Beta's colander

This is a classic titanic post, because (as so often) I don't really know what I'm talking about. Through the lockdown I made a half-hearted attempt to learn R, the leading statistical programme. As usual, what caught my eye was an ancient Greek. 

Specifically, Eratosthenes of Cyrene. Eratosthenes is one of the great figures of Hellenistic Alexandria. He was head of the library there, the highest intellectual position of the day. He's best remembered today for estimating the circumference of the earth to an astounding degree of accuracy, but he was also a polymath interested in literature, music, and, as we'll see, mathematics. He was called 'Beta' by his peers, not because he was on Reddit, but because he was the second best at everything.

Not too far into my studies in R, I came across Eratosthenes' sieve, a simple algorithm that's often used as an exercise for coding students. What follows is an attempt to unpack one way of coding Eratosthenes' sieve in R, with some help from a friend and some code I found online. I struggled with it a bit myself, which should hopefully put me in the ideal position to explain, since I can remember what confused me (everything), and nothing about it seems obvious to me. 

I'll start with the sieve itself. The point of it is to find all the prime numbers up to a certain point (let's say, up to 100). Starting with 2, what you do is to move through the rest of the numbers crossing out all of the multiples of 2 (since if they're multiples of any other number higher than 1 they can't be prime). Then you do the same with 3, 4, and so on, until you can't do it anymore, since all the multiples of the number you started with have already been crossed out (as multiples of an earlier number). Then you can look at all the numbers that haven't been crossed out. Those are your primes. 

I guess it makes sense as an exercise for beginner coders because it's a pretty easy to understand algorithm, a set of steps for a brain or computer to work through to get a set of outputs (here, prime numbers).

So - as previously advertized - below is one way of getting the R programme to become, for a thrilling and infinitely repeatable moment, the head librarian of Alexandria. After the code itself I'll break it all up into bits, accompanying each bit with some comments that hopefully a) are correct and b) help you understand what's going on. Code is bold, the commentary (a highly Alexandrian form) not. The commentary is talking to R and telling it what to do (imperative mood).

sieve <- function(n) {   

if (n < 2) return(NULL) 

  a <- rep(T, n) 

  a[1] <- F 

  for(i in seq(n)) { 

    if (a[i]) { 

      j <- i * i 

      if (j > n) return(which(a)) 

      a[seq(j, n, by=i)] <- F 

   }

  }

}

sieve <- function(n) { 

Make "sieve" a function that does the below to a number we'll call "n." (This just names our algorithm and makes it a function, a way of doing things to what's in the brackets that follow).

if (n < 2) return(NULL) 
a <- rep(T, n) 

If n is smaller than 2 then spit out NULL. (In other words, refuse to do this computation if it's on a number of numbers less than 2).

Make a vector called "a" and store in it n repetitions, all marked "true." (T stands for 'true.' This is like writing out the number in the grid above. Saying they're all true effectively means we're starting with the assumption that they're all prime.)

a[1] <- F 

Store the first item as false in "a." (Because we want to get rid of 1 immediately?)

for(i in seq(n)) { 

Start with i = 1, iterating the code between here and the closing }, incrementing i by one each time, until i = seq(n). (Note the opening curly bracket. The code within the for loop, contained between the curly brackets, will run a number of times equal to the number of elements in seq(n), starting with i = 1, and with the value of i increasing by one each time it runs.)

if (a[i]) { 

If a[i] is true, the code within the ensuing curly brackets will execute. If a[i] is false, it won't. (So if an individual number is false, it won't run the code. At this point that's just 1, as specified above, so I think this just stops R running this algorithm with the number 1. That's actually important, because if it did it would cross off all the numbers and we wouldn't get any primes; another way of looking at this is that being a multiple of 1 doesn't mean a number isn't a prime, and the algorithm needs to recognize this.)

j <- i * I

This finds the square of i and stores it in variable j. Remember we're in the for loop, with i incrementing each time the loop repeats. The first time round i = 1 and j = 1; the second, i = 2 and j = 4; the third, i = 3 and j = 9; and so on.

 if (j > n) 

If (and only if) the square of i is greater than the number of elements in the entire sequence...(i.e. that number times itself is larger than e.g. 100...)

return(which(a)) 

Return the sequential positions of all those numbers in a which are true (T). (The idea here is that if we've reach a number whose square is greater then the number of numbers in the sequence, e.g. 100, then we've found all the primes already. I guess this is just a separate assumption that happens to be true?)

a[seq(j, n, by=i)] <- F 

Mark as F (false, i.e. not prime - in other words, cross out) all the numbers in a between the jth and nth (the nth being the last) that are divisible by (that's what 'by' does) i (and thus aren't prime). This line is the heart and soul of the code, the steely essence of Beta's colander. 

