Technical editor needed
June 20, 2025 at 9:39 AM by Dr. Drang
I read this article from Scientific American about the GBU-57/B, the “bunker buster” bomb that Donald Trump will… or won’t… or will… or won’t allow Israel to use on Iran’s Fordo nuclear facility. The facility is buried deep within a mountain, and the GBU-57/B is the only non-nuclear bomb that may be able to destroy it. The article is worth reading, but if you do, you’ll probably notice some obvious errors.
The first error is related to concrete, which is why I picked up on it. Here’s the passage:
According to a 2012 Congressional Research Service briefing, the GBU-57/B has been reported to burrow through 200 feet of concrete or bedrock with a density of 5,000 pounds per square inch (comparable to the strength of bridge decks or parking-garage slabs).
The 5,000 psi figure refers to the compressive strength of concrete, not its density. Back when I was a student, 5,000 psi was kind of on the strong side for commercially available concrete; now it’s a garden-variety strength, as suggested by the parenthetical comment. The compressive strength of intact rock is often much higher than this, but natural rock formations tend to have joints and other defects that reduce their strength. By the way, even if you don’t have much experience with concrete or rock, you should know that something’s fishy with this passage. Density is weight or mass per unit volume—it can’t be measured in pounds per square inch.
Later, we see this:
About one fifth of the warhead’s 5,342-pound total weight is made up of two explosives: 4,590 pounds of AFX-757 plus 752 pounds of PBXN-114.
Since the sum of the two explosive weights—4,590 lbs and 752 lbs—is equal to 5,342 lbs, it’s hard to see how their sum could be one-fifth of that total. I’m guessing the intention here is to say that the combined explosive weight is about one-fifth of the missile’s total weight, which is given earlier in the article as about 30,000 lbs.
There’s also a discussion of how the ogive shape of the missile’s nose gives it both good aerodynamic and good penetrating properties. There’s nothing wrong with this, but it suggests the shape is something special. It isn’t. The ogive shape is common in rockets, missiles, and bullets. Maybe the GBU-57/B’s ogive is unusual in some way, but if it is, the article doesn’t say so.
I should say that this article isn’t in the Scientific American magazine proper, it’s just on the web, and maybe web articles aren’t given the same scrutiny as print articles. It does seem odd, though, that piece coming out under the SciAm name is edited at the same level as a blog post.
A polygon puzzle that really isn’t
June 17, 2025 at 9:13 PM by Dr. Drang
This is another post about a puzzle in Scientific American. I confess that this and my previous post have just been placeholders, things that I’m putting here because the post I really want to write is giving me trouble. It started when I read this article in Ars Technica about dropping eggs. The more I thought about it—and the paper it’s based on—the more I felt I should say, and now I have a couple of weeks’s worth of notes and calculations that I’m struggling to organize. Posts like this are much easier to write, so here we are.
The puzzle is this one. There are eight regular polygons with increasing numbers of sides, triangle through decagon. The first seven have numbers in them, and you’re supposed to find the number that goes in the last one.
Because of the house of cards puzzle I discussed several months ago, I decided to set up a difference table, like this:
The numbers in the difference column are obviously a series of prime numbers, so I figured the next difference would be the next prime, 29, and therefore the missing number would be 129. But I had no clue as to what that had to do with polygons.
It turns out that the number for the triangle is the sum of the first three prime numbers, the number for the square is the sum of the first four prime numbers, and so on. The (slight) geometric aspect of the problem is the number of sides being the number of prime numbers you should add. This is what makes the difference table work out the way it does and how I got the right answer without really solving the problem.
If you’re wondering, yes, this sequence is in the Online Encyclopedia of Integer Sequences: A007504. Mathematicians really like their prime numbers.
Dodecagon star puzzle
June 8, 2025 at 3:19 PM by Dr. Drang
Looking through the June edition of Scientific American this morning, I saw this puzzle (that’s an Apple News link; here’s a regular one):
The regular dodecagon and the blue star inside it both have a side length of 1 unit. What is the area of the star?
There’s a clever way to solve this, which is how SciAm does it. I like clever solutions as much as the next person, but being an engineer, I go for brute force when a clever solution doesn’t come to me. In this case, that’s getting the area of the dodecagon and subtracting off the twelve light blue equilateral triangles.
There’s a relatively simple formula for the area of any n-sided regular polygon:
where s is the side length.
We can prove to ourselves that this works by checking some areas we already know. For an equilateral triangle, the area is
which may not be immediately familiar, but you’d get the same answer by doubling the area of a 30-60-90 triangle with a hypotenuse of s.
For a square, it’s
And for a regular hexagon, it’s
This is the area of six equilateral triangles, which is how you can form a regular hexagon.
So the area of the star is
which would be nice if I had the tangent of 15° on the tip of my tongue. Luckily, I do know the trig function values for 30° and the half-angle formulas for tangent are easy to find.
So
Because tangent is in the denominator of the area formula, I’m choosing the half-angle formula that puts the square root in its denominator. That’ll put the square root in the numerator of the area formula.
Therefore,
which is pretty darned simple and is obviously the reason the puzzle exists. That this is equal to the area of six unit squares is a decent clue to the clever solution.
