## How to Tell if a Point Is Inside a Shape

As I continue to approach readiness for the Little Mathematics A-to-Z, let me share another piece you might have missed. Back in 2016 somehow two A-to-Z’s wasn’t enough for me. I also did a string of “Theorem Thursdays”, trying to explain some interesting piece of mathematics. The Jordan Curve Theorem is one of them.

The theorem, at heart, seems too simple to even be mathematics. It says that a simple closed curve on the plane divides the plane into an inside and an outside. There are similar versions for surfaces in three-dimensional spaces. Or volumes in four-dimensional spaces and so on. Proving the theorem turns out to be more complicated than I could fit into an essay. But proving a simplified version, where the curve is a polygon? That’s doable. Easy, even.

And as a sideline you get an easy way to test whether a point is inside a shape. It’s obvious, yeah, if a point is inside a square. But inside a complicated shape, some labyrinthine shape? Then it’s not obvious, and it’s nice to have an easy test.

This is even mathematics with practical application. A few months ago in my day job I needed an automated way to place a label inside a potentially complicated polygon. The midpoint of the polygon’s vertices wouldn’t do. The shapes could be L- or U- shaped, so that the midpoint wasn’t inside, or was too close to the edge of another shape. Starting from the midpoint, though, and finding the largest part of the polygon near to it? That’s doable, and that’s the Jordan Curve Theorem coming to help me.

## How to Make a Transcendental Number

I am, believe it or not, working ahead of deadline on the Little Mathematics A-to-Z for this year. I feel so happy about that. But that’s eating up time to write fresh stuff here. So please let me share some older material, this from my prolific year 2016.

Transcendental numbers, which I describe at this link, are nearly all the real numbers. We’re able to prove that even though we don’t actually know very many of them. We know some numbers that we’re interested in, like π and $e$, are. And that this has surprising consequences. π being a transcendental number means, for example, the Ancient Greek geometric challenge to square the circle using straightedge and compass is impossible.

However, it’s not hard to create a number that you know is transcendental. Here’s how to do it, with an easy step-by-step guide. If you want to create this and declare it’s named after you, enjoy! Nobody but you will ever care about this number, I’m afraid. Its only interesting traits will be that it’s transcendental and that you crafted it. Still, isn’t that nice anyway? I think it’s nice anyway.

## How To Find A Logarithm Without Much Computing Power

I don’t yet have actual words committed to text editor for this year’s little A-to-Z yet. Soon, though. Rather than leave things completely silent around here, I’d like to re-share an old sequence about something which delighted me. A lon while ago I read Edmund Callis Berkeley’s Giant Brains: Or Machines That Think. It’s a book from 1949 about numerical computing. And it explained just how to really calculate logarithms.

Anyone who knows calculus knows, in principle, how to calculate a logarithm. I mean as in how to get a numerical approximation to whatever the log of 25 is. If you didn’t have a calculator that did logarithms, but you could reliably multiply and add numbers? There’s a polynomial, one of a class known as Taylor Series, that — if you add together infinitely many terms — gives the exact value of a logarithm. If you only add a finite number of terms together, you get an approximation.

That suffices, in principle. In practice, you might have to calculate so many terms and add so many things together you forget why you cared what the log of 25 was. What you want is how to calculate them swiftly. Ideally, with as few calculations as possible. So here’s a set of articles I wrote, based on Berkeley’s book, about how to do that.

Machines That Think About Logarithms sets out the question. It includes some talk about the kinds of logarithms and why we use each of them.

Machines That Do Something About Logarithms sets out principles. These are all things that are generically true about logarithms, including about calculating logarithms.

Machines That Give You Logarithms explains how to use those tools. And lays out how to get the base-ten logarithm for most numbers that you would like with a tiny bit of computing work. I showed off an example of getting the logarithm of 47.2286 using only three divisions, four additions, and a little bit of looking up stuff.

Without Machines That Think About Logarithms closes it out. One catch with the algorithm described is that you need to work out some logarithms ahead of time and have them on hand, ready to look up. They’re not ones that you care about particularly for any problem, but they make it easier to find the logarithm you do want. This essay talks about which logarithms to calculate, in order to get the most accurate results for the logarithm you want, using the least custom work possible.

And that’s the series! With that, in principle, you have a good foundation in case you need to reinvent numerical computing.

## Is this mathematics thing ambiguous or confusing?

There is an excellent chance it is! Mathematicians sometimes assert the object of their study is a universal truth, independent of all human culture. It may be. But the expression of that interest depends on the humans expressing it. And as with all human activities it picks up quirks. Patterns that don’t seem to make sense. Or that seem to conflict with other patterns. It’s not two days ago I most recently saw someone cross that 0 times anything is 0, but 0! is 1.

Mathematicians are not all of one mind. They notice different things that seem important and want to focus on that. They use ways that make sense to their culture. When they create new notation, or new definitions, they use the old ones to guide them. When a topic’s interesting enough for many people to notice, they bring many trails of notation to describe it. Usually a consensus emerges, that there are some notations that work well to describe these concepts, and the others fall away. But it’s difficult to get complete consistency. Particularly when there are several major fields that don’t need to interact much, but do have some overlap.

Christian Lawson-Perfect has started something that might be helpful for understanding this. WhyStartAt.xyz is to be a collection of “ambiguous, inconsistent, or just plain unpleasant conventions in mathematical notation”. There’s four major categories already: inconsistencies, ambiguities, unpleasantness, and conflicting definitions. And there’s a set of references useful for anyone curious why something is a convention. (Nobody knows why we use ‘m’ for the slope in the slope-intercept or point-slope equations describing a line. Sometimes a convention is arbitrary.) It’s already great reading, though, not just for this line from our friend Thomas Hobbes.

## How June 2021 Treated My Mathematics Blog

It’s the time of month when I like to look at what my popularity is like. How many readers I had, what they were reading, that sort of thing. And I’m even getting to it earlier than usual in the month of July. Credit a hot Sunday when I can’t think of other things to do instead.

According to WordPress there were 2,507 page views here in June 2021. That’s down from the last couple months. But it is above the twelve-month running mean, leading up to June, which was of 2,445.9 views per month. The twelve-month running median was 2,516.5. This all implies that June was quite in line with my average month from June 2020 through May 2021. It just looks like a decline is all.

