The Summer 2017 Mathematics A To Z: Quasirandom numbers

Gaurish, host of, For the love of Mathematics, gives me the excuse to talk about amusement parks. You may want to brace yourself. Yes, this essay includes a picture. It would have included a video if I had enough WordPress privileges for that.

Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Quasirandom numbers.

Think of a merry-go-round. Or carousel, if you prefer. I will venture a guess. You might like merry-go-rounds. They’re beautiful. They can evoke happy thoughts of childhood when they were a big ride it was safe to go on. But they don’t often make one think of thrills.. They’re generally sedate things. They don’t need to be. There’s no great secret to making a carousel a thrill ride. They knew it a century ago, when all the great American carousels were carved. It’s simple. Make the thing spin fast enough, at the five or six rotations per minute the ride was made for. There are places that do this yet. There’s the Cedar Downs ride at Cedar Point, Sandusky, Ohio. There’s the antique carousel at Crossroads Village, a historical village/park just outside Flint, Michigan. There’s the Derby Racer at Playland in Rye, New York. There’s the carousel in the Merry-Go-Round Museum in Sandusky, Ohio. Any of them are great rides. Two of them have a special edge. I’ll come back to them.

Rye (New York) Playland Amusement Park’s is the fastest carousel I’m aware of running. Riders are warned ahead of time to sit so they’re leaning to the left, and the ride will not get up to full speed until the ride operator checks everyone during the ride. To get some idea of its speed, notice the ride operator on the left and how far she leans. She’s not being dramatic; that’s the natural stance. Also the tilt in the carousel’s floor is not camera trickery; it does lean like that. If you have a spare day in the New York City area and any interest in classic amusement parks, this is worth the trip.

Randomness is a valuable resource. We know it’s key to many things. We have major fields of mathematics built on it. We can understand the behavior of variables without ever knowing what value they have. All we need is to know than the chance they might be in some particular range. This makes possible all kinds of problems too complicated to do otherwise. We know it’s critical. Quantum mechanics would not work without randomness. Without quantum mechanics, matter doesn’t work. And that’s true randomness, the kind where something is unpredictable. It’s not the kind of randomness we talk about when we ask, say, what’s the chance someone was born on a Tuesday. That’s mere hidden information: if we knew the month and date and year of a person’s birth we would know whether they were born Tuesday or not. We need more.

So the trouble is actually getting a random number. Well, a sequence of randomly drawn numbers. We rarely need this if we’re doing analysis. We can understand how some process changes the shape of a distribution without ever using the distribution. We can take derivatives of a function without ever evaluating the original function, after all.

But we do need randomly drawn numbers. We do too much numerical work with them. For example, it’s impossible to exactly integrate most functions. Numerical methods can take a ferociously long time to evaluate. A family of methods called Monte Carlo rely on randomly-drawn values to estimate the integral. The results are strikingly good for the work required. But they must have random numbers. The name “Monte Carlo” is not some cryptic code. It is an expression of how randomly drawn numbers make the tool work.

It’s hard to get random numbers. Consider: we can’t write an algorithm to do it. If we were to write one, then we’d be able to predict that the sequence of numbers was. We have some recourse. We could set up instruments to rely on the randomness that seems to be in the world. Thermal fluctuations, for example, created by processes outside any computer’s control, can give us a pleasant dose of randomness. If we need higher-quality random numbers than that we can go to exotic equipment. Geiger counters watching the decay of a not-alarmingly-radioactive sample. Cosmic ray detectors watching the sky.

Or we can write something that produces numbers that look random enough. They won’t really be random, and if we wait long enough we’ll notice the sequence repeats itself. But if we only need, say, ten numbers, who cares if the sequence will repeat after ten million numbers? (We’ll surely need more than ten numbers. But we can postpone the repetition until we’ve drawn far more than ten million numbers.)

Two of the carousels I’ve mentioned have an astounding property. The horses in a file move. I mean, relative to each other. Some horse will start the race in front of its neighbors; some will start behind. The four move forward and back thanks to a mechanism of, I am assured, staggering complexity. There are only three carousels in the world that have it. There’s Cedar Downs at Cedar Point in Sandusky, Ohio; the Racing Downs at Playland in Rye, New York; and the Derby Racer at Blackpool Pleasure Beach in Blackpool, England. The mechanism in Blackpool’s hasn’t operated in years. The one at Playland’s had not run in years, but was restored for the 2017 season. My love and I made a trip specifically to ride that. (You may have heard of a fire at the carousel in Playland this summer. This was of part of the building for their other, non-racing, antique carousel. My last information was that the carousel itself was all right.)

