Who wouldn't want to have a wearable sun-dial as part of a time traveller's outfit?
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.
Don't worry, this will be over soon and there's no exam. Everything vital will be covered when you mark out the plastic.
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.
Step 2: Parts
Depending on your access to materials, this can cost very little, bordering on nothing at all.
In the image, the parts used are laid out. Don't worry if you can't see the scrawled biro, I'll go through everything as we go along.
First, you'll see a rather dead looking watch, this cost AU$10 new and has been sitting in my junk box for years. We'll only use the straps and the little sprung rods down in the bottom right.
The transparent plastic dome is-was a scoop from a protein shake (yuck) with the handle cut off and filed smooth. The closer to round this is, the more accuracy you can get in your dial.
The white circular piece of plastic is exactly that. This one is acrylic cut with a hack saw and a rasp (call it a workout or find a laser cutter). This will become the base plate.
The pen-scratchings are plans for a mount to attach the straps to, that comes later too.
Step 3: Marking Out
The base plate of the dial is where everything happens, so it makes sense to take your time getting it ready.
I scored little grooves into mine to absorb and protect the permanent marker (i don't know what from). You don't need to do this. Also, I have far too many lines on mine which makes it difficult to see the shadow.
First, you'll need to find the centre. I used a ruler but there are many ways of doing it.
From the centre, draw the little circle in the middle a third the size of the plate (a ruler and a pocket calculator is probably the simplest means of finding the right size).
In the little circle, draw a line from one side to the other, straight through the middle. You'll end up aligning the shadow with this line, so don't clutter it like I did.
You also need to put little marks at the edges of the plate in line with this line, these marks will become markers for the solstices (20th of dec and 20th of june).
In the end, you'll need marks for the equinoxes to line the dome up with; these are simply at right angles to the solstice marks.
On a piece of paper, draw a circle the same size as the base plate, this will help you to set your compass to the right distances to mark the plate neatly.
Going back to high school geometry, or Wikipedia, divide this circle into six arcs by tracing a circle the same size from one point on the edge to another point on the edge, all the way around the circle.
Then bisect one of these arcs, giving you marks for a twelfth of a circle.
*Remember, this is on paper, so you only need to get one of any length to set the compass.
You also need to find an arc that's a third the length of one of these 12 arcs (I had to resort to trial-and-error at this point)
Step 4: More Marking Out
The third-of-a-twelfth is an offset for the months, since the solstice occurs about a third of a month before new year's day. Use this length clockwise from a solstice mark to place the start of January.
Then, starting from January, use the twelfth lengths all the way around the plate to mark all the other months ( you'll get better accuracy if you mark out July as well: divide into six, then divide again)
From these twelve month marks, draw lines from the edges to the centre, but stop when you hit the circle. The less markings you put in there, the better.
The dome is a little harder, but since it's the same size as the plate, you've already got all of the lengths you'll want to use.
You need a line going all the way from the edge of the dome, over the very top, and back down to the exact opposite side.
You can use the freshly marked base plate as a guide for the edges but the line itself poses a fair problem. I had some success with using the edge of some sticky tape as a guide. It wasn't perfect, but it was better than free-hand.
This line will cast a shadow, so make it wide enough to do so - but fine enough to look neat.
 The second image is a template.
Step 5: Straps
It's small, it tells the time: let's put watch straps on it.
Cheap watches, like the one gutted for parts here, have these tiny sprung bars that go in the straps and lock into tiny pockets in the watch body. They compress along their length (about four moving parts in each bar) to go in and out of the pockets in the watch body. The ends of the rods have a little flange that you can press a needle against to compress it.
Find a thick pin.
The watch straps should slide sideways a tiny bit exposing the end of the rod, mine even had notch cut into the strap exposing the flange. push the end of the rod into the rod's body, and the strap should come off.
This mounting takes the watch straps, sits against your wrist and is glued or screwed to the underside of the base plate. The next few steps cover it's construction.
Step 6: Mount
While it might look like some fancy metalworking was involved, it really wasn't.
you'll need tin snips or a hack saw, and some files.
Mark out (on paper again) the shape of this mount. Mine was 5cm long, folded 1cm away from either end, with tabs from the fold to the edge 5mm wide (the first picture has 5mm grid paper still glued to it).
It needs to be wide enough so that when you fold it, the strap fits inside, and narrow enough for the sprung bars to be snug.