There were 1,753 unique visitors recorded by WordPress in June. That again fits between the running averages. There were a mean 1,728.4 unique visitors per month between June 2020 and May 2021. There was a median of 1,800 unique visitors each month over that same range.

The number of likes given collapsed, a mere 36 clicks of the like button given in June compared to a mean of 57.3 and median of 55.5. Given how many of my posts were some variation of “I’m struggling to find the energy to write”? I can’t blame folks not finding the energy to like. Comments were up, though, surely in response to my appeal for Mathematics A-to-Z topics. If you’ve thought of any, please, let me know; I’m eager to know.

I had nine essays posted in June, including my readership review post. These were, in the order most-to-least popular (as measured by page views):

In June I posted 7,852 words, my most verbose month since October 2020. That comes to an average of 981.5 words per posting in June. But the majority of them were in a single post, the exploration of MLX, which shows how the mean can be a misleading measure. This does bring my words-per-posting mean for the year up to 622, an increase of 70 words per posting. I need to not do that again.

As of the start of July I’ve had 1,631 posts here, which gathered 138,286 total views from 81,404 logged unique visitors.

If you’d like to be a regular reader, this is a great time for it, as I’ve almost worked my way through my obsession with checksum routines of 1980s computer magazines! And there’s the A-to-Z starting soon. Each year I do a glossary project, writing essays about mathematics terms from across the dictionary, many based on reader suggestions. All 168 essays from past years are at this link. This year’s should join that set, too.

If you’d like to get new posts without typos corrected, you can sign up for e-mail delivery. Or if you have a WordPress account, you can use “Follow NebusResearch” to add this page to your Reader. And I am @nebusj@mathstodon.xyz, the mathematics-themed instance of the Mastodon network. Thanks for reading, however you find most comfortable.

## How did Compute!’s Automatic Proofreader Work?

After that work on MLX, the programs that Compute! and Compute!’s Gazette used to enter machine language programs, I figured I was done. There was the Automatic Proofreader, used to catch errors in typing in BASIC programs. But that program was written in the machine language of the 6502 line of microchip. I’ve never been much on machine language and figured I couldn’t figure out how it worked. And then on a lark I tried and saw. And it turned out to be easy.

With qualifiers, of course. Compute! and Compute!’s Gazette had two generations of Automatic Proofreader for Commodore computers. The magazines also had Automatic Proofreaders for the other eight-bit computers that they covered. I trust that those worked the same way, but — with one exception — don’t know. I haven’t deciphered most of those other proofreaders.

Let me introduce how it was used, though. Compute! and Compute!’s Gazette offered computer programs to type in. Many of them were in BASIC, which uses many familiar words of English as instructions. But you can still make typos entering commands, and this causes bugs or crashes in programs. The Automatic Proofreader, for the Commodore (and the Atari), put in a little extra step after you typed in a line of code. It calculated a checksum. It showed that on-screen after every line you entered. And you could check whether that matched the checksum the magazine printed. So the listing in the magazine would be something like:

100 POKE 56,50:CLR:DIM IN$,I,J,A,B,A$,B$,A(7),N$ :rem 34
110 C4=48:C6=16:C7=7:Z2=2:Z4=254:Z5=255:Z6=256:Z7=127 :rem 238
120 FA=PEEK(45)+Z6*PEEK(46): BS=PEEK(55)+Z6*PEEK(56): H$="0123456789ABCDEF" :rem118  You would type in all those lines up to the :rem part. ‘rem’ here stands for ‘Remark’ and means the rest of the line is a comment to the programmer, not the computer. So they’d do no harm if you did enter them. But why type text you didn’t need? So after typing, say, 100 POKE 56,50:CLR:DIM IN$,I,J,A,B,A$,B$,A(7),N$you’d hit return and with luck get the number 34 up on screen. The Automatic Proofreader did not force you to re-type the line. You were on your honor to do that. (Nor were you forced to type lines in order. If you wished to type line 100, then 200, then 300, then 190, then 250, then 330, you could. The checksum would calculate the same.) And it didn’t only work for entering programs, these commands starting with line numbers. It would return a result for any command you entered. But since you wouldn’t know what the checksum should be for a freeform command, that didn’t tell you much. The first-generation Automatic Proofreader, which is what I’m talking about here, returned a number between 0 and 255. And it was a simple checksum. It could not detect transposed characters: the checksum for PIRNT was the same as PRINT and PRITN. And, it turns out, errors could offset: the checksum for PEEK(46) would be the same as that for PEEK(55). And there was one bit of deliberate insensitivity built in. Spaces would not be counted. The checksum for FA=PEEK(45)+Z6*PEEK(46) would be the same as FA = PEEK( 45 ) + Z6 * PEEK( 46 ). So you could organize text in whatever way was most convenient. Given this, and given the example of the first MLX, you may have a suspicion how the Automatic Proofreader calculated things. So did I and it turned out to be right. The checksum for the first-generation Automatic Proofreader, at least for the Commodore 64 and the Vic-20, was a simple sum. Take the line that’s been entered. Ignore spaces. But otherwise, take the ASCII code value for each character, and add that up, modulo 256. That is, if the sum is (say) 300, subtract 256 from that, that is, 44. I’m fibbing a little when I say it’s the ASCII code values. The Commodore computers used a variation on ASCII, called PETSCII (Commodore’s first line of computers was the PET). For ordinary text the differences between ASCII and PETSCII don’t matter. The differences come into play for various characters Commodores had. These would be symbols like the suits of cards, or little circles, or checkerboard patterns. Symbols that, these days, we’d see as emojis, or at least part of an extended character set. But translating all those symbols is … tedious, but not hard. If you want to do a simulated Automatic Proofreader in Octave, it’s almost no code at all. It turns out Octave and Matlab need no special command to get the ASCII code equivalent of text. So here’s a working simulation function retval = automatic_proofreader (oneLine) trimmedLine = strrep(oneLine, " ", ""); # In Matlab this should be replace(oneLine, " ", ""); retval = mod(sum(trimmedLine), 256); endfunction To call it type in a line of text: automatic_proofreader("100 POKE 56,50:CLR:DIM IN$,I,J,A,B,A$,B$,A(7),N$")  Capitalization matters! The ASCII code for capital-P is different from that for lowercase-p. Spaces won’t matter, though. More exotic characters, though, such as the color-setting commands, are trouble and let’s not deal with that right now. Also you can enclose your line in single-quotes, in case for example you want the checksum of a line that had double-quotes. Let’s agree that lines with single- and double-quotes don’t exist. I understand the way Commodore 64’s work well enough that I can explain the Automatic Proofreader’s code. I plan to do that soon. I don’t know how the Atari version of the Automatic Proofreader worked, but since it had the same weaknesses I assume it used the same algorithm. There is a first-generation Automatic Proofreader with a difference, though, and I’ll come to that. ## History of Philosophy podcast has another episode on Nicholas of Cusa A couple weeks ago I mentioned that Peter Adamson’s The History of Philosophy Without Any Gaps had an episode about Nicholas of Cusa. Last week the podcast had another one, a half-hour interview with Paul Richard Blum about him and his work. As with the previous podcast, there’s almost no mention of Nicholas of Cusa’s mathematics work. On the other hand, if you learn the tiniest possible bit about Nicholas of Cusa, you learn everything there is to know about Nicholas of Cusa. (I believe this joke would absolutely kill with the right audience, and will hear nothing otherwise.) The St Andrews Maths History site has a biography focusing particularly on his mathematical work. I’m sorry not to be able to offer more about his mathematical work. If someone knows of a mathematics-history podcast with a similar goal, please leave a comment. I’d love to know and to share with other people. ## I’m looking for topics for the Little 2021 Mathematics A-to-Z I’d like to say I’m ready to start this year’s Mathematics A-to-Z. I’m not sure I am. But if I wait until I’m sure, I’ve learned, I wait too long. As mentioned, this year I’m doing an abbreviated version of my glossary project. Rather than every letter in the alphabet, I intend to write one essay each for the letters in “Mathematics A-to-Z”. The dashes won’t be included. While I have some thoughts in minds for topics, I’d love to know what my kind readers would like to see me discuss. I’m hoping to write about one essay, of around a thousand words, per week. One for each letter. The topic should be anything mathematics-related, although I tend to take a broad view of mathematics-related. (I’m also open to biographical sketches.) To suggest something, please, say so in a comment. If you do, please also let me know about any projects you have — blogs, YouTube channels, real-world projects — that I should mention at the top of that essay. To keep things manageable, I’m looking for the first couple letters — MATH — first. But if you have thoughts for later in the alphabet please share them. I can keep track of that. I am happy to revisit a subject I think I have more to write about, too. Past essays for these letters that I’ve written include: ## M. ## A. ## T. ## H. The reason I wrote a second Tiling essay is because I forgot I’d already written one in 2018. I hope not to make that same mistake again. But I am open to repeating a topic, or a variation of a topic, on purpose.. ## Here’s some Matlab/Octave code for your MLX simulator I am embarrassed that after writing 72,650 words about MLX 2.0 for last week, I left something out. Specifically, I didn’t include code for your own simulation of the checksum routine on a more modern platform. Here’s a function that carries out the calculations of the Commodore 64/128 or Apple II versions of MLX 2.0. It’s written in Octave, the open-source Matlab-like numerical computation routine. If you can read this, though, you can translate it to whatever language you find convenient. function [retval] = mlxII (oneline) z2 = 2; z4 = 254; z5 = 255; z6 = 256; z7 = 127; address = oneline(1); entries = oneline(2:9); checksum = oneline(10); ck = 0; ck = floor(address/z6); ck = address-z4*ck + z5*(ck>z7)*(-1); ck = ck + z5*(ck>z5)*(-1); # # This looks like but is not the sum mod 255. # The 8-bit computers did not have a mod function and # used this subtraction instead. # for i=1:length(entries), ck = ck*z2 + z5*(ck>z7)*(-1) + entries(i); ck = ck + z5*(ck>z5)*(-1); endfor # # The checksum *can* be 255 (0xFF), but not 0 (0x00)! # Using the mod function could make zeroes appear # where 255's should. # retval = (ck == checksum); endfunction  This reproduces the code as it was actually coded. Here’s a version that relies on Octave or Matlab’s ability to use modulo operations: function [retval] = mlxIIslick (oneline) factors = 2.^(7:-1:0); address = oneline(1); entries = oneline(2:9); checksum = oneline(10); ck = 0; ck = mod(address - 254*floor(address/256), 255); ck = ck + sum(entries.*factors); ck = mod(ck, 255); ck = ck + 255*(ck == 0); retval = (ck == checksum); endfunction  Enjoy! Please don’t ask when I’ll have the Automatic Proofreader solved. ## How did Compute!’s and Compute!’s Gazette’s New MLX Work? A couple months ago I worked out a bit of personal curiosity. This was about how MLX worked. MLX was a program used in Compute! and Compute!’s Gazette magazine in the 1980s, so that people entering machine-language programs could avoid errors. There were a lot of fine programs, some of them quite powerful, free for the typing-in. The catch is this involved typing in a long string of numbers, and if any were wrong, the program wouldn’t work. So MLX, introduced in late 1983, was a program to make typing in programs better. You would enter in a string of six numbers — six computer instructions or data — and a seventh, checksum, number. Back in January I worked out finally what the checksum was. It turned out to be simple. Take the memory location of the first of your set of six instructions, modulo 256. Add to it each of the six instructions, modulo 256. That’s the checksum. If it doesn’t match the typed-in checksum, there’s an error. There’s weaknesses to this, though. It’s vulnerable to transposition errors: if you were supposed to type in 169 002 and put in 002 169 instead, it wouldn’t be caught. It’s also vulnerable to casual typos: 141 178 gives the same checksum as 142 177. Which is all why the original MLX lasted only two years. # What Was The New MLX? The New MLX, also called MLX 2.0, appeared first in the June 1985 Compute!. This in a version for the Apple II. Six months later a version for the Commodore 64 got published, again in Compute!, though it ran in Compute!’s Gazette too. Compute! was for all the home computers of the era; Compute!’s Gazette specialized in the Commodore computers. I would have sworn that MLX got adapted for the Atari eight-bit home computers too, but can’t find evidence it ever was. By 1986 Compute! was phasing out its type-in programs and didn’t run much for Atari anymore. The new MLX made a bunch of changes. Some were internal, about how to store a program being entered. One was dramatic in appearance. In the original MLX people typed in decimal numbers, like 32 or 169. In the new, they would enter hexadecimal digits, like 20 or A9. And a string of eight numbers on a line, rather than six. This promised to save our poor fingers. Where before we needed to type in 21 digits to enter six instructions, now we needed 18 digits to enter eight instructions. So the same program would take about two-thirds the number of keystrokes. A plausible line of code would look something like: 0801:0B 08 00 00 9E 32 30 36 EC 0809:31 00 00 00 A9 00 8D 20 3A 0811:D0 20 CF 14 20 1B 08 4C 96 0819:C7 0B A9 93 20 D2 FF A9 34  (This from the first lines for “Q-Bird”, a game published in the December 1986 Compute!’s Gazette.) And, most important, there was a new checksum. # What was the checksum formula? I had a Commodore 64, so I always knew MLX from its Commodore version. The key parts of the checksum code appear in it in lines 350 through 390. Let me copy out the key code, spaced a bit out for easier reading: 360 A = INT(AD/Z6): GOSUB 350: A = AD - A*Z6: GOSUB 350: PRINT":"; 370 CK = INT(AD/Z6): CK = AD - Z4*CK + Z5*(CK>27): GOTO 390 380 CK = CK*Z2 + Z5*(CK>Z7) + A 390 CK = CK + Z5*(CK>Z5): RETURN  Z2, Z4, Z5, Z6, and Z7 are constants, defined at the start of the program. Z4 equals 254, Z5 equals 255, Z6 equals 256, and Z7, as you’d expect, is 127. Z2, meanwhile, was a simple 2. A bit of Commodore BASIC here. INT means to take the largest whole number not larger than whatever’s inside. AD is the address of the start of the line being entered. CK is the checksum. A is one number, one machine language instruction, being put in. GOSUB, “go to subroutine”, means to jump to another line and execute commands from there, and then RETURN. That’s the command. The program then continues from the next instruction after the GOSUB. In this code, line 350 converts a number from decimal to hexadecimal and prints out the hexadecimal version. This bit about adding Z5 * (CK>Z7) looks peculiar. Commodore BASIC evaluates logical expressions like CK > 27 into a bit pattern. That pattern looks like a number. We can use it like an integer. Many programming languages do something like that and it can allow for clever but cryptic programming tricks. An expression that’s false evaluates as 0; an expression that’s true evaluates as -1. So, CK + Z5*(CK>Z5) is an efficient little filter. If CK is smaller than Z5, it’s left untouched. If CK is larger than Z5, then subtract Z5 from CK. This keeps CK from being more than 255, exactly as we’d wanted. But you also notice: this code makes no sense. Like, starting the checksum with something derived from the address makes sense. Adding to that numbers based on the instructions makes sense. But the last instruction of line 370 is a jump straight to line 390. Line 380, where any of the actual instructions are put into the checksum, never gets called. Also, there’s eight instructions per line. Why is only one ever called? And this was a bear to work out. One friend insisted I consider the possibility that MLX was buggy and nobody had found the defect. I could not accept that, not for a program that was so central to so much programming for so long. Also, not considering that it worked. Make almost any entry error and the checksum would not match. # Where’s the rest of the checksum formula? This is what took time! I had to go through the code and find what other lines call lines 360 through 390. There’s a hundred lines of code in the Commodore version of MLX, which isn’t that much. They jump around a lot, though. By my tally 68 of these 100 lines jump to, or can jump to, something besides the next line of code. I don’t know how that compares to modern programming languages, but it’s still dizzying. For a while I thought it might be a net saving in time to write something that would draw a directed graph of the program’s execution flow. It might still be worth doing that. The checksum formula gets called by two pieces of code. One of them is the code when the program gets entered. MLX calculates a checksum and verifies whether it matches the ninth number entered. The other role is in printing out already-entered data. There, the checksum doesn’t have a role, apart from making the on-screen report look like the magazine listing. Here’s the code that calls the checksum when you’re entering code: 440 POKE 198,0: GOSUB 360: IF F THEN PRINT IN$ PRINT" ";
560 FOR I=1 TO 25 STEP 3:
B$= MID$(IN$, I): GOSUB 320: IF I<25 THEN GOSUB 380: A(I/3)=A 570 NEXT: IF ACK THEN GOSUB 1060: PRINT "ERROR: REENTER LINE ": F = 1: GOTO 440 580 GOSUB 1080: [ several more lines setting up a new line of data to enter ]  Line 320 started the routine that turned a hexadecimal number, such as 7F, into decimal, such as 127. It returns this number as the variable named A. IN$ was the input text, part of the program you you enter. This should be 27 characters long. A(I/3) was an element in an array, the string of eight instructions for that entry. Yes, you could use the same name for an array and for a single, unrelated, number. Yes, this was confusing.

