Category Archives: Uncategorized

Stdout is a dead end

I haven’t written much about it on this blog but the project I’ve been working on full time for the last two years is a new kind of terminal, chorus. Current terminals are a relic of the past and chorus is my take on a modern replacement.

I’ve created a blog specifically about chorus, the chorus console blog. There I’ll be writing about the approach I’m taking and my thoughts about what terminals are and where they should be moving. The title of this post itself hints at where I see them evolving. But you can read more about that in the introductory post.

A simple way to pre-scramble a sequence

I have this problem where I need to try something a number of times, maybe 64 times, and each time with a different small seed, say a value between 0 and 1024. I’d like the seeds to be reasonably scrambled and I don’t want to see the same one more than once. I also need the same order every time, that’s what I mean by pre-scrambling – the scrambling isn’t dynamic, it’s hardcoded. How do I do this in a simple way without resorting to say just hard coding a list of numbers?

What I’d really like to do is use something like a random number generator but with pre-determined set parameters that I know will generate the numbers I want. The xorshift generator seems like a good candidate because it’s really simple but the quality is decent. It requires 4 parameters, x, y, z, and w, and then you can generate a sequence of values in the following way, where limit is the size of the set you want to generate:

t = x ^ (x << 11) & 0xFFFFFFFF
x = y
y = z
z = w
w = (w ^ (w >> 19) ^ t ^ (t >> 8)) & 0xFFFFFFFF
return w % limit

This is what I’d like to use — I just need some parameters. And it turns out that there are parameters that will make it behave like I want. Here’s an example,

0x31f12daf 0x6dd97342 0xda4daacd 0x41235c71: 85/256 85 (33.2% 1x)

This means: with these parameters x y z w and a limit of 256 you get 85 different values (that’s the 85 in “85/256”) if you execute the xorshift code 85 times (that’s the second 85), which means that you’re covering 33.2% of the possible values and executing the loop 1x per unique generated value. Here are some seeds for different limits that are powers of 2, that generate a pretty big set of unique values,

0x2735e8f4 0x9fcb0c3e 0xd0c18064 0x0f6b0093: 26/32 26 (81.2% 1x)
0x1ff079d6 0x54248059 0xc968e911 0x38c15c75: 39/64 39 (60.9% 1x)
0xf8efc789 0x88a40b37 0x86f01aa6 0x15384f52: 59/128 59 (46.1% 1x)
0x31f12daf 0x6dd97342 0xda4daacd 0x41235c71: 85/256 85 (33.2% 1x)
0xfd8193d0 0xc8b69d8f 0xa42b9d1c 0x7e69d727: 133/512 133 (26% 1x)
0x4617b049 0xf7a49270 0xa88b568c 0x32dff77c: 174/1024 174 (17% 1x)

As an example of how to use one of these, here’s how you’d use the first line, the one that generates values with a limit of 32, in python:

def generate():
  x = 0x2735e8f4
  y = 0x9fcb0c3e
  z = 0xd0c18064
  w = 0x0f6b0093
  for i in range(0, 26):
    t = (x ^ (x << 11)) & 0xFFFFFFFF
    x = y
    y = z
    z = w
    w = (w ^ (w >> 19) ^ t ^ (t >> 8)) & 0xFFFFFFFF
    yield w % 32

print list(generate())

This prints

[2, 18, 9, 30, 12, 0, 19, 17, 7, 5, 14, 16, 6, 24, 15, 4, 3,
31, 10, 29, 8, 23, 11, 22, 1, 26]

which, and you can check if you want, are 26 different values between 0 and 32.

If you can tolerate some duplicates you can get better coverage. If you can only tolerate a few, say about one duplicate in 16 values, you can get,

0xf852d2f8 0xaf137287 0x6a0d4795 0x326c8a41: 28/32 29 (87.5% 1x)
0xa8683488 0x361c1383 0xbc95298a 0xe4c57646: 46/64 49 (71.9% 1.1x)
0xca30a753 0x317aa350 0x1bc8bee2 0xce4830bf: 79/128 87 (61.7% 1.1x)
0xe079cdb3 0x7cb29574 0x45eebac8 0xeb8dcefc: 132/256 147 (51.6% 1.1x)
0xf41e4441 0x3345d823 0x608ff840 0x554f5cdd: 238/512 269 (46.5% 1.1x)
0x201ceef2 0x7f49ce17 0x0bd10615 0x139bfa7b: 423/1024 485 (41.3% 1.1x)

So for example in the 256 case, if you generate 147 values with the given seed you’ll hit 132 different values between 0 and 256, and the rest will be duplicates. That’s a lot better coverage than the 85 you could hit if you didn’t allow them.

If you can tolerate even more duplicates, say you don’t mind hitting roughly one on average per unique value, you can get close to full coverage

0xa9bb0026 0xe0178ff2 0x443fd052 0x9fe0d459: 32/32 41 (100% 1.3x)
0x4c19f4c7 0x1d253873 0x5c1f6f17 0x812944fc: 64/64 106 (100% 1.7x)
0x854fbffe 0x5aa965d5 0x06ff8c3e 0xd5057351: 127/128 252 (99.2% 2x)
0xeb03a458 0x26ae990e 0x6227aaee 0x192b3cb7: 242/256 496 (94.5% 2x)
0x97bbc402 0x20470416 0x09ce179b 0xbf276b7d: 469/512 980 (91.6% 2.1x)
0x6d36254c 0x19c5f46a 0x1614da1a 0xcfadbfc8: 915/1024 1939 (89.4% 2.1x)

As an example, running the same python code from above but now with parameters from this set you get,

[19, 27, 10, 12, 18, 16, 13, 15, 24, 4, 23, 11, 28, 30, 26,
19, 24, 1, 12, 15, 29, 2, 19, 8, 9, 20, 6, 5, 17, 3, 14, 31,
22, 25, 18, 27, 8, 20, 0, 7, 21]

In this list of length 41 all numbers between 0 and 31 occur, but 8, 12, 15, 18, 20, 24, and 27 are there twice and 19 is there three times.

