The Mandelbear's Musings

I have been down at Rest Stop since last Sunday afternoon. I'd been hoping to get in some practice time with m, but that didn't work out. So I did a little guitar and singing practice on my own -- not nearly enough, but I'm trying to get my finger calluses back in shape. Getting there, but feeling very discouraged over how little music I've done in the last few years. So I've been filling in some of the time listening to Talis Kimberley on YouTube, and the rest of it hanging out in the 4th, 8th, and 24th dimensions. Mostly the 8th, with the octonions and the E8 lattice, and the 24th, with the Leech lattice.

The Leech Lattice has nothing at all to do with blood-sucking invertebrates (a category which also includes many politicians), but it is definitely a time-sucker. I was also going to say that the E8 lattice has no connection to The Adventures of Buckaroo Banzai Across the 8th Dimension, but it turns out that there is (a rather tenuous) one. Which is absolutely mind-boggling.

Some day I may write a post about why I find this stuff mind-bogglingly beautiful even though I don't know nearly enough about group theory, string theory, geometry, particle physics, elliptic curves, or modular forms to be able to follow the math. You can get a lot of it from this lecture by John Baez (who happens to be Joan Baez's cousin). Many of the bizarre mathematical connections come from the fact that if you add up the squares of the numbers from 1 to 24, you get a perfect square. Which is why the Leech lattice is the best way to pack spheres in 24 dimensions.

The Leech lattice is also related to the Binary Golay code, which brings up all sorts of other connections because Marcel Golay also collaborated with my father on the Savitzky–Golay filter, in an article that marked "the dawn of the computer-controlled analytical instrument". Dad would have been fascinated by all this.

All of which has been incredibly addicting and has kept my mind (mostly) off my worries about the state of my health and the deplorable state of US politics.

I'll leave you with last Sunday's quote of the day:

c: I should know where my checkbook is.
me: Put it next to your towel.

( Notes & links, as usual )

ETA: Bathsheba Sculpture - E8 I might just need one of these. only $60

I really need to write my memoirs, preferably before my memory deteriorates to the point where I can't. (I am inspired by my mom, who published the third edition of hers last year.) I have, however, given up on the idea of following the King of Hearts' advice to "begin at the beginning, [...] and go on till you come to the end: then stop". (I note in passing that I haven't come to the end yet.) So I'm just going to dive in at whatever point seems interesting at the moment. I'll tag these by year, so that anyone interested (possibly as many as two of you) can sort them out later.

This particular point was suggested by somebody's mention of their Erdős number, so I suppose I ought to explain that first. Content Warning: contains math, which you can safely skip over if you're math-phobic. Deciding which parts to skip is left as an exrcise for the reader.

You have perhaps heard of the parlor game called "Six Degrees of Kevin Bacon", based on the concept of "six degrees of separation". The idea is to start with an actor, and figure out the shortest possible list of movies that links them with Kevin Bacon. The length of that list is the actor's "Bacon number", with Bacon himself having the number zero, anyone who acted in a movie with him having the number one, and so on. As far as I know I don't have a finite Bacon number, but it's not outside the realm of possibility if, as most people do, you include TV shows and so on. I think I've been in at least one brief local TV news item.

But sometime during my senior year at Carleton College, I co-authored a paper with one of my math professors, Ken Wegner, which gave me an Erdős number of 7. The paper, published in 1970 in The American Mathematical Monthly, was "Solutions of Φ(x) = n , Where Φ is Euler's Φ-Function" [Wegner, K., & Savitzky, S. (1970), The American Mathematical Monthly, 77(3), 287-287. doi:10.2307/2317715].

So now I have three things to explain: What is an Erdős number? What is Euler's Φ function? And finally, What was my contribution to the paper?