But here’s the logic. Line 440 starts work on your entry. It calculates the part of the checksum that comes from the location in memory that data’s entered in. Line 560 does several bits of work. It takes the entered instructions and converts the strings into numbers. Then it takes each of those instruction numbers and adds its contribution to the checksum. Line 570 compares whether the entered checksum matches the computed checksum. If it does match, good. If it doesn’t match, then go back and re-do the entry.

The code for displaying a line of your machine language program is shorter:

630 GOSUB 360:
B = BS + AD - SA;
FOR I = B TO B+7:
A = PEEK(I):
GOSUB 350:
GOSUB 380:
PRINT S$; 640 NEXT: PRINT ""; A = CK: GOSUB 350: PRINT  The bit about PEEK is looking into the buffer, which holds the entered instructions, and reading what’s there. The GOSUB 350 takes the number ‘A’ and prints out its hexadecimal representation. GOSUB 360 calculates the part of the checksum that’s based on the memory location. The GOSUB 380 contributes the part based on every instruction. S$ is a space. It’s used to keep all the numbers from running up against each other.

# So what is the checksum formula?

The checksum takes in two parts. The first part is based on the address at the start of the line. Let me call that the number $AD$. The second part is based on the entry, the eight instructions following the line. Let me call them $D_1$ through $D_8$. So this is easiest described in two parts.