Finally, if you don’t care how many times you try as long as you generate all the numbers that’s possible too. At this point you probably want to keep track of which value you’ve seen, maybe with a bit vector, and just filter the raw values you generate.

0xa9bb0026 0xe0178ff2 0x443fd052 0x9fe0d459: 32/32 41 (100% 1.3x)
0x4c19f4c7 0x1d253873 0x5c1f6f17 0x812944fc: 64/64 106 (100% 1.7x)
0xf3bd4634 0xb1442411 0xd536236f 0x3beae2fb: 128/128 291 (100% 2.3x)
0x1c608f31 0x34e54bbd 0xa799c7a8 0x78af6664: 256/256 729 (100% 2.8x)
0x10bace53 0x88bc0781 0xd815fe20 0xc77bcbec: 512/512 1786 (100% 3.5x)
0xb6c93670 0xb89f9de6 0x3ba3713c 0x78bcabc7: 1024/1024 4341 (100% 4.2x)

If you look closely you’ll notice that the parameters are the same as before for some limits, that’s because we already had perfect coverage before so there’s nothing to improve. And as the set grows you can see that the number of times you have to try grows, so at 1024 you need to loop 4341 times before you’ve seen all values. And note also that the duplicates aren’t evenly distributed, the further you get the more work you have to do to get a value you haven’t seen before.

Of limited use I guess but if this is the kind of thing you need, pretty convenient. I’ve dumped the code I used to generate these in a gist.

Piping into a desktop indicator

This post is about interacting with the graphical desktop environment from a shell script. Specifically: say you have a long-running bash script and you’d like to keep an eye on it through a unity app indicator. How do you make that work in an appropriately unix-y way? (If you’re not sure what I mean see the animation below).

Background. Over the weekend I finally changed from NFS to rsync for keeping my source code in sync across machines. Much better if my network connection is unreliable and fewer moving parts. After I was done with the basics though I thought: I really need to be able to keep an eye on how syncing is going. It’s just the worst to spend ages unsuccessfully trying to reproduce an issue on a virtual machine only to realize that it’s because the code I’m running isn’t being properly synced when I edit files on my laptop.

Here’s what I had in mind: when I run the code change monitor shell script I want to get an app indicator in the top bar that changes state when a sync is going on, so I can see that syncing actually happening and whether it’s slow. And I want it to show if there are any errors. This, in other words


The syncing logic itself is straightforward: one component uses inotifywait to wait for files to change (, then calls out to a script that does the actual sync (

How to show the indicator though? There is no command-line api for this kind of thing and I wasn’t sure what it would look like if there was. Also, the existing code was properly unix-y and I wanted it to stay that way. One component knows how to wait for changes but knows nothing about rsync; the rsync part knows how to do a single sync but doesn’t care when or why it is being called. They only meet at the top level, --root . ./

What I ended up with was a python script,, that was responsible for just the indicator. When it starts it creates the indicator and then waits around, reading input from stdin. When it sees a command in the output it recognizes, say [SYNC: syncing], it consumes that line and changes the state of the indicator. Anything that doesn’t match is just passes straight through to stdout. Here’s what it looks like,


This way the indicator logic doesn’t need to know about the watching or rsyncing, and those parts don’t need to know about the indicator. All you need to do is change the toplevel command that puts it all together to, --root . ./ | ./

where is a trivial wrapper around that looks like this,


echo "[SYNC: syncing]"
if ./; then
  echo "[SYNC: idle]"
  echo "[SYNC: error]"

You can even replace this with a generic command that takes another command as an argument and all it does is call that command surrounded with appropriate SYNC messages.

Pretty clean and unix-y I think, independent components and so on.  The python indicator script itself,, is straightforward, only 60ish lines with comments.

The ease of supporting this makes me wonder though. Why aren’t long-running commands better at interacting with the desktop?

Poly-ranges in haskell

I’ve been reading Learn You a Haskell and got stuck on a tiny point at the beginning, in the section about ranges: you can’t have a range with more than one step value. This post is about what it might mean to allow multiple steps and how Charles Babbage might have had an opinion on this particular aspect of haskell’s design.

Here’s the whole paragraph from Learn You a Haskell,

To make a list containing all the natural numbers from 1 to 20, you just write [1..20]. [… snip …] Ranges are cool because you can also specify a step. What if we want all even numbers between 1 and 20? Or every third number between 1 and 20?

ghci> [2,4..20]
ghci> [3,6..20]

It’s simply a matter of separating the first two elements with a comma and then specifying what the upper limit is. While pretty smart, ranges with steps aren’t as smart as some people expect them to be. You can’t do [1,2,4,8,16..100] and expect to get all the powers of 2. Firstly because you can only specify one step. And secondly because some sequences that aren’t arithmetic are ambiguous if given only by a few of their first terms.