Cut out the shape on the paper, and stick it to a chunk of metal, preferably stainless steel (dumpster dive for this one: a computer chassis, for example, is fair game).
This piece is aluminium about 1mm thick and came from a discarded plotter.
You only need to use a glue stick for this one, you want this paper to come off in the end.
Cut the metal with either the tin snips or the hack saw, being careful to avoid cutting the paper, but the closer you get, the less filing you do.
Now file it, rough for the meaty bits, fine for the edges and burrs.
At some point, before you bend it, you'll need to take to it with a drill. A 1mm bit was pretty much the perfect size for my watch parts.
The straps will hinge on this point, I put the holes a little under 2mm from the ends of the tabs.
Get it in the vice, and brute force it into the right shape, tabs last.
BE CAREFUL, if you bend it on the line, it could end up too narrow and you'll probably have to start again (I did, bending it back caused a tab to tear off.)
check for burrs again, file them smooth.
if you used a glue stick, soak the finished product in water to release the paper.
The sprung bars for the watch straps go in a LOT easier than they come out.
Step 7: Put It Together
Don't use super-glue.
You need to glue the dome to the base and fill the dome with water. leaving a little bubble is also extremely important.
I used super-glue on my first attempt but it frosted the inside of the dome so you couldn't see anything.
I also found that acetone does nothing for this frosting and melted the dome. Try to avoid that.
Luckily, I had a spare.
The most appropriate glue i can think of is actually hot glue. shoe-goo and bathroom caulk are good, but you really need something that sets fast.
Press the dome to the base plate, line up the equatorial line with the equinox marks from step3.
Run the glue gun around the seam, pressing the glue into the seam, but stop at both solstice marks. Leave a tiny patch un-glued at both ends, we still need to fill it with water.
The two patches left unsealed should allow water to get in when you immerse it. You want to have your glue gun handy.
Only leave a small bubble in the dial.
Press your finger over the bottom seam to stop the water getting out, and glue the top sealed. Turn it over and glue the other side so it's all sealed.
I wanted to use screws to fasten the straps, but i didn't have any handy. More hot glue.
If you want to use it as a spirit level, you'll need to file the bottom corners a little to level it, use a level surface to calibrate it.
Step 8: Making It Work
After all this, it does tell the time, and it does point north, but there's one thing missing.
You need to know your latitude (Perth, Australia is 32 degrees south).
Look at your dome as if it were a globe. You have an equator, and the solstice marks can substitute as poles. Place your mark in line with the top and a solstice mark, at the same position as your latitude.
Now position the dial such that the bubble is exactly under this mark.
At the December solstice, the sun is the furthest south so it should cast a shadow further north in your dial. That means that December is the north-most month in your dial.
If you're in the northern hemisphere, the bubble should be closer to the mark for the December solstice. Opposite in the southern hemisphere.
In step4, I said to place January a third of a month clockwise around the circle. If you did that, count your months clockwise around the circle. Follow that line (toward the centre) to the inner circle, then horizontally to the line in the middle of the dial.
(mine has heaps of horizontal lines to help with this, but i can't see the shadow the equatorial casts).
That gives you a height on the line which relates to how far the sun is away from the equator. (December and June are pretty much at the very ends of this line)
That spot on the line is what you need to align your equatorial shadow over. You do this by rotating the dial (and usually your body) while keeping the bubble under the spot at your latitude.
When the equatorial shadow is over that spot, the December solstice mark is pointing true north.
Then it's a matter of reading the marks in the shadow from one end to the centre line. The very edges are six o'clock morning and night. This picture shows the dial at 11:20 in Perth AU, December 22.
The mark for seven o'clock is not visible in this view because of the shape of the dome. However, you can see the bright spot surrounding the noon mark (the dome is actually quite distorted, you can see that in the lines on the base plate). The equatorial line doesn't need to be this thick unless you want the shadow to be photogenic.
There are a few offsets you need to account for, first is the equation of time which makes local solar time fast or slow, the next is daylight saving, the next is only if you're in a very large time zone (like China) and the sun isn't overhead at "noon".
The bubble sticks to the plastic a little bit making it hard lo line up properly (detergent might help).
The permanent marker runs and turns the water purple.
Taking it further:
A watch that only works from six to six isn't much help, but if you place the latitude mark and the noon mark further to the east, it works longer when the sun is in the west.
Taking it too far:
Because it's december and everyone seems to be feeling extremely festive, you could try putting 'snow' in your wearable snow dome. This will, of course, make it almost impossible to read.