The base of the checksum, which I’ll call $ck_{0}$, is:

$ck_{0} = AD - 254 \cdot \left(floor(AD \div 256)\right) \\ \mbox { [ subtract 255 if this is 256 or greater ] }$

## Reading the Comics, December 20, 1948: What is Barnaby’s friend’s name Edition?

Have a special one today. I’ve been reading a compilation of Crockett Johnson’s 1940s comic Barnaby. The title character, an almost too gentle child, follows his fairy godfather Mr O’Malley into various shenanigans. Many (the best ones, I’d say) involve the magical world. The steady complication is that Mr O’Malley boasts abilities beyond his demonstrated competence. (Although most of the magic characters are shown to be not all that good at their business.) It’s a gentle strip and everything works out all right, if farcically.

This particular strip comes from a late 1948 storyline. Mr O’Malley’s gone missing, coincidentally to a fairy cop come to arrest the pixie, who is a con artist at heart. So this sees the entry of Atlas, the Mental Giant, who’s got some pleasant gimmicks. One of them is his requiring mnemonics built on mathematical formulas to work out names. And this is a charming one, with a great little puzzle: how do you get A-T-L-A-S out of the formula Atlas has remembered?

I’m sorry the solution requires a bit of abusing notation, so please forgive it. But it’s a fun puzzle, especially as the joke would not be funnier if the formula didn’t work. I’m always impressed when a comic strip goes to that extra effort.

Johnson, who also wrote the Harold and the Purple Crayon books, painted over a hundred canvasses with theorem-based pictures. There’s a selection of them at the Smithsonian Institute’s web site, here.

## There’s a new tiny sci.math archive out there

My friend Porsupah Rhee — you might know her work from a sometimes-viral photo of rabbits fighting, available on some fun merchandise — tipped me off to this. It’s a new attempt at archiving Usenet, and also Fidonet and other bulletin boards. These are the things we used for communicating before web forums and then Facebonk took over everything everywhere. There were sprawling and often messy things, moderated only by the willingness of people to not violate social norms. Sometimes this worked; sometimes it didn’t.

Usenet was a most important piece of my Internet history; for many years it was very nearly the thing to use the Internet for. For several years it had a great archive, in the form of Deja News, which kept its many conversations researchable. Google bought this up, and as is their way, made it worse. Part of this was trying to confuse people about the difference between Usenet and their own Google Groups, a discussion-board system that I assume they have shut down. If it’s possible to search Usenet through Google anymore, I can’t find how to do it.

So I’m eager to see this archive at I Ping Therefore I Am. I don’t know where it’s getting its records from, or how new ones are coming in. What it has got is a bunch of messages from 1986. This makes for a great, weird peek at a time when the Internet was much smaller, and free of advertising, but still recognizable.

The archives do extend already to sci.math, a group for the discussion of mathematics topics. Also for discovering how people write out mathematics expressions when they don’t have LaTeX, or at least Word’s Equation Editor, to format things. This also covers two subordinate groups, sci.math.stat (for statistics) and sci.math.symbolic (for symbolic algebra discussions).

It would be bad form to join any of these conversations, even if you could figure a way how. But there may be some revealing pieces there now. And I hope the archive will grow, especially to cover the heights of 1990s Usenet. You do not have permission to look up anything I wrote longer than, oh, six weeks ago.

## No, You Can’t Say What 6/2(1+2) Equals

I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:

$6 \div 2\left(1 + 2\right) =$

Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.

That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:

$6 \div 2(3) =$

Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take $6\div 2$ and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by $2\cdot 3$, in which case the answer is 1?

And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.

I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.

This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.

But you also start to see why mathematicians tend to avoid the $\div$ symbol. We understand, for example, $a \div b$ to mean $a \cdot \frac{1}{b}$. Carry that out and then there’s no ambiguity about

$6 \cdot \frac{1}{2} \cdot 3 =$

I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in $6\div 2(1 + 2)$ also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.

(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)

And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.

And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.

## Reading the Comics Follow-up: Where Else Is A Tetrahedron’s Centroid Edition

A Reading the Comics post a couple weeks back inspired me to find the centroid of a regular tetrahedron. A regular tetrahedron, also known as “a tetrahedron”, is the four-sided die shape. A pyramid with triangular base. Or a cone with a triangle base, if you prefer. If one asks a person to draw a tetrahedron, and they comply, they’ll likely draw this shape. The centroid, the center of mass of the tetrahedron, is at a point easy enough to find. It’s on the perpendicular between any of the four faces — the equilateral triangles — and the vertex not on that face. Particularly, it’s one-quarter the distance from the face towards the other vertex. We can reason that out purely geometrically, without calculating, and I did in that earlier post.

But most tetrahedrons are not regular. They have centroids too; where are they?

Thing is I know the correct answer going in. It’s at the “average” of the vertices of the tetrahedron. Start with the Cartesian coordinates of the four vertices. The x-coordinate of the centroid is the arithmetic mean of the x-coordinates of the four vertices. The y-coordinate of the centroid is the mean of the y-coordinates of the vertices. The z-coordinate of the centroid is the mean of the z-coordinates of the vertices. Easy to calculate; but, is there a way to see that this is right?

What’s got me is I can think of an argument that convinces me. So in this sense, I have an easy proof of it. But I also see where this argument leaves a lot unaddressed. So it may not prove things to anyone else. Let me lay it out, though.

So start with a tetrahedron of your own design. This will be less confusing if I have labels for the four vertices. I’m going to call them A, B, C, and D. I don’t like those labels, not just for being trite, but because I so want ‘C’ to be the name for the centroid. I can’t find a way to do that, though, and not have the four tetrahedron vertices be some weird set of letters. So let me use ‘P’ as the name for the centroid.

Where is P, relative to the points A, B, C, and D?

And here’s where I give a part of an answer. Start out by putting the tetrahedron somewhere convenient. That would be the floor. Set the tetrahedron so that the face with triangle ABC is in the xy plane. That is, points A, B, and C all have the z-coordinate of 0. The point D has a z-coordinate that is not zero. Let me call that coordinate h. I don’t care what the x- and y-coordinates for any of these points are. What I care about is what the z-coordinate for the centroid P is.

The property of the centroid that was useful last time around was that it split the regular tetrahedron into four smaller, irregular, tetrahedrons, each with the same volume. Each with one-quarter the volume of the original. The centroid P does that for the tetrahedron too. So, how far does the point P have to be from the triangle ABC to make a tetrahedron with one-quarter the volume of the original?