Erdős number: As you might expect from the introduction about the Bacon Number, a mathematician's Erdős number is the smallest number of co-authored papers connecting them to Paul Erdős (1913–1996), an amazingly prolific (at least 1,525 papers) 20th Century mathematician. He spent the latter part of his life living out of a suitcase, visiting his over 500 collaborators (who thus acquired an Erdős number of 1. The Erdős number was first defined in print in 1969, so about the time I was collaborating with Wegner on Euler's Φ function.

Euler's Φ function, Φ(n), also called the Totient function, is defined as the number of positive integers less or equal to n that are relatively prime to n; or in other words the numbers in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n,k) = 1. (You will also see it written in lower-case as "φ", or in Latin as "phi".)

The totient function is pretty easy to compute, at least for sufficiently small numbers. The inverse is rather less straightforward, and has been the subject of a considerable number of StackExchange queries. (This answer includes a good set of links.) I was thinking of including some detail about that, and was barely able to keep myself from falling down the usual rabbit-hole, which almost always ends up somewhere in group theory. For example, φ(n) is the order of the multiplicative group of integers modulo n. See what I mean?

My contribution to the paper was not very closely related to the actual mathematics of the problem; what I did was write the computer program that computed and printed out the table of results. That involved a hack. A couple of hacks, actually.

In 1969, Carleton College's computer lab contained an IBM 1620 and a couple of keypunches. The 1620 was fairly primitive even by 1960s standards; its memory consisted of 20,000 6-bit words, with a cycle time of 20 microseconds. Each word contained one BCD-coded decimal digit, a "flag" bit, and a parity check bit. It did arithmetic digit-by-digit using lookup tables for addition and multiplication. It was not particularly fast -- about a million times slower than the CPU in your phone. But it was a lot of fun. Unlike a mainframe, it could sit in one corner of a classroom (if it was air-conditioned), it was (comparatively) inexpensive, and it could stand up to students actually getting their hands on it.

A lot of the fun came from the fact that the 1620's "operating system" was the human operator sitting at the console, which consisted mainly of an electric typewriter and a row of buttons and four "sense switches" that the program could read. If you wanted to run a program, you put a stack of punched cards into the reader and pushed the "load" button, which read a single 80-column card into the first 80 characters of memory, set the program counter to zero, and started running. My program was written in FORTRAN. Not even FORTRAN II. Just FORTRAN.

Computing the table that occupied most of the paper took about a week.

Here's where it gets interesting, because obviously I wasn't the only student who wanted to use the 1620 that week. So I wrote an operating system -- a foreground/background system with my program running in the background, with everyone else's jobs running in the "foreground". That would have been easy except that the 1620 could only run one program at a time. Think about that for a moment.

My "operating system" consisted mainly of a message written on the back of a Hollerith card that said something like: "Flip sense switch 1 and wait for the program to punch out a deck of cards (about a minute). When you're done, put the deck in the reader and press LOAD."

Every time the program went around its main loop, it checked Sense Switch 1, and if it was set, it sent the contents of memory to the card punch. Dumping memory only took one instruction, but it wasn't something you could do from FORTRAN, so I put in a STOP statement (which FORTRAN did have) and changed it to a dump instruction. By scanning the program's object code (remember this is a decimal machine; an instruction took up 12 columns on the card) and replacing the HALT instruction with DUMP.

It worked.

    MR: Collaboration Distance
     MR Erdos Number = 7
     S. R. Savitzky 	  coauthored with    Kenneth W. Wegner 		MR0260667
     Kenneth W. Wegner 	  coauthored with    Mark H. Ingraham 		MR1501805
     Mark H. Ingraham 	  coauthored with    Rudolph E. Langer 		MR1025350
     Rudolph E. Langer 	  coauthored with    Jacob David Tamarkin 	MR1501439
     Jacob David Tamarkin coauthored with    Einar Hille 	        MR1555331
     Einar Hille 	  coauthored with    Gábor Szegő 	        MR0008279
     Gábor Szegő 	  coauthored with    Paul Erdős 	        MR0006783
     MR0260667 points to: K. W. Wegner and S. R. Savitzky, (1970)
     Solutions of φ (x) = n, Where φ is Euler's φ-Function on JSTOR,
     The American Mathematical Monthly, 77(3), 287-287.
     DOI: 10.1080/00029890.1970.11992471.

