which is one of those cool advanced tools you get a couple years into a mathematics or physics major. It is, like many of these cool advanced tools, about solving differential equations. Differential equations turn up anytime the current state of something affects how it’s going to change, which is to say, anytime you’re looking at something not boring. It’s one of mathematics’s uses of “duals”, letting you swap between the function you’re interested in and what you know about how the function you’re interested in changes.

On the linked page, Jonathan Manton tries to present reasons behind the Legendre transform, in ways he likes better. It might not explain the idea in a way you like, especially if you haven’t worked with it before. But I find reading multiple attempts to explain an idea helpful. Even if one perspective doesn’t help, having a cluster of ideas often does.

## How April 2017 Treated My Mathematics Blog

Didn’t think I’d forget to post my readership review, did you? I just ran out of good posting slots for it last week, as I didn’t want to put off my return to Why Stuff Can Orbit more.

So, my page views dropped back below a thousand for the month. I knew it would happen sooner or later. It just barely missed a thousand, too: WordPress says I had 994 pages viewed over the month. It’s not significantly different to March’s 1,026, although given that I posted one more thing over the month than I did the previous month it hurts. At its rate I’d have had 1,027.1 views were April a 31-day month. That sort of excuse won’t fly; in so-short February there were 1,063 page views here. I’m just in a thousand-view groove right now.

The number of distinct visitors was technically down, but I can’t say it’s by anything significant. There were 696 visitors in April, down from 699 in March, and up from 680 in February. That’s no difference at all.

The number of likes were up slightly, to 90 in April. There had been 85 in March and 77 in February. There’s no disapproving of that trend, although as happened on my humor blog I noticed the long-term trend and realized my likability peaked sometime around June of 2015. That was the month of my first-ever A To Z project, but I’m surprised to not have had anything near that peak (518 likes!) since.

Not pictured: the rising curve of how much my Mom likes my posts.

Comments were flat again, 16 for April. There were 15 in March and 18 in February. I’m going to screw up my month-to-month comparisons again. I’m switching my links to previous posts back to the way that gets counted as a comment. So I’ll lose track of how often people actually say something versus how often I point people to my own writing. And I need to ask more engaging questions anyway.

So what articles were popular here in April? Two perennials, a couple of did-you-read-this bits, and then the answer to a trivia question people sometimes think to ask. Have some fun, if you’re a loyal reader: take your guesses what they are and see if you’re right. I’ll put the answers underneath the Big Geography Table, where I list the countries that sent readers here and how many they sent.

So the most-read stuff for April was:

So I make out that 45 separate countries sent me readers in April, down from 56 in March and 64 in February. “European Union” remains strangely absent. There were ten single-reader countries, down from 26 in March and 22 in February. Finland, Portugal, Switzerland, Thailand, and Vietnam were single-reader countries last month, and Vietnam is on a three-month streak.

The month started at 48,218 page views from 21,550 logged unique visitors. And I’m listed at having 655 followers on WordPress. You could be among them, by clicking the ‘Follow on WordPress’ button. There’s also a smaller number of e-mail followers, who followed by e-mail instead. There are advantages to following by e-mail, such as that then I don’t know if I’m read at all, and I can’t fix the typos and grammatical messes that I notice only after a post has gone live.

WordPress insights say the most popular day for readership around here in Sunday, when 16 percent of views come in. That’s what I would expect, except that in March the most popular day was Tuesday (18 percent of views), and in February it was Monday (16 percent). These are so close to one-seventh — 14.3 percent — that I figure there’s not any real difference. The most popular hour was that of 6 pm, which is when I normally schedule things to appear. 11 percent of page views came between 6 and 7 pm (Universal Time), down from March’s 12 percent but back to February’s number.

So allow me now to close with some of the search terms bringing people here:

• comic script of the apple by plato (Huh?)
• how many grooves are on a cd
• origin is the gateway to your entire gaming universe. (thank goodness)
• worst ways to pack (there’s a lot of room to do badly!)
• puzzle pool ball large table frictionless
• particle theory comic strip (fair enough)

Thanks for being around, though, and thanks for reading this. I’m hoping to get to some more interesting stuff in the Why Stuff Can Orbit series this month and then I’m figuring what I want my big summer project to be. Stick around for updates, please.

• #### Thys 1:40 pm on Wednesday, 10 May, 2017 Permalink | Reply

I’m one of the 7 from South Africa! :D

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• #### Joseph Nebus 6:05 pm on Thursday, 11 May, 2017 Permalink | Reply

Aw, nice to see you! Thanks. I just melt into the many United States readers myself.