The answer comes from the same trick used last time. The volume of a cone is one-third the area of the base times its altitude. The volume of the tetrahedron ABCD, for example, is one-third times the area of triangle ABC times how far point D is from the triangle. That number I’d labelled h. The volume of the tetrahedron ABCP, meanwhile, is one-third times the area of triangle ABC times how far point P is from the triangle. So the point P has to be one-quarter as far from triangle ABC as the point D is. It’s got a z-coordinate of one-quarter h.

Notice, by the way, that while I don’t know anything about the x- and y- coordinates of any of these points, I do know the z-coordinates. A, B, and C all have z-coordinate of 0. D has a z-coordinate of h. And P has a z-coordinate of one-quarter h. One-quarter h sure looks like the arithmetic mean of 0, 0, 0, and h.

At this point, I’m convinced. The coordinates of the centroid have to be the mean of the coordinates of the vertices. But you also see how much is not addressed. You’d probably grant that I have the z-coordinate coordinate worked out when three vertices have the same z-coordinate. Or where three vertices have the same y-coordinate or the same x-coordinate. You might allow that if I can rotate a tetrahedron, I can get three points to the same z-coordinate (or y- or x- if you like). But this still only gets one coordinate of the centroid P.

I’m sure a bit of algebra would wrap this up. But I would like to avoid that, if I can. I suspect the way to argue this geometrically depends on knowing the line from vertex D to tetrahedron centroid P, if extended, passes through the centroid of triangle ABC. And something similar applies for vertexes A, B, and C. I also suspect there’s a link between the vector which points the direction from D to P and the sum of the three vectors that point the directions from D to A, B, and C. I haven’t quite got there, though.

I will let you know if I get closer.

## In Our Time podcast has episode on Pierre-Simon Laplace

I have another mathematics-themed podcast to share. It’s again from the BBC’s In Our Time, a 50-minute program in which three experts discuss a topic. Here they came back around to mathematics and physics. And along the way chemistry and mensuration. The topic here was Pierre-Simon Laplace, who’s one of those people whose name you learn well as a mathematics or physics major. He doesn’t quite reach the levels of Euler — who does? — but he’s up there.

Laplace might be best known for his work in celestial mechanics. He (independently of Immanuel Kant) developed the nebular hypothesis, that the solar system formed from the contraction of a great cloud of dust. We today accept a modified version of this. And for studying the question of whether the solar system is stable. That is, whether the perturbations every planet has on one another average out to nothing, or to something catastrophic. And studying probability, which has more to do with these questions than one might imagine. And then there’s general mechanics, and differential equations, and if that weren’t enough, his role in establishing the Metric system. This and more gets discussion.

## How March 2020 Treated My Mathematics Blog

March was the first time in three-quarters of a year that I did any Reading the Comics posts. One was traditional, a round-up of comics on a particular theme. The other was new for me, a close look at a question inspired by one comic. Both turned out to be popular. Now see if I learn anything from that.

I’d left the Reading the Comics posts on hiatus when I started last year’s A-to-Z. Given the stress of the pandemic I did not feel up to that great a workload. For this year I am considering whether I feel up to an A-to-Z again. An A-to-Z is enjoyable work, yes, and I like the work. But I am still thinking over whether this is work I want to commit to just now.

That’s for the future. What of the recent past? WordPress’s statistics page suggests that the comics were very well-received. It tells me there were 2,867 page views in March. That’s the greatest number since November, the last full month of the 2020 A-to-Z. This is well above the twelve-month running average of 2,199.8 views per month. And as far above the twelve-month running median of 2,108 views per month. Per posting — there were ten postings in March — the figures are even greater. There were 286.7 views per posting in March. The running mean is 172.9 views per posting, and the running median 144.8.

There were 1,993 unique visitors in March. This is well above the running averages. The twelve-month running mean was 1,529.4 unique visitors, and the running median 1,491.5. This is 199.3 unique visitors per March posting, not a difficult calculation to make. The twelve-month running mean was 121.1 viewers per posting, though, and the mean a mere 99.8 viewers per posting. So that’s popular.

Not popular? Talking to me. We all feel like that sometimes but I have data. After a chatty February things fell below average for March. There were 30 likes given in March, below the running mean of 56.7 and median of 55.5. There were 3.0 likes per posting. The running mean for the twelve months leading in to this was 4.2 likes per posting. The running median was 4.0.

And actual comments? There were 10 of them in March, below the mean of 14.3 and median of 10. This averaged 1.0 comments per posting, which is at least something. The running per-post mean is 1.6 comments, though, and median is 1.4. It could be the centroids of regular tetrahedrons are not the hot, debatable topic I had assumed.

Pi Day was, as I’d expected, a good day for reading Pi Day comics. And miscellaneous other articles about Pi Day. I need to write some more up for next year, to enjoy those search engine queries. There are some things in differential equations that would be a nice different take.

As mentioned, I posted ten things in March. Here they are in decreasing order of popularity. I would expect this to be roughly a chronological list of when things were posted. It doesn’t seem to be, but I haven’t checked whether the difference is statistically significant.

In March I posted 5,173 words here, for an average 517.3 words per post. That’s shorter than my average January and February posts were. My average words-per-posting for the year has dropped to 558. And despite my posts being on average shorter, this was still my most verbose month of 2021. I’ve had 12,844 words posted this year, through the start of April, and more than two-fifths of them were March.

As of the start of April I’ve posted 1,605 things to the blog here. They’ve gathered 129,696 page views from an acknowledged 75,266 visitors.

If you have a WordPress account you can use the “Follow NebusResearch” button to add me to your Reader. If you have Twitter, congratulations; I don’t exactly. My account at @nebusj is still there, but it only has an automated post announcement. I don’t know when that will break. If you’re on Mastodon, you can find me as @nebusj@mathstodon.xyz.

One last thing. WordPress imposed their awful, awful, awful ‘Block’ editor on my blog. I used to be able to us the classic, or ‘good’, editor, where I could post stuff without it needing twelve extra mouse clicks. If anyone knows hacks to get the good editor back please leave a comment.

I’ve been reading The Disordered Cosmos: A Journey Into Dark Matter, Spacetime, and Dreams Deferred, by Chanda Prescod-Weinstein. It’s the best science book I’ve read in a long while.