There are two other numbers of interest: the Shūsaku Number, measuring a Go player's distance from the famous 19th-Century Go player Hon'inbō Shūsaku, and the Sabbath Number, measuring a musician's distance from the band Black Sabbath. I'm pretty sure I have a Sabbath number through filkdom. I definitely have a Shūsaku number of 5 from having lived down the hall from Jim Kerwin, Shūsaku Number 4, my sophomore and junior years at Carleton. That's another story.

And if I expect to write more journal entries about math, I'm going to have to extend my posting software to allow entries written in LaTeX. Hmm.

The Mandelbear's Memoirs

You may remember from my previous post about Hyperviewer that I’d been plagued by a mysterious bug. The second time the program tried to make a simplex (the N-dimensional version of a triangle (N=2) or tetrahedron (N=3), a whole batch of “ghost edges” appeared and the program (quite understandably) blew up. I didn’t realize it until somewhat later, but there were ghost vertices as well, and that was somewhat more fundamental. Basically, nVertices, the field that holds the number of vertices in the polytope, was wildly wrong.

Chasing ghosts

Eventually I narrowed things down to someplace around here, which is where things stood at the end of the previous post.

        let vertices = [];
        /* something goes massively wrong, right here. */
        for (let i = 0; i < dim; ++i) {
            vertices.push(new vector(dim).fill((j) => i === j? 1.0 : 0));
        }

I found this by throwing an error, with a big data dump, right in the middle if nVertices was wrong (it’s supposed to be dim+), or if the length of the list of vertices was different from nVertices.

         let vertices = [];
         /* something goes massively wrong, right here. */
         if (this.nVertices !== (dim + 1) || this.nEdges !== ((dim + 1) * dim / 2) ||
             this.vertices.length !== 0 || this.edges.length !== 0 ) {
             throw new Error("nEdges = " + this.nEdges + " want " +  ((dim + 1) * dim / 2) +
                             "; nVertices = " + this.nVertices + " want " + dim +
                             "; vertices.length = " + this.vertices.length +
                             ' at this point in the initialization, where dim = ' + dim +
                             " in " + this.dimensions + '-D ' + this.name 
                           );
        } 
        for (let i = 0; i < dim; ++i) {

It appeared that nVertices was wildly wrong at that point. If I’d looked carefully and thought about what nVertices actually was, I would probably have found the bug at that point. Or even earlier. Instead, what clinched it was this:

         this.vertices = vertices;
         this.nVertices = vertices.length;  // setting this to dim+1 FAILS:
         // in other words, this.nVertices is getting changed between these two statements!
         if (this.vertices.length !== this.nVertices || this.edges.length !== 0) {
             throw new Error("expect " + this.nVertices + " verts, have " + this.vertices.length +
                             " in " + this.dimensions + '-D ' + this.name +
                             "; want " + this.nEdges + " edges into " + this.edges.length
                            );
         }

The code that creates the list of vertices produces the right number of vertices. If I set nVertices equal to the length of that list, everything was fine.

If instead I set

        this.nVertices = dim+1;

it was wrong. Huh? For example, in four dimensions, the number of vertices is supposed to be five, and that was the length of the list. When is 4+1 not equal to 5?

At this point a light bulb went off, because it was clear that dim+1 was coming out equal to 41. In three dimensions it was 31. When is 4+1 not equal to 5? When it’s actually "4"+1. In other words, dim was a string. JavaScript “helpfully” converts a string to a number when you do anything arithmetical to it, like multiply it by something or raise it to a power. But + isn’t always an arithmetic operation! In JavaScript (and many other languages) it’s also used for string concatenation.

What went wrong, and a rant

The problem was that, the second time I tried to create a simplex, the number of dimensions was coming from the user interface. From an <input element in a web form. And every value that you get from a web form is a string. HTML knows nothing about numbers, and it has no way to know what you’re going to do with the input you get.