Liked by 1 person

## Reading the Comics, May 2, 2017: Puzzle Week

If there was a theme this week, it was puzzles. So many strips had little puzzles to work out. You’ll see. Thank you.

Bill Amend’s FoxTrot for the 30th of April tries to address my loss of Jumble panels. Thank you, whoever at Comic Strip Master Command passed along word of my troubles. I won’t spoil your fun. As sometimes happens with a Jumble you can work out the joke punchline without doing any of the earlier ones. 64 in binary would be written 1000000. And from this you know what fits in all the circles of the unscrambled numbers. This reduces a lot of the scrambling you have to do: just test whether 341 or 431 is a prime number. Check whether 8802, 8208, or 2808 is divisible by 117. The integer cubed you just have to keep trying possibilities. But only one combination is the cube of an integer. The factorial of 12, just, ugh. At least the circles let you know you’ve done your calculations right.

Steve McGarry’s activity feature Kidtown for the 30th plays with numbers some. And a puzzle that’ll let you check how well you can recognize multiles of four that are somewhere near one another. You can use diagonals too; that’s important to remember.

Mac King and Bill King’s Magic in a Minute feature for the 30th is also a celebration of numerals. Enjoy the brain teaser about why the encoding makes sense. I don’t believe the hype about NASA engineers needing days to solve a puzzle kids got in minutes. But if it’s believable, is it really hype?

Marty Links’s Emmy Lou from the 29th of October, 1963 was rerun the 2nd of May. It’s a reminder that mathematics teachers of the early 60s also needed something to tape to their doors.

Mel Henze’s Gentle Creatures rerun for the 2nd of May is another example of the conflating of “can do arithmetic” with “intelligence”.

Mark Litzler’s Joe Vanilla for the 2nd name-drops the Null Hypothesis. I’m not sure what Litzler is going for exactly. The Null Hypothesis, though, comes to us from statistics and from inference testing. It turns up everywhere when we sample stuff. It turns up in medicine, in manufacturing, in psychology, in economics. Everywhere we might see something too complicated to run the sorts of unambiguous and highly repeatable tests that physics and chemistry can do — things that are about immediately practical questions — we get to testing inferences. What we want to know is, is this data set something that could plausibly happen by chance? Or is it too far out of the ordinary to be mere luck? The Null Hypothesis is the explanation that nothing’s going on. If your sample is weird in some way, well, everything is weird. What’s special about your sample? You hope to find data that will let you reject the Null Hypothesis, showing that the data you have is so extreme it just can’t plausibly be chance. Or to conclude that you fail to reject the Null Hypothesis, showing that the data is not so extreme that it couldn’t be chance. We don’t accept the Null Hypothesis. We just allow that more data might come in sometime later.

I don’t know what Litzler is going for with this. I feel like I’m missing a reference and I’ll defer to a finance blogger’s Reading the Comics post.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 3rd is another in the string of jokes using arithmetic as source of indisputably true facts. And once again it’s “2 + 2 = 5”. Somehow one plus one never rates in this use.

Aaron Johnson’s W T Duck rerun for the 3rd is the Venn Diagram joke for this week. It’s got some punch to it, too.

Je Mallett’s Frazz for the 5th took me some time to puzzle out. I’ll allow it.

## Why Stuff Can Orbit, Part 8: Introducing Stability

Previously:

I bet you imagined I’d forgot this series, or that I’d quietly dropped it. Not so. I’ve just been finding the energy for this again. 2017 has been an exhausting year.

With the last essay I finished the basic goal of “Why Stuff Can Orbit”. I’d described some of the basic stuff for central forces. These involve something — a planet, a mass on a spring, whatever — being pulled by the … center. Well, you can call anything the origin, the center of your coordinate system. Why put that anywhere but the place everything’s pulled towards? The key thing about a central force is it’s always in the direction of the center. It can be towards the center or away from the center, but it’s always going to be towards the center because the “away from” case is boring. (The thing gets pushed away from the center and goes far off, never to be seen again.) How strongly it’s pulled toward the center changes only with the distance from the center.

Since the force only changes with the distance between the thing and the center it’s easy to think this is a one-dimensional sort of problem. You only need the coordinate describing this distance. We call that ‘r’, because we end up finding orbits that are circles. Since the distance between the center of a circle and its edge is the radius, it would be a shame to use any other letter.

Forces are hard to work with. At least for a lot of stuff. We can represent central forces instead as potential energy. This is easier because potential energy doesn’t have any direction. It’s a lone number. When we can shift something complicated into one number chances are we’re doing well.

