All the symmetrical watch faces (and code to generate them)

If you ever look at pictures of clocks and watches in advertising they are set to roughly 10:10 which is meant to be the most attractive (smiling!) position for the hands. They are actually set to 10:09.14 if the hands are truly symmetrical.

CC BY 2.0 image by Shinji

I wanted to know what all the possible symmetrical watch faces are and so I wrote some code using Processing. Here's the output (there's one watch face missing, 00:00 or 12:00, because it's very boring):

The key to writing this is to figure out the relationship between the hour and minute hands when the watch face is symmetrical. In an hour the minute hand moves through 360° and the hour hand moves through 30° (12 hours are shown on the watch face and 360/12 = 30).

The core loop inside the program is this:

  for (int h = 0; h <= 12; h++) {
    float m = (360-30*float(h))*2/13;
    int s = round(60*(m-floor(m)));
    int col = h%6;
    int row = floor(h/6);
    draw_clock((r+f)*(2*col+1), (r+f)*(row*2+1), r, h, floor(m), s);
  }

h is the hour number, m the number of minutes past the hour and s the number of seconds past the minute. As you can see, the loop looks at the hours 0 to 12 and then calculates the minutes and seconds using this formula:

    float m = (360-30*float(h))*2/13;
    int s = round(60*(m-floor(m)));

The s part is simple, it's just the decimal part of m turned into seconds. m is the interesting calculation and gives the number of minutes past the hour h (expressed as a decimal to also capture the seconds). Here are the details of how m is calculated from h.

If you look back at the watch face above it's not actually showing 10:09.14, it's showing 10:11.39. I think this is in part because it puts the second hand in a pleasing location. If I modify my program to show the location of the second hand you can see that perfect symmetry between hour and minute hands gets messed up by its presence.

 

The difference between parentheses and curly braces in GNU Make

One of the problems/perks of having written a book about GNU Make is that people ping me with questions. This morning someone said to me: "Especially curly braces vs parentheses is something that always confuses me".

As always the first port of call with GNU Make questions should be the FSF's manual. It says the following: "To substitute a variable’s value, write a dollar sign followed by the name of the variable in parentheses or braces: either $(foo) or ${foo} is a valid reference to the variable foo."

And that seems to work well:

$ cat Makefile

foo := hello.world

$(info $(foo))

$(info ${foo})

$(info $(foo:world=everyone))

$(info ${foo:world=everyone})

$(info $(foo:hello.%=good morning.%))

$(info ${foo:hello.%=good morning.%})

$ make

hello.world

hello.world

hello.everyone

hello.everyone

good morning.world

good morning.world

make: *** No targets.  Stop.

You can see that simple variable references work, as do substitutions (where I changed world to everyone) using substitution references.

But a quick search of the GNU Make manual shows that braces are mentioned once again when talking about function calls: "The delimiters which you use to surround the function call, whether parentheses or braces, can appear in an argument only in matching pairs; the other kind of delimiters may appear singly. If the arguments themselves contain other function calls or variable references, it is wisest to use the same kind of delimiters for all the references; write $(subst a,b,$(x)), not $(subst a,b,${x}). This is because it is clearer, and because only one type of delimiter is matched to find the end of the reference."

The only problem with that description is that the example given using $(subst) works just fine. Here are four possible combinations of parentheses and curly braces:

$ cat Makefile

foo := hello.world

$(info $(subst o,0,$(foo)))

$(info $(subst o,0,${foo}))

$(info ${subst o,0,$(foo)})

$(info ${subst o,0,${foo}})

$ make

hell0.w0rld

hell0.w0rld

hell0.w0rld

hell0.w0rld

make: *** No targets.  Stop.

That worked fine. If you need to work with literal parentheses or braces then the different forms can be helpful. Suppose you want to change every ) to (, or } to {. Here's a Makefile containing two emojis and we use substitution to flip the ) to ( and } to {:

$ cat Makefile

smile := :-)

curly-smile := :-}

$(info $(curly-smile))

$(info $(smile))

$(info $(curly-smile:}={))

$(info $(smile:)=())

$(info ${curly-smile:}={})

$(info ${smile:)=(})

$ make

:-}

:-)

:-{

=()

={}

Makefile:11: *** unterminated variable reference.  Stop.

Wait! That didn't 100% work. The first two $(info) calls did output the value of $(smile) and $(curly-smile) just fine.  Then $(info $(curly-smile:}={)) changed the } to { without a problem because the substitution was made using parentheses (see the start of $(curly-smile:).

But what happened to $(info $(smile:)=()) and why was the output =()? Well, GNU Make interpreted $(smile:) as a variable reference to a variable called smile: and not smile! You can verify that by defining a variable called smile: using this trickery (you have to do that to make a variable called smile: because if you try smile: := foo GNU Make will think you are making a rule to make a file called smile!)

colon := :

smile$(colon) := foo

But in my Makefile above smile: isn't defined so it evaluates to an empty string and we're left with the rest of $(info $(smile:)=()) which is just =() which GNU Make outputs as a string literal. What happened here was GNU Make saw the ) at the end of $(smile:) as closing $(smile:) and not as a literal ). 

If you want to work with characters that GNU Make treats as special, it's best to define variables to use in their place. Since GNU Make will parse functions calls, variable references etc. before substituting the variables you'll get the output you expect.

$ cat Makefile

smile := :-)

curly-smile := :-}

$(info $(curly-smile))

$(info $(smile))

open-curly := {

close-curly := {

open-paren := (

close-paren := )

$(info $(curly-smile:$(open-curly)=$(close-curl)))

$(info $(smile:$(open-paren)=$(close-paren)))

$(info ${curly-smile:$(open-curly)=$(close-curl)})

$(info ${smile:$(open-paren)=$(close-paren)})

$ make

:-}

:-)

:-}

:-)

:-}

:-)

make: *** No targets.  Stop.

But back to the Makefile above. The last two lines are:

$(info ${curly-smile:}={})

$(info ${smile:)=(})

and the last two lines of output are:

={}

Makefile:11: *** unterminated variable reference.  Stop.

The first line of output is easy to understand. It's the same problem that occurred with $(smile:) but with ${curly-smile:} being interpreted as a reference to a non-existent variable called curly-smile:. But what about the last line? 

Once again GNU Make got confused and parses $(info ${smile:)=(}) as $(info ${smile:) (because the ) terminated the $(info)) followed by =(}).

The moral of this story is... don't mix parentheses and braces in GNU Make, and if you need to work with them as literals then define variables that contain them and use variable references.