The number twelve is quite prolific. Here are some examples which
feature twelve as a central theme.
### Time

*N.B.*I hope to include a lot more diagrams and pictures here
which show off the splendour of twelve in these settings but for now
it's just text.

The venerable clock has twelve hours displayed on its face. There are
sixty minutes in an hour, and sixty seconds in a minute.
#### Sixty

### Degrees

Sixty is twelve times five:

sixty twelve five o o o o o o o o o o o o

o o o o o o o o o o o o

o o o o o o o o o o o o

o o o o o o o o o o o o

o o o o o o o o o o o o

There are twenty four hours in a day, which of course is two chunks of twelve. And twelve months in a year.

Once you get outside the realm of the day, time gets rather inconsistent in the Gregorian calender with no particular pattern to the number of days in a month or even number of days in a year (it leaps every four, except every hundred and then there's the leap second complication...). However, the Mayans had a calender that was at once both more and less regular than the Gregorian.

There are three hundred and sixty degrees in a circle. Three hundred and
sixty has an enormous number of divisors (including twelve!) which means
there are a large number of ways to carve it up. This is of course
exactly what you want: degrees are in essence a way of specifying a
chunk of a circle so the more convenient it is to use degrees to carve
up a circle the better.
### Summary

### References & more

There are a couple of other "common" ways to measure angles, one is radians and the other is grads (or grades). Radians are primarily used in advanced math and trigonometry and we won't talk about them here.

Grads however make for an interesting comparison with degrees. A grad is one hundredth of a right-angle, in other words there are four hundred grads in a full circle. So you could talk about percentages of right-angles and the grad would be the percentage. This has a certain appeal to it (half a right angle is fifty % or fifty grads) but in practice what if you wanted to say "a third of a right-angle"? You would immediately bump into the same problem as writing out the decimal for "one third", you can without using fractions. It's (in base 10) 33.333333....%. Whereas in degrees half a right-angle is forty five ° and a third of a right-angle is thirty °

Not surprisingly, and I believe it's for these reasons, you almost never hear of grads in practice although they will probably have an enduring spot on calculators for the rest of time. (If you have heard anything beyond what's mentioned here please do drop me a note!)

Twelve appears in common systems we use every day, and we saw how
attempts to use systems based on ten quickly run into problems with
thirds. But first, what exactly are bases?