The bold highlight is mine. This makes sense – how can the language guess that what you have in mind when you write [1, 2, 4, 8, 16..] is the powers of 2?

On the other hand though, just as a fun exercise, if you were to allow multiple step values how would you define them? Even though this is a highly specialized and modern language design question I know Charles Babbage would have understood it and I suspect he would have had an opinion.

A while ago I wrote a post about Babbage’s difference engine. The difference engine is basically an adder that calculates successive values of a polynomial by doing simple addition, using a technique called divided differences. This is described in full detail in that post but here’s a summary. Take a sequence of values of values of a polynomial, for instance the first four squares,

  \begin{tabular}{lcl}  f(0) & = & 0 \\  f(1) & = & 1 \\  f(2) & = & 4 \\  f(3) & = & 9 \\  & \vdots &  \end{tabular}

You can then calculate the next values using just addition by calculating the differences between the initial values. To get the differences for a sequence you subtract each pair of values,

  \begin{tabular}{l|l}  0 \\  & 1 \\  1 \\  & 3 \\  4 \\  & 5 \\  9  \end{tabular}

Do it once and you get the first differences. Do the same thing to the first differences and you get the second differences,

  \begin{tabular}{l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 \\  4 && 2\\  & 5 \\  9  \end{tabular}

Do the same thing to the second differences and you get the third differences,

  \begin{tabular}{l|l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 && 0\\  4 && 2\\  & 5 \\  9  \end{tabular}

In this case there’s just one and it’s 0 so we can ignore it. Once you have all the differences you can calculate the next value of the polynomial by adding them together from right to left,

  \begin{tabular}{l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 \\  4 && 2 \\  & 5 \\  9  \end{tabular}  \quad\to\quad  \begin{tabular}{l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 \\  4 && 2 \\  & 5 & \tiny{+0} \\  9 && \bf{2}  \end{tabular}  \quad\to\quad  \begin{tabular}{l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 \\  4 && 2 \\  & 5 \\  9 & \tiny{+2} & \bf{2} \\  & \bf{7} \\  \end{tabular}  \quad\to\quad  \begin{tabular}{l|l|l}  0 \\  & 1 \\  1 && 2\\  & 3 \\  4 && 2 \\  & 5 \\  9 && \bf{2} \\  \tiny{+7} & \bf{7} \\  \bf{16} \\  \end{tabular}

There it is, 16, the next value in the sequence. At this point we can just keep going,

  \begin{tabular}{l|l|l}  \multicolumn{1}{r}{}&&2 \\  & 7 & \tiny{+0}\\  16 &\tiny{+2}& 2 \\  \tiny{+9} & 9 \\  \bf{25} \\  \end{tabular}  \quad\to\quad  \begin{tabular}{l|l|l}  \multicolumn{1}{r}{}&&2 \\  & 9 & \tiny{+0}\\  25 &\tiny{+2}& 2 \\  \tiny{+11} & 11 \\  \bf{36} \\  \end{tabular}  \quad\to\quad  \begin{tabular}{l|l|l}  \multicolumn{1}{r}{}&&2 \\  & 11 & \tiny{+0}\\  36 &\tiny{+2}& 2 \\  \tiny{+13} & 13 \\  \bf{49} \\  \end{tabular}  \quad\to\quad\dots

This technique works for any polynomial. The difference engine was given its name because it is a mechanical implementation of it.

Already you may suspect how this can be relevant to ranges in haskell. You can generalize ranges to n step values by using divided differences on the step values to generate the rest of the sequence. With two step values the result of this is a linear sequence, as you would expect (polyEnumFrom is my name for the generalized range function),

ghci> polyEnumFrom [0, 1]
[0, 1, 2, 3, 4, 5, 6, ...]
ghci> polyEnumFrom [4, 6]
[4, 6, 8, 10, 12, 14, 16, ...]
ghci> polyEnumFrom [10, 9]
[10, 9, 8, 7, 6, 5, 6, ...]

Give it three elements, for instance the squares from before, and the result is a second-order polynomial,

ghci> polyEnumFrom [0, 1, 4]
[0, 1, 4, 9, 16, 25, 36, ...]

Similarly with the cubes,

ghci> polyEnumFrom [0, 1, 8, 27]
[0, 1, 8, 27, 64, 125, 216, ...]

There is nothing magical going on here to recognize the squares or cubes here, it’s a completely mechanical process that uses subtraction to get the differences and addition to then continue the sequence past the step values. It won’t give you the squares though, only polynomials. If you try you get

ghci> polyEnumFrom [1, 2, 4, 8, 16]
[1, 2, 4, 8, 16, 31, 57, ...]

For any list of 3 step values there is exactly one polynomial of order 2 or less that yields those values; an order-2 polynomial has 3 free variables so there’s no room for more than one solution. The result of a 3-step range is the sequence of values produced by that order-2 polynomial. In general, the result of a multi-step range with n steps is the sequence of values generated by the unique polynomial of order at most n-1 whose first n values are the steps. Conveniently, if the steps are integers the polynomial is guaranteed to only produce integers.

Babbage built the difference engine because it was really important then be able to produce tables of function values by calculating successive values of polynomials. I think he would have appreciated the power of a language where you can condense the work of generating an approximate table of sine values into a single line,

take 10 (polyEnumFrom [0, 0.0002909, 0.0005818, 0.0008727])

which yields,


(The four values given to polyEnumFrom are the first four sine values from the table.) Of course, Babbage did lose interest in the difference engine himself and preferred to work on the more general analytical engine. If he did see haskell he would probably quickly lose interest in ranges and move on to more general advanced features. But I digress.

