Digital sundial

I appreciated the math, as someone with measure theory and analysis experience, and the result is very cool, but it's a bit funny how it's just thrown in there.

That said, assuming the reader knows the requisite math it's a very concise explanation of the necessary theory. I tried to explain the math somewhere else in the comments but I'm afraid I didn't do a great job.

I think it's a very similar principle to this:

http://en.wikipedia.org/wiki/Parallax_barrier http://en.wikipedia.org/wiki/Lenticular_printing

The theorem says something like this: imagine you have a 2D plane, with the usual x and y axes. For any θ between 0 and pi, there is a unique line passing through the origin. That's L_θ [1]. Now for each line, say we pick out some subset of the line, called G_θ [2]. In the picture I've made G_θ thicker for emphasis, but if you think of the whole line as a bunch of points, then G_θ is just some of those points. So for each θ, we have a line, and a corresponding subset of the line. If we add all the lines together we get the entire 2D plane, and if we add all the G_θ's together we get some set of points in that plane. The important condition in the theorem that needs to hold for the results to be true is that if we add all the G_θ's together, we get a set which we can say has some area.

Now proj_θ F is sort of like the "shadow" of F on L_θ, for some set F. See this picture [3]. The perpendicular projection takes a 2D set and projects it onto a 1D set (the line). Analagously, if the sun was in the sky above your head and you were standing on a 2D plane, then your shadow would be the perpendicular projection of a 3D set (you) onto a 2D set (the ground).

Anyway, if we add all the G_θ's together and get some set which is suitably "nice", then there is some other 2D set F such that if we project F onto any* line L_θ, the "shadow" on L_θ covers G_θ completely, and the part of the shadow that isn't covering G_θ is negligibly small [4]. So applied to the sundial, this means there exists some shape such that its shadow at some time of day will be that time.

* Technically, it's "almost any", which means, informally, for all but a negligible number of lines. The stuff about measure, measurable and almost all is all from measure theory [5]. I can explain some of the measure theory concepts if you'd like.

[1] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#0

[2] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#1

[3] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#2

[4] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#3

[5] http://en.wikipedia.org/wiki/Measure_(mathematics)

As the sun moves (or as the earth spins) the light from each window will change its angle.

So if we add another wall with more windows, we can filter our filtered light. Then you'd say, "when the light shines in window 4 it's say 6pm".

Now add a board person with a degree in mathematics, and a grid of windows. And you get single "pixels/windows" that show light based on the minute and hour, and BOOM a digital clock.

Fun fact: Only with the advent of railroads did time begin to be synchronized in larger areas. Until then, each city and town set it's own time, based on the local solar time.

https://en.wikipedia.org/wiki/Railway_time

There exists public clocks that track both local and standard time: https://en.wikipedia.org/wiki/The_Exchange,_Bristol#Clock

This website sells small versions, my guess is that larger ones (as seen here[1]) are made on order, if at all.

[1]http://www.atlasobscura.com/places/genk-sundial-park

I won one. It is a very weird object.

http://en.wikipedia.org/wiki/Digital_data

Though I have to say in this specific context the definition specific to clocks/watches makes just as much sense:

>(of a clock or watch) showing the time by means of displayed digits rather than hands or a pointer

This is also a fun display of sig figs. You can build a sundial that tries to display to single minutes, but that doesn't mean it'll be accurate unless you have a rather elaborate microcontroller almost continually adjusting the angles from day to day, not to mention daylight savings time. I think if you'd have to adjust the mounting angle thru the day would depend on latitude and month of year.

On the equinox, a quarter of an angular degree is a minute of chronological time. This varies a bit thru the year, and depends on your lattitude too. Ask your local celestial navigation guy. I have just enough experience with celestial navigation and sailboats to know I shouldn't be doing it, or at least I'd have to be super careful if I tried it for real.

Note that solar time will differ from UTC depending on where you are in your timezone.