These racing derbies have the horses in a file move forward and back in a “random” way. It’s not truly random. If you knew exactly which gears were underneath each horse, and where in their rotations they were, you could say which horse was about to gain on its partners and which was about to fall back. But all that is concealed from the rider. The horse patterns will eventually, someday, repeat. If the gear cycles aren’t interrupted by maintenance or malfunctions. But nobody’s going to ride any horse long enough to notice. We have in these rides a randomness as good as what your computer makes, at least for the purpose it serves.

The racing nature of Playland’s and Cedar Point’s derby racers mean that every ride includes exciting extra moments of overtaking or falling behind your partners to the side. It also means quarreling with your siblings about who really won the race because your horse started like four feet behind your sister’s and it ended only two feet behind so hers didn’t beat yours and, long story short, there was some punching, there was some spitting, and now nobody is gonna be allowed to get ice cream at the Carvel’s (for Playland) or cheese on a stick (for Cedar Point). This is the Cedar Downs ride at Cedar Point, and focuses on the poles that move the horses.

What does it mean to look random? Some things seem obvious. All the possible numbers ought to come up, sooner or later. Any particular possible number shouldn’t repeat too often. Any particular possible number shouldn’t go too long without repeating. There shouldn’t be clumps of numbers; if, say, ‘4’ turns up, we shouldn’t see ‘5’ turn up right away all the time.

We can make the idea of “looking” random quite literal. Suppose we’re selecting numbers from 0 through 9. We can draw the random numbers we’ve picked. Use the numbers as coordinates. Say we pick four digits: 1, 3, 9, and 0. Then draw the point that’s at x-coordinate 13, y-coordinate 90. Then the next four digits. Let’s say they’re 4, 2, 3, and 8. Then draw the point that’s at x-coordinate 42, y-coordinate 38. And repeat. What will this look like?

If it clumps up, we probably don’t have good random numbers. If we see lines that points collect along, or avoid, there’s a good chance our numbers aren’t very random. If there’s whole blocks of space that they occupy, and others they avoid, we may have a defective source of random numbers. We should expect the points to cover a space pretty uniformly. (There are more rigorous, logically sound, methods. The eye can be fooled easily enough. But it’s the same principle. We have some test that notices clumps and gaps.) But …

The thing is, there’s always going to be some clumps. There’ll always be some gaps. Part of randomness is that it forms patterns, or at least things that look like patterns to us. We can describe how big a clump (or gap; it’s the same thing, really) is for any particular quantity of randomly drawn numbers. If we see clumps bigger than that we can throw out the numbers as suspect. But … still …

Toss a coin fairly twenty times, and there’s no reason it can’t turn up tails sixteen times. This doesn’t happen often, but it will happen sometimes. Just luck. This surplus of tails should evaporate as we take more tosses. That is, we most likely won’t see 160 tails out of 200 tosses. We certainly will not see 1,600 tails out of 2,000 tosses. We know this as the Law of Large Numbers. Wait long enough and weird fluctuations will average out.

What if we don’t have time, though? For coin-tossing that’s silly; of course we have time. But for Monte Carlo integration? It could take too long to be confident we haven’t got too-large gaps or too-tight clusters.

This is why we take quasi-random numbers. We begin with what randomness we’re able to manage. But we massage it. Imagine our coins example. Suppose after ten fair tosses we noticed there had been eight tails turn up. Then we would start tossing less fairly, trying to make heads more common. We would be happier if there were 12 rather than 16 tails after twenty tosses.

Draw the results. We get now a pattern that looks still like randomness. But it’s a finer sorting; it looks like static tidied up some. The quasi-random numbers are not properly random. Knowing that, say, the last several numbers were odd means the next one is more likely to be even, the Gambler’s Fallacy put to work. But in aggregate, we trust, we’ll be able to enjoy the speed and power of randomly-drawn numbers. It shows its strengths when we don’t know just how finely we must sample a range of numbers to get good, reliable results.

To carousels. I don’t know whether the derby racers have quasirandom outcomes. I would find believable someone telling me that all the possible orderings of the four horses in any file are equally likely. To know would demand detailed knowledge of how the gearing works, though. Also probably simulations of how the system would work if it ran long enough. It might be easier to watch the ride for a couple of days and keep track of the outcomes. If someone wants to sponsor me doing a month-long research expedition to Cedar Point, drop me a note. Or just pay for my season pass. You folks would do that for me, wouldn’t you? Thanks.