So the fix was simple (and you can see it here on GitHub: convert the value from a string to a number right off before trying to use it as a number of dimensions. But… But… But cubes and octohedrons were right!

That’s because the number of vertices in a N-cube is 2**N, and in an N-orthoplex (octohedron in three dimensions) it’s N*2 (and multiplication is always an arithmetic operator in JavaScript). And it worked when I was creating the simplex’s vertices because it was being compared against in a for loop. And so on.

If I’d been using a strongly-typed language, the compiler would have found this two weeks ago.

There are two main ways of dealing with data in a programming language, called “strong typing” and “dynamic typing”. In a strongly-typed language, both values and variables (the boxes you put values into) have types (like “string” or “integer”), and the types have to match. You can’t put a string into a variable with a type of integer. Java is like that (mostly).

Some people find this burdensome, and they prefer dynamically-typed languages like JavaScript. In JavaScript, values have types, but variables don’t. It’s called “dynamic” typing because a variable can hold anything, and its type is that of the last thing that was put into it.

You can write code very quickly in a language where you don’t have to declare your variables and make sure they’re the right type for the kind of values you want to put into them. You can also shoot yourself in the foot much more easily.

There are a couple of strongly-typed variants on JavaScript, for example CoffeeScript and TypeScript, and a type-checker called “Flow”. I’m going to try one of those next.

There was one more problem with simplexes

(simplices?) … but that was purely geometrical, and just because I was trying to do all the geometry in my head instead of on paper, and wasn’t thinking things through.

If you’re in N dimensions, you can create an N-1 dimensional simplex by simply connecting the points with coordinates like [1,0,0], [0,1,0], and [0,0,1] (in three dimensions – it’s pretty easy to see that that gives you an equilateral triangle). Moreover, all the vertices are on the unit sphere, which is where we want them. The last vertex is a bit of a problem.

A fair amount of googling around (or DuckDuckGoing around, in my case) will eventually turn up this answer on mathoverflow.net, which says that in N dimensions, the last vertex has to be at [x,...,x] where x=-1/(1+sqrt(1+N)). Cool! And it works. Except that it’s not centered – that last vertex is a lot closer to the origin than the others. It took me longer than it should have to get this right, but the center of the simplex is its “center of mass”, which is simply the average of all the vertices. So that’s at y=(1+x)/(N+1) because there are N+1 vertices. Now we just have to subtract y from all the coordinates to shift it over until the center is at the origin.

Then of course we have to scale it so that all the vertices are back on the unit sphere. You can find the code here, on GitHub.

Another fine post from The Computer Curmudgeon.

The only major news is that we have a firm date for when the previous owner of our house gets all his stuff moved out: June 30th. That's about three weeks sooner than the original worst-case plan, so Colleen and I will have the entire month to get moved in, rather than a week. Yay!

I don't seem to have done much this week. I did get the car charged, and deposited a bunch of checks (including some old enough that I'm not sure they're still good -- I need to get a lot better at that). Mostly I sat around the apartment exploring an assortment of math topics on Wikipedia and YouTube.

It turns out that, thanks to a paper I wrote back at Carleton with one of my math professors, it can easily be determined that my Erdős number is officially 7. Unofficially, if one includes patents as well as actual math papers, it's 4. That still probably exceeds the number of people reading this who knew what an Erdős number is before reading this. The official value almost certainly does.

I did some actual programming yesterday (which I made more progress on today), aimed at bringing my song formatting and typesetting into the 21^st Century. Mostly that means switching from postscript files to PDFs everywhere, upgrading to LaTeX2e, and simplifying the build process. There are still a few formatting issues that need to be dealt with; I will be having some fun this week refactoring my horrible old style files into classes.

There was some discussion in comments elsewidth about finding a therapist; I did a little link chasing. Not going to do anything about it until after we move.

( Notes & links, as usual )

There's that moment when everything changes,
But really it's just you,
Seeing things differently.