But we are describing something in space. Something in three-dimensional space, although it turns out we’ll only need two. We don’t care about stuff that plunges right into the center; that’s boring. We like stuff that loops around and around the center. Circular orbits. We’ve seen that second dimension in the angular momentum, which we represent as ‘L’ for reasons I dunno. I don’t think I’ve ever met anyone who did. Maybe it was the first letter that came to mind when someone influential wrote a good textbook. Angular momentum is a vector, but for these problems we don’t need to care about that. We can use an ordinary number to carry all the information we need about it.

We get that information from the potential energy plus a term that’s based on the square of the angular momentum divided by the square of the radius. This “effective potential energy” lets us find whether there can be a circular orbit at all, and where it’ll be. And it lets us get some other nice stuff like how the size of the orbit and the time it takes to complete an orbit relate to each other. See the earlier stuff for details. In short, though, we get an equilibrium, a circular orbit, whenever the effective potential energy is flat, neither rising nor falling. That happens when the effective potential energy changes from rising to falling, or changes from falling to rising. Well, if it isn’t rising and if it isn’t falling, what else can it be doing? It only does this for an infinitesimal moment, but that’s all we need. It also happens when the effective potential energy is flat for a while, but that like never happens.

Where I want to go next is into closed orbits. That is, as the planet orbits a sun (or whatever it is goes around whatever it’s going around), does it come back around to exactly where it started? Moving with the same speed in the same direction? That is, does the thing orbit like a planet does?

(Planets don’t orbit like this. When you have three, or more, things in the universe the mathematics of orbits gets way too complicated to do exactly. But this is the thing they’re approximating, we hope, well.)

To get there I’ll have to put back a second dimension. Sorry. Won’t need a third, though. That’ll get named θ because that’s our first choice for an angle. And it makes too much sense to describe a planet’s position as its distance from the center and the angle it makes with respect to some reference line. Which reference line? Whatever works for you. It’s like measuring longitude. We could measure degrees east and west of some point other than Greenwich as well, and as correctly, as we do. We use the one we use because it was convenient.

Along the way to closed orbits I have to talk about stability. There are many kinds of mathematical stability. My favorite is called Lyapunov Stability, because it’s such a mellifluous sound. They all circle around the same concept. It’s what you’d imagine from how we use the word in English. Start with an equilibrium, a system that isn’t changing. Give it a nudge. This disrupts it in some way. Does the disruption stay bounded? That is, does the thing still look somewhat like it did before? Or does the disruption grow so crazy big we have no idea what it’ll ever look like again? (A small nudge, by the way. You can break anything with a big enough nudge; that’s not interesting. It’s whether you can break it with a small nudge that we’d like to know.)

One of the ways we can study this is by looking at the effective potential energy. By its shape we can say whether a central-force equilibrium is stable or not. It’s easy, too, as we’ve got this set up. (Warning before you go passing yourself off as a mathematical physicist: it is not always easy!) Look at the effective potential energy versus the radius. If it has a part that looks like a bowl, cupped upward, it’s got a stable equilibrium. If it doesn’t, it doesn’t have a stable equilibrium. If you aren’t sure, imagine the potential energy was a track, like for a toy car. And imagine you dropped a marble on it. If you give the marble a nudge, does it roll to a stop? If it does, stable. If it doesn’t, unstable.

A phony effective potential energy. Most are a lot less exciting than this; see some of the earlier pieces in this series. But some weird-shaped functions like this were toyed with by physicists in the 19th century who were hoping to understand chemistry. Why should gases behave differently at different temperatures? Why should some combinations of elements make new compounds while others don’t? We needed statistical mechanics and quantum mechanics to explain those, but we couldn’t get there without a lot of attempts and failures at explaining it with potential energies and classical mechanics.

Stable is more interesting. We look at cases where there is this little bowl cupped upward. If we have a tiny nudge we only have to look at a small part of that cup. And that cup is going to look an awful lot like a parabola. If you don’t remember what a parabola is, think back to algebra class. Remember that curvey shape that was the only thing drawn on the board when you were dealing with the quadratic formula? That shape is a parabola.

Who cares about parabolas? We care because we know something good about them. In this context, anyway. The potential energy for a mass on a spring is also a parabola. And we know everything there is to know about masses on springs. Seriously. You’d think it was all physics was about from like 1678 through 1859. That’s because it’s something calculus lets us solve exactly. We don’t need books of complicated integrals or computers to do the work for us.

