My sister and I always thought a wrist-mounted sun-dial would be completely impractical, Not so.
(although it's still nowhere near as convenient as a normal watch)
To have a functioning sun-dial outside of your garden, you need to know where north is..
Also, you need to keep it level. How do you combine a shadow, a compass and a spirit-level, and fit it on a wristband?
Researching existing sun-dial designs, one type in particular caught my fancy: the equatorial dial.
It can function at any latitude (if you know what it is) and some of them actually give you true north when you line them up to read the time.
An afternoon later, I had a modified design that closely resembled a snow dome.. how could i resist?
This dial can read to within five minutes of local solar time if you etch the components accurately.
Step 1: The Maths, yuck.
If you lived a few hundred years ago, you'd know that the earth is flat and the sun goes around the earth. While wrong, it's a good starting point.
The sun passes the roughly same spot in the sky every 24 hours so it spends 12 hours above the base plate. We need to divide the dome into 12 equal increments: one for each hour. This is the equatorial line.
The circle on the base plate is a bit harder to grasp. Every year, the sun wanders north and south, about 24 degrees above and below the equator. As it turns out, this is very close to a sine wave so we'll call upon the unit circle:
The base plate is divided into 12 sections, one for each month. Each month has a height above or below the equatorial line, this height follows the same sine wave the sun follows year after year.
The size of the circle is important too, the sun's rays have to hit the very top or very bottom of the circle at the solstice, making them about 24 degrees north or south of the very top of the dome.
Unfortunately, since this is basically a snow dome, we need to account for the refraction angle into the water.
A bit of mucking about with algebra tells you that the diameter of the circle needs to be about a third the diameter of the dome (0.312 times the size, to be precise).
(If you actually do the algebra, a plot of the sine of the incident angle vs the sine of the refracted light is slightly non linear, but it's only out by less than 1%)
 This is a PDF of some of the algebra.