Is a shorthand for sequences of polynomial values actually a useful feature in a language? I doubt it. But I like the idea that a corner of a modern language  intersects with Babbage and the difference engine. Maybe it would make sense as a little easter-egg tribute to Babbage, a difference engine hiding behind a familiar language construct. I quite like the idea of that.

If you want to play around with it I’ve put an implementation in a gist. For something that might seem fairly complex the implementation is simple – polyEnumFrom and its helpers is just 13 lines. Haskell is an elegant language.

Z-Quads: Conversions

This is the third post in a series about z-quads, a spatial coordinate system. The first one was about the basic construction, the second about how to determine which quads contain each other. This one is about conversions: how to convert positions to quads and back.

squares-with-grid163241First, a quick reminder. A quad is a square(-ish) area on the surface of the earth. Most of the time we’ll use the unit square instead of the earth because it makes things simpler and you can go back and forth between lat/longs and unit coordinates just by scaling the value up and down. (There’s a section at the end about that.)

So, when I’m talking about encoding a point as a quad, what I mean more specifically is: given a point on the unit square (x, y) and a zoom level z, find the quad at zoom level z that contains the point. As it turns out, the zoom level doesn’t really matter so we’ll always find the most precise quad for a given point. If you need a different zoom level you can just have a step at the end that gets the precise quad’s ancestor at whatever level you want.


So far we’ve focused on quads as an abstract mathematical concept and there’s been no reason to limit how far down you can zoom. In practice though you’ll want to store quads in integers of finite size, typically 32-bit or 64-bit. That limits the precision.

It turns out that you can fit 15 zoom levels into 32 bits and 31 into 64. At zoom level 15 the size of a quad is around 1.5 km2. At zoom level 31 it’s around 3 cm2. So yeah, you typically don’t need more than 31 levels. A third case is if you want to store your quad as a double – JavaScript, I’m looking at you here. A double can hold all integers up to 53 bits, after which some can’t be represented. That corresponds to 26 zoom levels which is also plenty precise for most uses.

Encoding points

Okay, now we’re ready to look at encoding. What we’ll do is take a unit point (x, y) and find the level 31 quad that contains it. Since we’ve fixed the zoom level we’ll immediately switch to talking about the scalar rather than the quad. That makes everything simpler because within one level we’re dealing with a normal z-order curve and conversion to and from those are well understood.

sumThe diagram on the right illustrates something we’ve seen before: that scalars are numbered such that the children at each level are contiguous. In this example, the first two green bits say which of the four 2×2 divisions we’re in. The next two say which 1×1 subdivision of that we’re in. It’s useful to think of this the same way as the order of significance within a normal integer: the further to the left a pair of bits is in a scalar the more significant it is in determining where the scalar is.

Now things will have to get really bit-fiddly for a moment. Let’s look at just a single 2×2 division. Each subdivision can be identified with an (x, y) coordinate where each ordinate is either 0 or 1, or alternatively with a scalar from 0 to 3. The way the scalars are numbered it happens that the least significant scalar bit gives you the x-coordinate and the most significant one gives you y:


(This is by design by the way, it’s why the z-shape for scalars is convenient). This means that given point x and y, each 0 or 1, you get the scalar they correspond to by doing

x + 2 y

What’s neat about that is that because of the way pairs of bits in the scalar increase in significance from right to left matches the significance of bits in an integer, you can do this for each bit of x and y at the same time, in parallel. To illustrate how this works I’ll take an example.

Let’s start with the point (2/5, 2/3) and find the level 5 scalar that contains it. First we’ll multiply both values by 32 (=25) and round the result to an integer,

x = (2/5) × 32 ≈ 12 = 01100b
y = (2/3) × 32 ≈ 21 = 10101b

Now we’ll perform x + 2 y for each bit in parallel like so,


Looks bizarre right, but it does work. We spread out the bits from each input coordinate so they can be matched up pairwise and then add them together. The result is the scalar we’re looking for because the order of significance of the bit-pairs in the scalar match the significance of the individual bits in each of the inputs.

Now, how can you tell that the result, scalar 626 on zoom level 5, is the right answer? This diagram suggests that it’s about right:


The bit-pairs in the scalar are [2, 1, 3, 0, 2]  so we first divide and take subdivision 2, then divide that and take 1, then divide that, and so on. Ultimately we end up at the green dot. In this case the last step would be to add the appropriate bias, b5, to get a proper quad and then we’re all done.

Because this conversion handles all the levels in parallel there’s little benefit to converting to a higher zoom level than the most precise, typically 31. If you want level 5 do the conversion to level 31 and then get the 26th ancestor of the result.

Most of the work in performing a conversion is spreading out the scaled 31-bit integer coordinates. One way to spread the bits is using parallel prefix which takes log2 n steps for an n-bit word,

→ 0000abcd0000efgh
→ 00ab00cd00ef00gh
→ 0a0b0c0d0e0f0g0h

That’s just one way to do it, there are others.

Decoding quads

The operation above gets you into quad-land and obviously you’ll want to be able to get out again. This turns out to just be a matter of running the encoding algorithm backwards. The main difference is that here we’ll stay on whatever zoom level the quad’s on instead of always going to level 31.