So here’s what we do. It’s something I did not get clearly when I was first introduced to these concepts. This left me badly confused and feeling lost in my first physics and differential equations courses. We are taking our original physics problem and building a new problem based on it. This new problem looks at how big our nudge away from the equilibrium is. How big the nudge is, how fast it grows, how it changes in time will follow rules. Those rules will look a lot like those for a mass on a spring. We started out with a radius that gives us a perfectly circular orbit. Now we get a secondary problem about how the difference between the nudged and the circular orbit changes in time.

That secondary problem has the same shape, the same equations, as a mass on a spring does. A mass on a spring is a central force problem. All the tools we had for studying central-force problems are still available. There is a new central-force problem, hidden within our original one. Here the “center” is the equilibrium we’re nudged around. It will let us answer a new set of questions.

## Reading the Comics, April 29, 2017: The Other Half Of The Week Edition

I’d been splitting Reading the Comics posts between Sunday and Thursday to better space them out. But I’ve got something prepared that I want to post Thursday, so I’ll bump this up. Also I had it ready to go anyway so don’t gain anything putting it off another two days.

Bill Amend’s FoxTrot Classics for the 27th reruns the strip for the 4th of May, 2006. It’s another probability problem, in its way. Assume Jason is honest in reporting whether Paige has picked his number correctly. Assume that Jason picked a whole number. (This is, I think, the weakest assumption. I know Jason Fox’s type and he’s just the sort who’d pick an obscure transcendental number. They’re all obscure after π and e.) Assume that Jason is equally likely to pick any of the whole numbers from 1 to one billion. Then, knowing nothing about what numbers Jason is likely to pick, Paige would have one chance in a billion of picking his number too. Might as well call it certainty that she’ll pay a dollar to play the game. How much would she have to get, in case of getting the number right, to come out even or ahead? … And now we know why Paige is still getting help on probability problems in the 2017 strips.

Jeff Stahler’s Moderately Confused for the 27th gives me a bit of a break by just being a snarky word problem joke. The student doesn’t even have to resist it any.

Sandra Bell-Lundy’s Between Friends for the 29th of April, 2017. And while it’s not a Venn Diagram I’m not sure of a better way to visually represent that the cartoonist is going for. I suppose the intended meaning comes across cleanly enough and that’s the most important thing. It’s a strange state of affairs is all.

Sandra Bell-Lundy’s Between Friends for the 29th also gives me a bit of a break by just being a Venn Diagram-based joke. At least it’s using the shape of a Venn Diagram to deliver the joke. It’s not really got the right content.

Harley Schwadron’s 9 to 5 for the 29th is this week’s joke about arithmetic versus propaganda. It’s a joke we’re never really going to be without again.

## Reading the Comics, April 24, 2017: Reruns Edition

I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.

Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.

One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.

So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.

There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.

So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.

Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.

Ted Shearer’s Quincy from the 28th of February, 1978. So, that FoxTrot problem I did? The conceptually-easy-but-tedious way is not too hard to do if you have a calculator. It’s a buch of typing but nothing more. If you don’t have a calculator, though, the desire not to do a whole bunch of calculating could drive you to the conceptually-harder-but-less-work answer. Is that a good thing? I suppose; insight is a good thing to bring. But the less-work answer only works because of a quirk in the problem, that Mary Lou is supposed to have a 50 percent chance of getting a question right. The low-insight-but-tedious problem will aways work. Why skip on having something to do the tedious part?

Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.

Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?

Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.

I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.

## Reading the Comics, April 22, 2017: Thought There’d Be Some More Last Week Edition

Allison Barrows’s PreTeena rerun for the 18th is a classic syllogism put into the comic strip’s terms. The thing about these sorts of deductive-logic syllogisms is that whether the argument is valid depends only on the shape of the argument. It has nothing to do with whether the thing being discussed makes any sense. This can be disorienting. It’s hard to ignore the everyday meaning of words when you hear a string of sentences. But it’s also hard to parse a string of sentences if the words don’t make sense in them. This is probably part of why on the mathematics side of things logic courses will skimp on syllogisms, using them to give an antique flavor and sense of style to the introduction of courses. It’s easier to use symbolic representations for logic instead.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 20th is the old joke about arithmetic being different between school, government, and corporate work. I haven’t looked at the comments — the GoComics redesign, whatever else it does, makes it very easy to skip the comments — but I’m guessing by the second one someone’s said the Common Core method means getting the most wrong answer.

