The limits of autodidacticism

When I was 10 years old (5th grade), circa 1963, I thought I might have discovered some new principle of numbers, but it was almost impossible for me to find out. There was no one to ask, and no references I could find.

My father would take me to his office (attached to a flour milling operation) on the weekend to give my mother a break, and there I would wander around entertaining myself by admiring the sailfish mounted on one man’s wall, and the manual plug-in switchboard. There were, of course, many desks in an open area for clerical work, and many of those desks had calculators.

These were not, perhaps, what you think of when you think of calculators. These were Friden machines, with a particular odor of lubricant familiar to those of us old enough to remember early data centers. They had a peculiarly comforting chug-a-lug sound as they performed their operations, and the answer appeared in a row of windowed numbers at the top, the entire carriage of which would clack along until it finished its operation. (Ah, the good old days, when calculators had gears. Lots of them.)

There was an option to see not only a long array of decimal places for a division operation, but you could also see the remainder, and keep dividing, so you could generate indefinitely long decimal remainders.

What I discovered is something I had no name for at the time, or for many years afterward. Here’s the simple version:

If you take the reciprocal of prime numbers other than 2 and 5 (in base 10), you will generate a repeating decimal. In other words, 1/7 = 0.142857142857142857…  Think of those 6 digits as arranged on a circle, like a wheel. Now, 2/7 = 0.285714285714285714…  Same wheel, but starting in a different place. Each of the 7ths uses the same digits and starts at a different place on the wheel.

Logically enough, there were 6 digits in the pattern, because there are 6 7ths. Or so I thought…

1/11 was disappointing, with only 2 repeating digits. Then I looked at 1/13. The repeating decimal for 1/13 also had 6 digits, not 12 as I had expected: 0.076823. But when I looked at all the 13ths, I discovered there were two patterns — two wheels, as it were — and each 13th found its starting place on one wheel or the other, and that distribution also followed a visible pattern (for wheels “a” and “b”): abaabb bbaaba.

It turned out that 1/11 wasn’t disappointing after all; it just had 6 wheels of 2 digits each (sort of a minimal case). You could consider 1/3 to be 2 wheels of 1 digit, trivially. But the real fun was in the larger primes.

I was hooked. I worked out the repeating decimal reciprocals for all the primes up to 100 and a good deal further and found no violation of the structure I could observe. I couldn’t see a way of predicting either the number of “wheels” a prime would have, or the distribution of which wheel would be used by which reciprocal, though they always fell into clear patterns

While I had no clear understanding of the history of mathematical discoveries at that age, I had some dim impression of it, and I thought it very unlikely that I was the first person to have noticed this. But how could I find out more? What tools would I need to learn to deal with the concepts?

I knew about the Sieve of Erastothenes (from George Gamow books, like One, Two, Three… Infinity, not my trivial grade school textbooks) and the difficulty of identifying primes, and wondered if it might be the case that “primes and only primes” manifested this property, in which case could one divide the length of the decimal repeat into a prime-candidate and if the number of digits evenly divided the prime-candidate minus one, might that be a useful prime test? (e.g., 13 = prime, because (13-1) is evenly divided by 6). I could observe that, but I lacked the tools to prove it.

And that’s when I hit a wall. I was admittedly a very odd 10-year-old, and I devoured the math books at my school (such as they were) the first week of each year. Alas, there was nothing then to supplement them. Certainly there was a math teacher, but this wasn’t anything he could deal with — he did pedagogy for conventional youngsters. I couldn’t find any adult who both understood what I was describing and was interested in helping me find out more.

Someone muttered “modular arithmetic”, which left me no wiser than I was before. I took a stab at looking through the local university library at math books for modular arithmetic, and other than concluding they were concerned with properties across base 10 and other bases, there was no place for me to start to get a handle on the issues.

Self-help was inadequate to the task.

I put the query aside and ultimately went to college at 17 determined to major in math. Whereupon I prudently dropped the advanced math classes I took freshman year when I realized, with horror, that the watershed courses which sorted the people who were fine with calculus from the real math students had left me on the wrong side of the divide. (I went on to get a real education in dead languages and related subjects, since I was at a good school, and made my career in software and related services.)

This has festered somewhat with me over the decades, as childhood frustrations do, and I was moved today to see what the real description of this concept is. Couldn’t have been more easy to find — first Google search, and there it was.

My old observations fall into the categories of “repeating decimals” and “cyclic numbers“.

I would have killed for access to these articles and their references back in 1963. We forget what a wonderful world we are living in now.


God is in the dice

Escher-NatureWe are part of nature, an animal like other animals, and our minds preserve that heritage.

I’m conscious of this every time that I’m startled by a grouse shaking the bush along my path when it erupts and I take an involuntary step backward, preparing to fight or run, once I determine what threat is coming my way. My rational mind knows it’s ridiculous, but I’m the product of thousands of ancestors who decided that sometimes it really is a lion, and it’s better to treat it that way, and laugh about it afterward. The ones who didn’t, after all, were sometimes wrong enough that they left no trace.

What do animals do? They observe. They watch the behavior of their mother and siblings, the way their herd, or pack, or flock treats its members. They watch their prey, if they are predators. And vice versa.

Humans, well, we watch everything. We’re fascinated by animals from our very earliest age. Not just our own family and tribe, but all animals. We can spend hours and days just watching them.