Given a quad q this is how you’d decode it:

  1. Subtract the bias to get the quad’s scalar.
  2. Mask out all the even-index bits to get the spread x value and the odd-index bits to get the spread y value. Shift the y value right by 1.
  3. Pack the spread-x and –y values back together to get a zq-bit integer value for each. Packing can be similarly to spreading but in reverse.
  4. Floating-point divide the integer values by 2zq. You now have your (x, y) coordinates between 0 and 1.

A quad is an area but the result here is a point, the top left corner of the quad. This is often what you’re interested in, for instance if you’re calculating is the full area of the quad. There you can just add the width and height of the quad (which are both 2zq) to get the other corners. Alternatively you might be interested in the center of the quad instead. To get you can modify the last step of the algorithm to do,

Apply (2 v + 1) / 2zq+1 as a floating-point operation to each of the integer coordinates.

This is just a circuitous way to add 0.5 to v before dividing by 2zq but this way we can keep the value integer until the final division. If you want to minimize the loss of precision when round-tripping a point through a quad you should use the center since it’s the point with the smallest maximum distance to the points that belong to that quad.

So, now we can convert floating-point coordinates to quads and back again with ease. Well, relative ease. In the last part I’ll talk briefly about something I’ve been hand-waving until now: converting from lat/long to the unit square and back.

Latitude and longitude

mapWhen you encounter a lat/long it is typically expressed using the WGS 84 standard. The latitude, a value between -90 and 90, gives you the north-south position. The longitude, a value between -180 and 180, gives you east-west.

A typical world map will look something like the one on the right here with America to the left and Asia to the right. The top left corner, northwest of North America, is lat 90 and long -180. The bottom right, south of New Zealand, has lat -90 and long 180. You could convert those ranges to an (x, y) on the unit square in any number of ways but the way I’ve been doing it follows the z-order curve the same way I’ve been using in this post: North America is in division 0, Asia in division 1, South America in division 2, and Australia in division 3. The conversion that gives you that is,

x = (180 + long) / 360
y = (90 – lat) / 180

This puts (0, 0) in the top left corner and (1, 1) in the bottom right and assigns the quads in that z-shaped pattern.

There’s a lot to be said about map projections at this point. I limit myself to saying that while it’s convenient that there’s a simple mapping between the space most commonly used for geographic coordinates and the space of quads, the Mercator projection really is a wasteful way to map a sphere onto a square. For instance, a full third of the space of quads is spent covering the Southern Ocean and Antarctica even though they take up much less than a third of the surface of the globe. I’d be interested in any alternative mappings that make more proportional use of the space of quads.

Okay, so now there’s a way to convert between WGS 84 and quads. I suspect there’s a lot more you can do with z-quads but this, together with the operations from the two previous posts, what I needed for my application so we’re at to the end of my series of posts about them. Well, except for one final thing: string representation. I’ve always been unhappy with how unmemorable geographic positions are so I made up a conversion from quads to sort-of memorable strings which I’ll describe in the last post.

In the meantime I made a little toy site that allows you to convert between positions and quads at a given zoom level. Here is an example,

This is quad 167159423 at zoom level 14 which covers, as always, Århus. The URL is,10.2062&zoom=14

which kind of explains itself. You can also do

Finally to see where a particular quad is try

If you want to explore the space of quads you can play around with the parameters yourself.

Z-Quads: Construction

This post is about a spatial coordinate system I’ve been playing around with, z-quads. Unlike most commonly used coordinate systems a z-quad doesn’t represent a point but a square with an area.

163241637To the right are some examples of quads of increasing accuracy, gradually zooming in on Århus. (The quads are the red rectangles). The first one that covers northern Germany, some of England, and half of Denmark, is quad 637 which has zoom level 5. 171171340006668638046The one that covers all of Århus is 163241 which has zoom level 9. Going further down to zoom level 15 is 668638046 which covers part of the center of Århus, and finally at zoom level 19 is 171171340006 which covers just a couple of houses.

This should give a rough sense of how the system works. A quad is an integer, typically 64 bits, which represents some square area on the earth. Each quad has a zoom level associated with it and higher numbers have higher zoom levels. There’s a lot more to it than that but that’s the core of it.

Below I’ll describe how z-quads are constructed, which is pretty straightforward. Then I’ll look at how to operate on them: how to convert to and from them, zoom in and out, that kind of thing.

Divide, divide, divide some more

squares-with-gridSquaresFor a hobby project I’m working on I needed to represent a large set of geographic coordinates. They happen to represent positions of bus and train stops but that’s not super important. To the right is a diagram that illustrates what I had in mind when I started playing around with this. What I wanted really was a way to divide the world evenly into squares of different sizes that I could divide my positions into. (For simplicity I’ve scaled the world down to a 1×1 unit square). Each of the three colored squares here should have some unique integer id. When I say “divide the world evenly” I mean that they’re not arbitrary squares, they’re on the grid you get if you divide the whole square into in four equal parts, then divide each of those into four as well, and so on and so on.

It seemed intuitively reasonable that you’d want the entire space to be represented by 0.


This is zoom level 0. Zooming down one level, you might divide 0 into four parts and call them 1-4,


This is zoom level 1. Then you can keep going and do the same again to get zoom level 2,


At each new zoom level you add 4 times as many new quads as you had on the previous level so where level 2 has 16, level 3 will have 64 and level 4 will have 256.