OK, I'm still not sure I understand all of that, so if you want to try to help out in the comments be my guest. 

Stairway walks in Wellington #1: Oriental Parade to Mount Vic Lookout (and back again)

This post is a tribute to Ada Bakalinsky and her classic Stairway Walks in San Francisco, which was once gifted to me by a fellow Hellenist. Anyone who's lived in both cities (as I've had the good fortune to do) or has even been to both immediately notices the similarity - both are compact, hilly cities build around a bay (with all the weather that that brings). And, though SF's obviously a lot larger, I'd wager that windy Welly can more than hold its own in terms of unexpected shortcuts from one vertical plane to another, and in terms of stairways in particular. So I thought I'd write up a Bakalinskian 'stairway walk' in Wellington here, partly just because it's an enjoyable thing to do. It may be the first of many, or the last of one. We'll see.

But for now, here's a walk that'll take you from down by the harbour on Oriental Bay up to the commanding heights of Mount Victoria Lookout and back again, with an embarrassment of prospects along the way. It think it took me about an hour and a half, but I could imagine it filling out up to two hours if you lingered a bit longer at the top to luxuriate in the panoramic views. (The Google Maps estimate doesn't seem to account for the incline, and it's pretty much all inclines).

The Freyberg Pool is a good place to start - it's within striking distance from Courtenay Place, the city's entertainment hub, and is easy to find (it's a massive sports complex jutting out into the habour). Walk back down the road toward the city with the harbour on your right until you get to the cafe Lola Stays. Just to the left of that is a long steep row of steps. Go up those and turn right up another little stairway. Go down the road with the Copthorne Hotel on your right, then turn left then left again onto McFarlane (trust me, it's the only way). Keep going up McFarlane as it dips down and then soars up until the boats below start looking like toys.

On your right you'll start to see a old (well, early 20th century) brick monastery looming above you - that's St. Gerard's. There's a stairway just to the right of it. Go up that with the monastery now looming above you to the left.

Then just continue upwards along the road till you come to the start of the town belt on Mount Victoria. Once you've entered into the forrested realm (where the Dark Riders once roamed), things can get a bit confusing. There are so many little paths! Try to stick to the main one, and if you stray off it (as you probably will), just keep going up and to the right, way from the water sparkling away to your left through the trees.

Near the top at least you should be able to re-join the main walk-way, and start getting a sense of how things are about to open up into light and sky.

At this point you'll see Mt. Vic lookout itself, with its monuments, artifacts, and 360 views of Wellington Harbour (as well as its glimpse of the Cook Straight).

And somewhere down there you'll see St. Gerrard's again, this time with you looming above it.

At this point (if you want to follow this walk) look out for the Lookout Road (with the radio tower on the right of it), and walk along that, looking out along the way for look-outs, e.g. one onto the Westpac Stadium (a.k.a. 'The Cake Tin').

At the end of this there's a little lookout nook with a statue of the city's eponymos. And more views (e.g., if you look the other way from the last picture, it's like you're not really in a city at all).

From here there's another little stairway to the right. If you go down this you'll walk through or past a little green area/lawn/park. Here's the view back up the hill once you've done that.

As the sign suggests, you're now on Thane Road. Follow this on its winding downhill trip (with more views on the right over Evans Bay) until you turn a sharpish right corner with another green area to your right. (And a sign for the Southern Walkway). Get on this. It's not a stairway but it probably should be. Instead it's a virtually vertical concrete path that'll takes you like a glass elevator down as you face the harbour.

There are a couple of turns off here. You can basically turn left and even more sharply downhill whenever you feel like getting back to Oriental Parade. I think this one turns off onto Hay Street, but I kept going onto Glass. Here's the NuZild-shabby little park marking the turnoff onto Glass.

Just down the road from here there's also a secret meditation spot where, the day I was there, someone had left their passport - and then returned to get it just as I was wondering what to do. Anyway, once you've had your moment of reflection in the face of the harbour you can turn down to your final staircase, at which point Oriental Bay will be there to welcome you back.

And then? Get an ice-cream, lay on the beach, go for a swim - whatever you want. As with Bakalinsky's bountiful book, there are plenty of possible add-ons and variations, as well as completely different routes in the same area. This was just to get you started in case you needed an idea. Oh, and here's that last area on a slightly better map. If you want you can not turn down Glass but instead keep going to the end of the Southern Walkway, which would probably add 15 minutes or so.