You could argue that solving the problem by brute force makes it not a puzzle at all—the whole point of puzzles is to be clever. But doing it this way reminded me of some formulas I’d forgotten1 and, as I said, gave me a clue to the clever solution.
-
If you woke me from a dead sleep and asked me for the double angle formulas, I’d be able to tell you—at least for sine and cosine. But I’ve never had a good grasp on the half-angle formulas. ↩
Math symbol frequencies
June 4, 2025 at 5:54 PM by Dr. Drang
I checked out a copy of Raúl Rojas’s The Language of Mathematics: The Stories behind the Symbols at my local library this morning. As the subtitle says, it covers the history and eventual standardization of many many mathematical symbols. The book is several years old, but the English translation (by Eduardo Aparicio from the original Spanish) is new. I first read about it in this Scientific American article (that’s an Apple News link).
The book has nine chapters on different aspects of mathematics, and each chapter has several short sections covering one or two symbols. Rojas says in the introduction that the sections are more or less self-contained, so you can skip around to the symbols that most interest you. At least for now, I’m starting at the beginning and reading sequentially.
An early section that brought me up short was “How Do We Use Mathematical Symbols?” It includes this table, which shows the frequencies of the most-used symbols (20 identifiers and 20 operators) from a set of arXiv math papers and engineering textbooks.
I had never before seen anyone do this. It’s obviously modeled on the character and word frequency tables that are pretty common and which form the basis for dissociated press and similar computer diversions.1 The tables were built to help with the development of mathematical handwriting recognition software. Software like Apple’s newish Math Notes, but this frequency analysis was done 20 years ago.
I started going through the table, trying to explain to myself why the symbols were in this order, when I ran into some questionable entries, which I’ve highlighted.
First, there are two as in the first column. While I can understand a being a popular symbol, there’s no reason for it to be there twice. More curious were the boxes in the last pair of columns. One of them has an overbar, so that could mean any symbol with an overbar (although that should be an identifier, not an operator), but the other one is just a plain box. Because boxes are often used to fill in for a missing glyph, I began to wonder if symbols in the table weren’t present in the font used in the book. But it doesn’t make sense for a book about mathematical symbols to use a deficient font. Also, the title page says the book uses STIX Two, which has a pretty damned complete set of glyphs. There’s no way it’s lacking a top twenty symbol.
So off I went to the bibliography to see where this table came from. Two publications were listed for this section:
- “An Analysis of Mathematical Expressions Used in Practice” by Clare M. So.
- “Mathematical Document Classification via Symbol Frequency Analysis” by Stephen M. Watt.
So’s paper is her master’s thesis, and Watt was her thesis advisor. Watt is also one of the original authors of the MathML spec and a contributor to the Maple computer algebra system. A heavy hitter when it comes to math symbols.
So’s thesis was written in 2005 and does the analysis of the arXiv material (about 19,000 papers) but not the engineering textbooks. Watt’s paper was written a few years later; it excerpts So’s work and adds the engineering books. Or should I say “engineering” books.
Here are the three books Watt analyzed:2
- Advanced Engineering Mathematics by Erwin Kreysig (available free online through O’Reilly if your library gives you a subscription).
- Advanced Engineering Mathematics by Michael Greenberg.
- Advanced Engineering Mathematics by Peter O’Neil.
I can understand, I guess, why a mathematician and computer scientist would see these as engineering textbooks (engineering is right there in the title), but they’re really math books aimed at engineers. They cover about what you’d expect: ordinary and partial differential equations, vector calculus, linear algebra, complex analysis, and numerical analysis. But they’re not representative of the mathematics seen in engineering publications.
Anyway, Watt’s paper has the table that Rojas’s was taken from, and it’s easy to see where the anomalies crept in.
The second a in Rojas’s table should be an alpha, . Maybe a secretarial error; maybe an overzealous editor.
As for the boxes in the engineering section of the table, they’re not two different symbols, they’re a single symbol meant to represent the horizontal bar of a fraction. This is indeed an operator, and the boxes are whatever the numerator and denominator are.
In looking at Watt’s table, we also see that the semicolon in Rojas’s table (which I highlighted in yellow) is supposed to be two symbols: a period with 16,213 entries and a prime with 12,401 entries.
The specious semicolon raises a question about the comma and period. Should they be in these lists? Commas and periods appear in display equations in most properly punctuated mathematical texts, but they really aren’t mathematical operators. You see them in passages like this:
Newton’s second law can be expressed as an equation,
where is the force, is the mass, and is the acceleration.
If you’re entering that equation in LaTeX, you may type
Newton's second law can be expressed as an equation,
$$ F = ma, $$
where…
which puts the comma inside the code for the display equation, but it isn’t truly part of the equation. My guess is that this is why we see commas appearing in these lists.
(You may have noticed from previous posts that I don’t add punctuation to display equations here on ANIAT. I’ve always assumed context will tell the reader how the equation fits into its sentence, and I think it’s less confusing to leave the punctuation out when writing for an audience that doesn’t spend much time reading text with equations.)
Looks like I’ve gone pretty far afield here. But that’s what happens when you find something that’s both interesting and odd.