Bil Keane and Jeff Keane’s Family Circus for the 21st of April, 2017. In fairness, there aren’t a lot of things we need all of 6, 7, and 8 for and you can just use whatever one of those you’re good at for any calculations with the others. Promise.

Bil Keane and Jeff Keane’s Family Circus for the 21st I don’t know is a rerun. But a lot of them are these days. Anyway, it looks like a silly joke about how nice mathematics would be without numbers; Dolly has no idea. I can sympathize with being intimidated by numerals. At the risk of being all New Math-y, I wonder if she wouldn’t like arithmetic more if it were presented as a game. Like, here’s a couple symbols — let’s say * and | for a start, and then some rules. * and * makes *, but * and | makes |. Also | and * makes |. But | and | makes |*. And so on. This is binary arithmetic, disguised, but I wonder if making it look like something inconsequential would make it more pleasant to learn, and if that would transfer over to arithmetic with 1’s and 0’s. Normal, useful arithmetic would be harder to play like this. You’d need ten symbols that are easy to write that aren’t already numbers, letters, or common symbols. But I wonder if it’d be worth it.

Tom Thaves’s Frank and Ernest for the 22nd is provided for mathematics teachers who need something to tape to their door. You’re welcome.

## What Do I Need To Get A B This Semester? (May 2017 Edition)

That said, everyone wants numbers. So here’s my posts. This is the original, about how to calculate exactly the score you need on your final to get whatever course grade you want. It allows for different sorts of weighting and extra credit and all that. If you don’t want to worry about extra credit here are some tables for common final-exam weightings with which you can approximate your needed score.

Also: review the syllabus. Read and understand any study guides you have. Review the in-course exams and homework assignments. Eat regularly and sleep as fully as you can the week or so before the exam; you do not have any problems that sleep deprivation will make better.

(Yes, this post is early. The schools I’m loosely affiliated with started early this term.)

## Reading the Comics, April 18, 2017: Give Me Some Word Problems Edition

I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.

Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.

Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.

(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)

J C Duffy’s Lug Nuts for the 16th is a Pi Day joke that ran late.

Jef Mallett’s Frazz for the 17th starts a little thread about obsolete references in story problems. It’s continued on the 18th. I’m sympathetic in principle to both sides of the story problem debate.

Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, \$2.25 per pound had to be divided by three.

But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.

And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.

So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?

Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.

Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of April, 2017. Before you ask what exactly the old theory of avocado intelligence was remember that Edison Lee’s lab partner there is a talking rat. Just saying.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.

## In Which I Offer Excuses Instead Of Mathematics

I’d been hoping to get back into longer-form essays. And then the calculations I meant to do on one problem turned out more complicated than I’d wanted. And they’re hard to square with the approach I used in some earlier work. Not that the results I was looking at were wrong, mind, just that an approach I’d used as “convenient for this sort of problem” turned inconvenient here.

So while I have the whole piece back in the shop for re-thinking, which is harder than even thinking, let me give you some other stuff to read. Or look at. One is from regular Singaporean correspondent MathTuition88. If you know anything about topology it’s because you’ve heard about Möbius strips. Surfaces with a single side are neat, and form the base of 95 percent of all science fiction stories in which the mathematics is the fantastic element. Klein bottles are often mentioned as a four-dimensional analogue to the Möbius strip, a solid object with no distinguishable interior or exterior. And a Klein bottle can be divided into two Möbius strips. MathTuition88 showcases a picture about how to turn two strips into a bottle. Or at least the best approximation of a bottle we can do; the actual Klein bottle is a four-dimensional structure and we can just make a three-dimensional imitation of the thing.

For something a bit more vector-analytic Joe Heafner’s Tensor Time has an essay about vectors. It’s about Heafner’s dislike for the way some vector problems are presented. Some common and easy ways to solve vector equations lead to spurious solutions that have to be weeded out by ad hoc reasoning; can’t we do better? Heafner argues that we can and should. The suggested alternative looks a little stuffy, but as often happens, spending more time on the setup means one spends less time confused later on. Worth pondering.

And this is a late addition, but I couldn’t resist.

Now I have a new favorite first chapter for a calculus text.

• #### sheldonk2014 7:43 pm on Monday, 1 May, 2017 Permalink | Reply

Thank you for visiting Joseph
Sorry it has taken me so long to get back to you
But I know you understand
As Sheldon Always

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• #### Joseph Nebus 3:23 am on Tuesday, 9 May, 2017 Permalink | Reply

Oh yes. Nothing you ever need to apologize for. I don’t check in on everyone often enough myself. Just glad you’re here and basically all right.

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