This is how z-quads are constructed. Pretty simple right? Already, by the way, we know the id of the blue square from the beginning because it’s there at zoom level 2: it’s quad 14. And just as a reminder, at this point we’re playing around on the unit square but it’s just a matter of scaling the values back up to get coordinates on the earth. Here’s 14 again, scaled back up:


One thing you may notice if you look closely at how the quads at zoom level 2 are numbered is that they look of out of sequence. You might expect them to be numbered 5-8 on the top row, 9-12 on the next, and so on. Instead they’re numbered in zig-zag, like this:

zigzagSo they zig-zag within each of the four parts of the big square, and also when going between the four parts. That’s because it turns out to be really convenient when you divide a quad to have all the quads in the subdivisions be contiguous. So the children of 1 are 5-8, of 2 are 9-12, and so on. We’ll end up relying on that constantly later on. (The numbering within each level it known as a Z-order or Moreton order curve).

Okay, so now I’ve postulated way to divide the unit square, by which I mean the surface of the earth, into regular parts. The next step is to figure out how to operate on them which is where it’ll become clear why this number is is particularly appealing.

Zoom bias

Given a quad there are many operations that you might conceivably want to perform. You might want to know which quad is its parent (that is, the quad that contains it at the zoom level above) or the parent’s parent, etc. Given two quads you might want to know whether one contains the other. Or you might want to find the most specific quad that contains both of them.

The first thing we’ll look at is how to get the parent of a quad. To explain that I need to introduce two contepts, zoom bias and a quad’s scalar. The zoom levels we’ve seen so far (and a few more) and the values of their quads are

  • Zoom 0: 0
  • Zoom 1: 1-4
  • Zoom 2: 5-20
  • Zoom 3: 21-84
  • Zoom 4: 85-341

In general, at level z you have 4z quads, so at level 2 you have 42=16, the first one being 5. At level 3 you have 43=64, the first one being 21, and so on.

The value of the first quad for a given zoom level turns out to appear again and again when doing operations so I’ll give that a name: the zoom bias. When I talk about the zoom-3 bias, which I’ll typically write as b3, what I’m referring to is 21 since that is the value of the first quad at zoom level 3.

The scalar of a quad is its value minus the bias of its zoom level. Here are the biases and scalars for zoom levels 1 to 3:


Parents and children

Now we’re ready to look at how you get a quad’s parent. As mentioned, quads are numbered such that the subquads of each parent are contiguous. That obviously also holds for the scalars. That means that if you divide the scalar value by 4, the number of children in each parent, this happens:


You get the scalar of the parent. So given a quad q at zoom level z, the parent quad can be computed as

parent(q) = ((qbz) / 4) + bz-1

Unbias the quad to get the scalar, divide it by 4 to get the parent’s scalar, then add the parent’s zoom level bias to get back a quad. And in fact this can be simplified if you use the fact that the zoom biases are related like this,

bz+1 = 4bz + 1

I’ll justify this in a moment but for now you can just take it as a given. If you plug this in and simplify you get that

parent(q) = (q – 1) / 4

(By the way, except when stated explicitly all values are integers so division implicitly rounds down). What’s neat about the simplified version is that bz no longer appears so you can apply this without explicitly knowing the zoom level. It’s pretty easy to calculate the zoom level for a quad, I’ll go over that in a bit, but it’s even easier if you don’t have to.

How about going the other way? Given a quad, finding the value of one of its children? That’s the same procedure just going the other way: unbias the parent, multiply by 4, add the child’s index (a value from 0 to 3), rebias:

child(q, i) = 4(qbz) + i + bz+1

Again this can be simplified by using the relation between the zoom biases to

child(qi) = 4q + i + 1

If instead of the child’s index (from 0 to 3) you use its quad value (from 1 to 4) it becomes even simpler,

child(qi) = 4q + i

This is part of why they’re called z-quads: it’s from all the z-words. The numbering zig-zags and it’s easy to zoom in and out, that is, get between a quad and its parent and children. (If, by the way, you’ve ever used a d-ary heap you may have noticed that this is the same operation used to get the child node in a 4-ary heap).

Since you can calculate the value of a child quad given its parent it also means that you can calculate the value of a quad one zoom level at a time, starting from 0. For instance, in the original example with the colored squares the blue square was child 2 of child 3 of 0. That means the value is

child(child(0, 3), 2) = 4 (4 0 + 3) + 2 = 4 3 + 2 = 12 + 2 = 14

Which we already knew from before of course. This isn’t how you would typically convert between positions and quads in a program, there’s a smarter way to do that, but I’ve found it useful for pen-and-paper calculations.

Ancestors and descendants

Now we know how to go between a quad and its parent and children we can generalize it to going several steps. When going multiple steps, instead of talking about parents and children we’ll talk about ancestors and descendants. An ancestor of a quad is its parent or one of the parent’s ancestors. A descendant of a quad is one of its children or one of the children’s descendants.

It turns out that just as there’s a simple formula for getting a quad’s parent there is one for getting any of its ancestors:

ancestor(q, n) = (qbn) / 4n

Given a quad this gives you the ancestor n steps above it. The special case where n is 1 is the parent function from before, b1 being 1. Even though it might not seem sensible now it’s worth noting for later that this is also well-defined for n=0 where the result is q. So a quad is its own 0th ancestor.