Part of it is a pop-science discussion of particle physics and cosmology, as they’re now understood. It may seem strange that the tiniest things and the biggest thing are such natural companion subjects. That is what seems to make sense, though. I’ve fallen out of touch with a lot of particle physics since my undergraduate days and it’s wonderful to have it discussed well. This sort of pop physics is for me a pleasant comfort read.

The other part of the book is more memoir, and discussion of the culture of science. This is all discomfort reading. It’s an important discomfort.

I discuss sometimes how mathematics is, pretensions aside, a culturally-determined thing. Usually this is in the context of how, for example, that we have questions about “perfect numbers” is plausibly an idiosyncrasy. I don’t talk much about the culture of working mathematicians. In large part this is because I’m not a working mathematician, and don’t have close contact with working mathematicians. And then even if I did — well, I’m a tall, skinny white guy. I could step into most any college’s mathematics or physics department, sit down in a seminar, and be accepted as belonging there. People will assume that if I say anything, it’s worth listening to.

Chanda Prescod-Weinstein, a Black Jewish agender woman, does not get similar consideration. This despite her much greater merit. And, like, I was aware that women have it harder than men. And Black people have it harder than white people. And that being open about any but heterosexual cisgender inclinations is making one’s own life harder. What I hadn’t paid attention to was how much harder, and how relentlessly harder. Most every chapter, including the comfortable easy ones talking about families of quarks and all, is several needed slaps to my complacent face.

Her focus is on science, particularly physics. It’s not as though mathematics is innocent of driving women out or ignoring them when it can’t. Or of treating Black people with similar hostility. Much of what’s wrong is passively accepting patterns of not thinking about whether mathematics is open to everyone who wants in. Prescod-Weinstein offers many thoughts and many difficult thoughts. They are worth listening to.

## Reading the Comics, April 1, 2021: Why Is Gunther Taking Algebraic Topology Edition

I’m not yet looking to discuss every comic strip with any mathematics mention. But something gnawed at me in this installment of Greg Evans and Karen Evans’s Luann. It’s about the classes Gunther says he’s taking.

The main characters in Luann are in that vaguely-defined early-adult era. They’re almost all attending a local university. They’re at least sophomores, since they haven’t been doing stories about the trauma and liberation of going off to school. How far they’ve gotten has been completely undefined. So here’s what gets me.

Gunther taking vector calculus? That makes sense. Vector calculus is a standard course if you’re taking any mathematics-dependent major. It might be listed as Multivariable Calculus or Advanced Calculus or Calculus III. It’s where you learn partial derivatives, integrals along a path, integrals over a surface or volume. I don’t know Gunther’s major, but if it’s any kind of science, yeah, he’s taking vector calculus.

Algebraic topology, though. That I don’t get. Topology at all is usually an upper-level course. It’s for mathematics majors, maybe physics majors.  Not every mathematics major takes topology.   Algebraic topology is a deeper specialization of the subject. I’ve only seen courses listed as algebraic topology as graduate courses. It’s possible for an undergraduate to take a graduate-level course, yes. And it may be that Gunther is taking a regular topology course, and the instructor prefers to focus on algebraic topology.

But even a regular topology course relies on abstract algebra. Which, again, is something you’ll get as an undergraduate. If you’re a mathematics major you’ll get at least two years of algebra. And, if my experience is typical, still feel not too sure about the subject. Thing is that Intro to Abstract Algebra is something you’d plausibly take at the same time as Vector Calculus.  Then you’d get Abstract Algebra and then, if you wished, Topology.

So you see the trouble. I don’t remember anything in algebra-to-topology that would demand knowing vector calculus. So it wouldn’t mean Gunther took courses without taking the prerequisites. But it’s odd to take an advanced mathematics course at the same time as a basic mathematics course. Unless Gunther’s taking an advanced vector calculus course, which might be. Although since he wants to emphasize that he’s taking difficult courses, it’s odd to not say “advanced”. Especially if he is tossing in “algebraic” before topology.

And, yes, I’m aware of the Doylist explanation for this. The Evanses wanted courses that sound impressive and hard. And that’s all the scene demands. The joke would not be more successful if they picked two classes from my actual Junior year schedule. None of the characters have a course of study that could be taken literally. They’ve been university students full-time since 2013 and aren’t in their senior year yet. It would be fun, is all, to find a way this makes sense.

This and my other essays discussing something from the comic strips are at this link.

## When is Easter likely to happen?

It’s a natural question to wonder this time of year. The date when Easter falls is calculated by some tricky numerical rules. These come from the desire to make Easter an early-spring (in the Northern hemisphere) holiday, while tying it to the date of Passover, as worked out by people who did not know the exact rules by which the Jewish calendar worked. The result is that some dates are more likely than others to be Easter.

A few years ago I wrote a piece finding how often Easter would be on different dates, in the possible range from the 22nd of March through the 25th of April. And discussed some challenges in the problem. Calendars are full of surprising subtle problems. Easter creates a host of new challenges.

## The 145th Playful Math Education Blog Carnival is posted

John Golden, MathHombre, was host this month for the Playful Math Education Blog Carnival. And this month’s collection of puzzles, essays, and creative mathematics projects. Among them are some quilts and pattern-block tiles, which manifest all that talk about the structure of mathematical objects and their symmetries in easy-to-see form. There’s likely to be something of interest there.

Among the wonderful things I discovered there is Math Zine Fest 2021. It’s as the name suggests, a bunch of zines — short printable magazines on a niche topic — put together for the end of February. I had missed this organizing, but hope to get to see later installments. I don’t know what zine I might make, but I must have something I could do.

Denise Gaskins, who organizes the carnival, has hosting slots available for later this year. Hosting is an exciting challenge I encourage people to try at least the once.

## Reading the Comics, March 16, 2021: Where Is A Tetrahedron’s Centroid Edition

Comic Strip Master Command has not, to appearances, been distressed by my Reading the Comics hiatus. There are still mathematically-themed comic strips. Many of them are about story problems and kids not doing them. Some get into a mathematical concept. One that ran last week caught my imagination so I’ll give it some time here. This and other Reading the Comics essays I have at this link, and I figure to resume posting them, at least sometimes.

Ben Zaehringer’s In The Bleachers for the 16th of March, 2021 is an anthropomorphized-geometry joke. Here the centroid stands in for “the waist”, the height below which boxers may not punch.