When you realize that the solid bench you're sitting on
Is mostly empty space between particles.
When you learn that even the particles
Aren't really particles, and that light isn't entirely waves either.

When you see the way special relativity views velocity
As simple rotation in four-space, 
And you study general relativity and realize
That it's geometry all the way down.

When you suddenly get recursion,
Reading the Algol 60 Report, with its crystalline prose
And elegantly compact rules.
When Goedel blows the top of your head off,
And you understand that some things simply can't be proved.
When you see how elegantly Turing applies the same trick.

When you realize that a little of the Unknowable
Isn't part of the Unknown anymore,
Because now you know why you can't know it.

First published in a comment in the October 2014 Crowdfunding Creative Jam, on the theme "Paradigm Shifting Without a Clutch."

This is entirely autobiographical, though the sequence has been messed with a little to give artistic verisimilitude to an otherwise bald and unconvincing narrative.

Mirrored from steve.savitzky.net. My poetry there is in really rough shape; hopefully I'll get a little work done on it soon.

Also adopted by ysabetwordsmith as part of her Schrodinger's Heroes series, which makes it unintentionally canonical fanfic for an imaginary TV show. Talk about shifting without a clutch! At least it has synchromesh. Or was that synchrotron?

Today's xkcd is screamingly funny, at least to anyone who knows the Banach–Tarski Theorem.

This is just plain wrong.

... unless you're familiar with "A Planet Named Shayol". In which case it is very, very wrong.

Just noticed that tomorrow is 2³/3³...

And that Thursday will be perfect...

Today is, among other things, Newton's Birthday, which makes the flower_cat's gift to me of a copy of God Created the Integers: the Mathematical Breakthroughs that Changed History by Stephen Hawking particularly appropriate.

It's an anthology of some of the most important papers in mathematics, with Hawking's introductions, starting with extracts from Euclid's Elements and ending with Turing's On Computable Numbers. I'm planning to just read the introductions and skim the rest -- it's the sort of book in which I could easily lose myself for months.

Thank you, love!

An actual tesseract is best described as a four dimensional cube...and is kind of confusing. So, in memory of L'Engle, we met up with Physicist David Morgan who took a little time out of his day to talk tesseracts with the BPP. Put your measley three-dimensional brains to work on this (Via Boing Boing)

I was out walking a couple of days ago and the phrase "Moebius Bandsaw" popped into my head. Turns out you don't always need a double edge to cut both ways.

The adventure starts with this post by jenkitty containing directions for knitting a Klein Bottle hat, ganked from this page on one-sided surfaces.

That, of course, immediately led me to remember this hat at the Acme Klein Bottle online store. Somewhere around here I recalled my friend Ted's amusing account of trying to commission a Klein Bottle from a glassblower in Germany. (He eventually succeeded.)

A few more links lead to the geometric topology section of the Geometry Junkyard, and an online book called Math That Makes You Go Wow: A Multi-Disciplinary Exploration of Non-Orientable Surfaces.

Finally, combining non-orientable surfaces with another of jenkitty's interests, I tracked down this appendix to The Ultimate Science Fiction Poetry Guide, which contained the correct text and attribution for a limerick I remembered reading in the December 1968 issue of Martin Gardner's Mathematical Games column of Scientific American: it's by the late science fiction author Cyril Kornbluth:

 A burleycue dancer, a pip
 Named Virginia, could peel in a zip;
     But she read science fiction
     And died of constriction
 Attempting a Mobius strip.

(The icon, BTW, is a 120-cell ganked from this Wikipedia article

Just in case you thought needlework necessarily had anything to do with reality, take a look at the rochet Hyperbolic Coral reef Project at The Institute For Figuring. For a deeper look, check out their Hyperbolic Space online exhibit. (From Make: Blog)

And now, if you'll excuse me, I think I need to go re-read Mount Analogue: A Novel of Symbolically Authentic Non-Euclidean Adventures in Mountain Climbing.