It’s easy to prove that the ancestor function is correct inductively by using a second relation that holds between the bzs:

bz+1 = 4z + bz

I won’t do the proof here but maybe we should take a short break and talk about why these relations between the biases hold. The bias at a zoom level is the index of the first quad at that level. At level 0 there is 1 (=40) quad so b1, the next available index, is 1. At level 1 there are 4 (= 41) which start from 1 so the next bias is

b2 = 40 + 41 = b1 + 41 = 5

If we keep going, the next level adds 16 (= 42) quads in addition to the 5 we already have which makes the next bias

b3 = 40 + 41 + 42 = b2 + 42 = 21

You can see how each bias is a sum of successive powers of 4 which gives you the relation I just mentioned above, that to get the next bias you just add the next power of 4 to the previous bias.

b3 = b2 + 42

It also gives us the other relation from earlier,

b3 = 40 + 41 + 42 = 40 + 4(40 + 41) = 4 b2 + 1

Okay so back to ancestors and descendants. As with ancestors you can also get to arbitrarily deep descendants in one step,

descendant(q, c, n) = 4n q + c

This gives you the descendant of q which relates to q the same way quad c, which has zoom level n, relates to the entire space. This can again be proven by induction with the base case being the child operation from before. And again this is well-defined for n=0 where the result is q.

So now we can jump directly between a quad and the quads related to it at zoom levels above and below it. This will be the building blocks for many other operations. But before we go on to the next part I’ll round off the ancestor/descendant relation with the concept of descendancy.

When you go from a quad to one of its ancestors you’re in some sense discarding information. If you start with a quad that covers the center of Århus from the initial example and zoom out to the one covering northern Europe, you’ve lost the information about where specifically in northern Europe you were. You can add that information back by using the descendant operation, but for that you need to know c, the quad that describes where within northern Europe the Århus quad is. That is, you need to know the descendancy of the Århus quad within the northern Europe quad. There is a simple way to calculate the descendancy of a quad within the ancestor n steps above it,

descendancy(q, n) = ((qbn) mod 4n) + bn

The descendancy is a little less intuitive than the others, that’s also why I’m not spending more time on how it’s derived or proving it correct, but it turns out that it’s actually quite useful when you work with quads in practice. And it definitely feels like the model would be incomplete without it. Note again, as always, this is well-defined when n=0 where it yields 0 (as in, any quad spans the entirety of itself).

Now we have a nice connection between the three operations: for all q and n where the operations are well-defined it holds that,

q = descendant(ancestor(q, n), descendancy(q, n), n)

With these basic building blocks in place we’re ready to go on to some higher-level operations, the first ones will be the two containment operations. Given quads q and s, is s contained in q? And given two quads, determine the most specific ancestor that contains both of them. For those you’ll have to go on to the next post, z-quads: containment.

(Update: hacker news discussion)

(Update: added some references mentioned in the discussion: z-order, d-ary heaps. Remove part where I couldn’t find other non-point coordinate systems)


I’m not convinced classboxes actually work.

I was reading up on safer approaches to monkeypatching and while looking into refinements in ruby I reread the classboxes article by Bergel, Ducasse, and Wuyts. This time I had some concrete uses in mind so I tried applying their lookup algorithm to those, and the way they behaved I’m not sure would work well in practice. This post assumes you’re already familiar with the classboxes article, but maybe it’s been a while so here’s a quick reminder.

Monkeypatching, as used in practice in languages like ruby, means adding methods to existing classes such that they’re visible across the whole program. This is obviously dangerous. Classboxes is intended as a more well-behaved way of doing the same thing. The extension is locally scoped and only visible to the parts of the program that explicitly import it.

The way classboxes work, basically, is that instead of the method dictionary of a class being a mapping from the name of a method to its implementations, it becomes a mapping from the name and the classbox that defined it to the implementation. So you’re still actually monkeypatching the entire program, but because methods are now tagged with their classbox they can be considered or ignored as appropriate, based on where the method you’re looking for is being called. If you’re making a call in a particular class box, and you’ve imported some other classbox, directly or indirectly, then the lookup will consider methods from that other classbox. Otherwise not. (Well, sort of, more or less). Okay, so here’s a concrete example of how it works and maybe doesn’t work.

Imagine you have a simple core library in one classbox that, among other basic types, defines a string type: (Don’t get hung up on the syntax, it’s just pseudocode).

classbox core {

  class String {
    # Yields the i'th character of this string.
    def this[i] => ...

    # The number of characters in this string.
    def this.length => ...



Let’s say you also have a simple collection library in another classbox that defines a simple array type,

classbox collection {

  import core;

  class Array {
    # Yields the i'th element of this array.
    def this[i] => ...

    # The number of elements in this array.
    def this.length => ...

    # Call the given block for each element.
    def this.for_each(block) => ...

    # Return the concatenation of this and that.
    def this.concat(that) {
      def result := new Array();
      for e in this do
      for e in that do
      return result;



So far so good, this is the basics. Now, it’s easy to imagine someone thinking: a string sure looks a lot like an array of characters, it just needs for_each and a concat method and then I can use my full set of array functions on strings as well. I know, I’ll make an extension:

classbox strings_are_arrays {

  import core.String;

  refine core.String {
    # Call the given block for each character.
    def this.for_each(block) => ...



Now we can use a string as if it were an array, for instance

classbox main {

  import array;
  import strings_are_arrays;

  def main() {
    def arr := ['f', 'o', 'o'];
    def str := "bar";
    # prints ['f', 'o', 'o', 'b', 'a', 'r']


Neat right? This is how I would expect extension methods to work. Except, as far as I can tell, it doesn’t. This won’t print an array, it will error inside Array.concat.