The centroid is good geometry, something which turns up in plane and solid shapes. It’s a center of the shape: the arithmetic mean of all the points in the shape. (There are other things that can, with reason, be called a center too. Mathworld mentions the existence of 2,001 things that can be called the “center” of a triangle. It must be only a lack of interest that’s kept people from identifying even more centers for solid shapes.) It’s the center of mass, if the shape is a homogenous block. Balance the shape from below this centroid and it stays balanced.

For a complicated shape, finding the centroid is a challenge worthy of calculus. For these shapes, though? The sphere, the cube, the regular tetrahedron? We can work those out by reason. And, along the way, work out whether this rule gives an advantage to either boxer.

The sphere first. That’s the easiest. The centroid has to be the center of the sphere. Like, the point that the surface of the sphere is a fixed radius from. This is so obvious it takes a moment to think why it’s obvious. “Why” is a treacherous question for mathematics facts; why should 4 divide 8? But sometimes we can find answers that give us insight into other questions.

Here, the “why” I like is symmetry. Look at a sphere. Suppose it lacks markings. There’s none of the referee’s face or bow tie here. Imagine then rotating the sphere some amount. Can you see any difference? You shouldn’t be able to. So, in doing that rotation, the centroid can’t have moved. If it had moved, you’d be able to tell the difference. The rotated sphere would be off-balance. The only place inside the sphere that doesn’t move when the sphere is rotated is the center.

This symmetry consideration helps answer where the cube’s centroid is. That also has to be the center of the cube. That is, halfway between the top and bottom, halfway between the front and back, halfway between the left and right. Symmetry again. Take the cube and stand it upside-down; does it look any different? No, so, the centroid can’t be any closer to the top than it can the bottom. Similarly, rotate it 180 degrees without taking it off the mat. The rotation leaves the cube looking the same. So this rules out the centroid being closer to the front than to the back. It also rules out the centroid being closer to the left end than to the right. It has to be dead center in the cube.

Now to the regular tetrahedron. Obviously the centroid is … all right, now we have issues. Dead center is … where? We can tell when the regular tetrahedron’s turned upside-down. Also when it’s turned 90 or 180 degrees.

Symmetry will guide us. We can say some things about it. Each face of the regular tetrahedron is an equilateral triangle. The centroid has to be along the altitude. That is, the vertical line connecting the point on top of the pyramid with the equilateral triangle base, down on the mat. Imagine looking down on the shape from above, and rotating the shape 120 or 240 degrees if you’re still not convinced.

And! We can tip the regular tetrahedron over, and put another of its faces down on the mat. The shape looks the same once we’ve done that. So the centroid has to be along the altitude between the new highest point and the equilateral triangle that’s now the base, down on the mat. We can do that for each of the four sides. That tells us the centroid has to be at the intersection of these four altitudes. More, that the centroid has to be exactly the same distance to each of the four vertices of the regular tetrahedron. Or, if you feel a little fancier, that it’s exactly the same distance to the centers of each of the four faces.

It would be nice to know where along this altitude this intersection is, though. We can work it out by algebra. It’s no challenge to figure out the Cartesian coordinates for a good regular tetrahedron. Then finding the point that’s got the right distance is easy. (Set the base triangle in the xy plane. Center it, so the coordinates of the highest point are (0, 0, h) for some number h. Set one of the other vertices so it’s in the xz plane, that is, at coordinates (0, b, 0) for some b. Then find the c so that (0, 0, c) is exactly as far from (0, 0, h) as it is from (0, b, 0).) But algebra is such a mass of calculation. Can we do it by reason instead?

That I ask the question answers it. That I preceded the question with talk about symmetry answers how to reason it. The trick is that we can divide the regular tetrahedron into four smaller tetrahedrons. These smaller tetrahedrons aren’t regular; they’re not the Platonic solid. But they are still tetrahedrons. The little tetrahedron has as its base one of the equilateral triangles that’s the bigger shape’s face. The little tetrahedron has as its fourth vertex the centroid of the bigger shape. Draw in the edges, and the faces, like you’d imagine. Three edges, each connecting one of the base triangle’s vertices to the centroid. The faces have two of these new edges plus one of the base triangle’s edges.

The four little tetrahedrons have to all be congruent. Symmetry again; tip the big tetrahedron onto a different face and you can’t see a difference. So we’ll know, for example, all four little tetrahedrons have the same volume. The same altitude, too. The centroid is the same distance to each of the regular tetrahedron’s faces. And the four little tetrahedrons, together, have the same volume as the original regular tetrahedron.

What is the volume of a tetrahedron?

If we remember dimensional analysis we may expect the volume should be a constant times the area of the base of the shape times the altitude of the shape. We might also dimly remember there is some formula for the volume of any conical shape. A conical shape here is something that’s got a simple, closed shape in a plane as its base. And some point P, above the base, that connects by straight lines to every point on the base shape. This sounds like we’re talking about circular cones, but it can be any shape at the base, including polygons.

So we double-check that formula. The volume of a conical shape is one-third times the area of the base shape times the altitude. That’s the perpendicular distance between P and the plane that the base shape is in. And, hey, one-third times the area of the face times the altitude is exactly what we’d expect.

So. The original regular tetrahedron has a base — has all its faces — with area A. It has an altitude h. That h must relate in some way to the area; I don’t care how. The volume of the regular tetrahedron has to be $\frac{1}{3} A h$.

The volume of the little tetrahedrons is — well, they have the same base as the original regular tetrahedron. So a little tetrahedron’s base is A. The altitude of the little tetrahedron is the height of the original tetrahedron’s centroid above the base. Call that $h_c$. How can the volume of the little tetrahedron, $\frac{1}{3} A h_c$, be one-quarter the volume of the original tetrahedron, $\frac{1}{3} A h$? Only if $h_c$ is one-quarter $h$.

This pins down where the centroid of the regular tetrahedron has to be. It’s on the altitude underneath the top point of the tetrahedron. It’s one-quarter of the way up from the equilateral-triangle face.

(And I’m glad, checking this out, that I got to the right answer after all.)

So, if the cube and the tetrahedron have the same height, then the cube has an advantage. The cube’s centroid is higher up, so the tetrahedron has a narrower range to punch. Problem solved.

I do figure to talk about comic strips, and mathematics problems they bring up, more. I’m not sure how writing about one single strip turned into 1300 words. But that’s what happens every time I try to do something simpler. You know how it goes.