What do we learn when we watch animals? We learn, species by species and circumstance by circumstance, just what they are likely to do — depending on the season, the environment, the weather, their state of health, or lust, or maternity. We know they make their own choices, within circumscribed limits. We might not know why, exactly, but we know what — they’re like us, or they seem to be, even the alien ones like bugs. We learn how to predict what they will do. We are wired, I would say, to pay attention to animals this way, just as they pay attention to each other. They have agency, and we want to understand how they work. … Read more…

The hidden personal cost of your computer ecosystem

TechEcosystemI’m an early adopter of technology, esp. software. It’s an essential component of my self-image that began with mathematics in grade school. Back in the 70’s I entered early computer businesses after college and made my entire career in a variety of young companies in the software, support, and consulting wings of the tech industry.

Outside of business, in my personal tool kit, I eagerly embraced home computers for general use, and specialist devices and software for hobbies like music and photography. I immersed myself in evolving standards for good user design and knowledge management philosophies. I studied the engineering principles of mainframe operating systems. By today’s standards I may not be a tech expert, but I am, by god, an experienced technology user.

And I am paying for it. Every day. With the only currency that matters — time.

And so are you. … Read more…

Devices vs Brains: Musical Memory

Hins Anders (Anders Zorn)
Hins Anders (Anders Zorn)

Thousands of tunes

Among other things, I’m a fiddler of folk music — have been for 25 years or more. Unusually, for an American, I specialize in traditional music from Scandinavia — Sweden, Norway, Denmark, and Finland — instead of Irish, Scottish, or Appalachian. It’s a very rich tradition, with roots in the Baroque and earlier, and a number of interesting bowed string instruments besides the violin, namely the nyckelharpa and the hardingfele.

Except for a handful who grew up in a Scandinavian musical tradition in America (very few), we almost all come from outside the ethnic culture, inspired by the love of the music itself.

Most of us play for dancers or are associated with people who do so. There are a number of thriving dance groups in America — the ones that have local musicians dance to live music, and the rest dance to recordings. These are social dance groups, by and large, not performance groups — it’s like having a local square dance. Both the musicians and the dancers are tightly connected with their counterparts in Sweden and Norway, and instructors travel regularly to America to lead workshops and teach at dance and music camps.

At a guess, there are maybe a couple of hundred musicians who dabble seriously in Scandinavian fiddle music in the USA, and perhaps a thousand or so dancers. That’s a small community, small enough that the musicians who’ve been around a while pretty much all know each other, as do the leaders of the dance groups.

In a traditional community in Sweden, the locals would have had a dozen or so dances, and the musicians would have played tunes in those genres. In America, where it’s an adopted tradition, the dancers tend to have a collector’s mentality: “Ooh, let’s learn that dance next!”  So, while a fiddler tied to a village in Sweden might have mastered a few genres of tunes for dances (and many tunes for each type), a fiddler for a modern American dance group needs to be able to cover, say, forty dance genres, with at least two tunes each. A typical free-for-all dance party for Scandinavian dancers might require a basic repertoire of eighty tunes, to just barely cover an evening (80 x 3 minutes each = 240 minutes = 4 hours), and that’s assuming all the ad hoc musicians know all the same tunes.

One of my early Swedish mentors recommended that I specialize in the tunes of a particular district (almost any area has dozens or hundreds of tunes). I explained to him that, as an American Scandie fiddler, I was already specializing — I wasn’t playing Irish or Scottish. Different worlds.

Read more…

Gödel, Escher, Bach

An Eternal Golden Braid

Thirty-six years ago, one of the greatest books of its decade was published — Douglas Hofstadter’s Gödel, Escher, Bach.

It’s hard to describe how this book affected a certain kind of person. You had to have an interest in logic, recursion, and self-reference in the fields of mathematics, music, and the visual arts, for starters. But if it caught you at the right stage in your life, when these interests were important to you and you had some familiarity with at least Escher and J. S. Bach, then there was nothing like this book. Anywhere.

And that’s still true. Even Hofstadter’s other books, fascinating and improbable as they all are, don’t come close.

It’s a tour de force — both witty and compelling.

There’s little original I can write on the topic, but here are some highlights…

The basics — Wikipedia and Amazon.

Here’s the reasonable (if not entirely sympathetic) review from the New York Times in 1979 by Brian Hayes. (Some readers just don’t seem to appreciate the linguistic humor, the playfulness with which Hofstadter approaches his subjects.)

In looking for good reviews, I was especially interested in recent ones, to see how the book is still received. I was quite happy to find this gem: A discarded review of ‘Godel, Escher Bach: an Eternal Golden Braid’. Not only do I advise you to read it (the comments section illuminates the variation in fans vs non-fans), but inside I found a surprise — a reference to a paean by Eliezer Yudkowsy, whose deliciously intelligent fan-fiction Harry Potter and the Methods of Rationality is a treat in its own right:

This is simply the best and most beautiful book ever written by the human species…

I’m not alone in this opinion, by the way. For one thing, Gödel, Escher, Bach won a Pulitzer Prize. Or just pick a random scientist and ask ver what vis favorite book is, and 1 out of 5 will say: “Gödel, Escher, Bach“. No other book even comes close.

It is saddening to contemplate that every day, 150,000 humans die without reading what is indisputably one of the greatest achievements of our species. Don’t let it happen to you.

Sure, if you’re just an average person, you might not understand everything in this book – but when you’re done reading, you won’t be an average person any more.

Read more…