Classbox method lookup uses the classbox that started the lookup to determine which methods are available. A method call in main will be able to see all the classboxes: itself, array and strings_are_arrays because it imports them, and core because strings_are_arrays imports it. So it can call String.for_each. It doesn’t call it directly though but indirectly through Array.concat. The classboxes visible from Array is collection because it’s in it and core because it imports it, not strings_are_arrays. So the attempt to call for_each on the string will fail.

Maybe this is a deliberate design choice. I suspect that design won’t work well in practice though. This way, extensions are second-class citizens because they disappear as soon as your object flows into code in another classbox. It also makes code sharing less effective: there may be a method that does what you want but you have to copy it into your own classbox for the extensions you need to be visible. It may also fail the other way round: someone passes you an object of a type you have extended and suddenly it changes behavior in a way neither you nor the caller could not have expected.

There’s of course also the chance that I’ve misunderstood the mechanism completely. I haven’t been able to find a full formal semantics so some of this is based on sort-of guesswork about how it works. But if this is how it does work I’m not convinced it’s a good idea.


Came across an article about information overload on google+, which was very apropos of my recent efforts to not be so stressed at work. I’ve made some changes to how I work over the last few months, some of them just tweaks actually, and I feel like they’ve improved my ability to focus and made work much less stressful. YMMV, obviously.

  • I use virtual desktops to minimize distraction. I have three virtual desktops: one desktop with just what I’m working on right now, nothing else, one with email, code reviews, and other things that might distract me, and one with personal stuff like personal email. I can easily switch desktops and check emails, but the only thing that can impose on me are chat popups, and people don’t chat me a lot. I occasionally check for new important email or code reviews but I can choose when, typically while I’m waiting for something on my work desktop to complete.
  • I keep the tasks I’m working on written down and categorized according to when I want to complete them. I pick one task from the stuff-I-should-do-soon pile and work just on that. When that pile is empty, or it’s been a while since the last time, I look over all my tasks, usually a quarter’s worth, and move some out of and some into the soon-pile. This means I don’t have to worry about all the things I’m not working on right now because I know I’ve considered what I could be working on and chosen what I’m doing right now as the most important. This has been very helpful in making me less stressed. This is apparently a well-known approach, I first saw it in this video about how to use Asana (Using Asana for Individual Task Management).
  • I make an effort to only work 8 hours a day. This is a tough one because it sometimes makes me feel like I’m slacking off, especially because I come in early so I’m almost always the first to go home. However, there’s plenty of research suggesting that even though you may feel you’re getting more work done working 10 hours a day it’s an illusion and you’re actually doing less than if you worked 8 hours (see why crunch mode doesn’t work). At first I just trusted the research, against my own intuition, but at this point I’m convinced — I feel much more focused and efficient and I see no evidence that I’m being less productive. I will occasionally work more for a while but I always feel a “overwork hangover” afterwards where I’m tired, less focused, more likely to space out. But at this point I know that and can make a conscious choice about whether it’s worth it.

Some of these require a bit of motivation; the one really easy thing is to clear any distractions from your workspace, especially if like me you were silly enough to have a browser window there. It’s easy and it makes a big difference.


A few weeks ago I needed a few integer values whose hexadecimal values were clearly recognizable when debugging — magic numbers. I did a search on the web for a list of good magic number, which I was sure would exist, but couldn’t find any. I ended up using 0xBADDEAD and 0xBEEFDAD which I thought were pretty good.

Then I made a mistake: I decided that I should make that list of magic numbers if no one else would. It wasn’t until I was done that that I realized the depth of pointlessness of the thing I had just done. But here it is: a list of all hexadecimal words that are meaningful in the english language, in the form of a google spreadsheet. As proof of the words’ meaningfulness each word has a link to a dictionary or encyclopedia describing the meaning. Most of them are pretty obscure — like dace, a small fish, or eba, a food eaten in West Africa. However, some are pretty meaningful (scroll down to see the list, for some reason blogger messes up the formatting):

ace A playing card
bad Not good; unfavorable; negative
bead A small decorative object
bee A flying insect
beef Culinary name for meat from bovines
cab A type of public transport
cafe A coffee-shop
dad Father
dead Bereft of life
deaf Insensitive to sound
decade A period of 10 years
decaf Decaffeinated coffee
fabaceae A family of flowering plants
facade The front of a building
face The front part of the head

The next time I or anyone else has to come up with a magic number it should be a lot easier. I’ll try to ignore the fact that I probably spent ten times as long compiling this list as any programmer spends inventing magic numbers in a lifetime.

In Brief

A few short messages.

Completion: I’m beginning to lose interest in the API gadget so I’m thinking of removing it. If you’re using it and don’t want me to remove it feel free to let me know.

Pattern Matching: A while back I wrote a post about pattern matching in the Scala language. I also had a discussion with Burak Emir on the Scala mailing list. The post was not too well written and I was my usual annoying self on the mailing list so I didn’t actually manage to change anyone’s opinion on the subject. Since, however, the Scala folks have whipped up a new paper on the subject, Matching Objects With Patterns which should be of interest to anyone who finds that sort of thing interesting.

Monkey’s @: Back in the day when I was working on wildcards in Java, before javac was open sourced, you could introduce all the silly variable names you wanted without anyone noticing. Now anyone can rummage around in the javac source and discover stuff like the MONKEYS_AT value. Neal breaks down and explains where that came from.

That’s it. Carry on.