Why is the site so dead today? It must be a weekend or something. Time to post an idea!

The commonly used units of angle are the degree, of which there are 360 per full rotation, and the radian, of which there are tau per
full rotation. There is also the grad, of which there are 400 per full rotation,
but it's not commonly used.

Degrees are supposedly nice because 360 has a lot of factors relative to most other numbers of similar magnitude, meaning you can
divide it by a lot of different numbers and still end up with integers. For instance, you can divide a circle in one (360°), two (180°),
three (120°), four (90°), five (72°), six (60°), eight (45°), nine (40°), ten (36°), twelve (30°), fifteen (24°), eighteen (20°), twenty
(18°), twenty-four (15°), thirty (12°), thirty-six (10°), forty (9°), forty-five (8°), sixty (6°), seventy-two (5°), ninety (4°), one and a
fifth hundred (3°), one and a half hundred (2°), and three and three fifths hundred (1°). But this is not necessarily intuitive if you're
more used to metric units with their decimals and tens than American units with their fractions and twelves.

So why not use the metric unit of angle, the radian? It is commonly used in science, after all. Well, that's even less decimal than the
degree. Any real-world measurement using radians will inevitably be an irrational number, which is just silly. The American system of
units is called "irrational" for not having simple ratios between units of different dimensions, but I'd call it more fundamentally irrational
to base one of your units on an irrational number.

Therefore I propose a new unit of angle, the circle (symbol: cir). There is one circle per full rotation. A right angle, or 90°, or pi/2
radians, is equal to 250 millicircles (mcir). This should be a lot more intuitive than either degrees or radians for people who are used to
the metric system. 1000 is also easy enough to divide by factors that metric users are likely to want to divide it by, and they already
have experience doing so.

I also propose a corresponding unit of solid angle, the sphere (symbol: sph). There is one sphere per whole sphere's worth of solid angle.
A hemisphere is 500 msph. This should also be better than the current square degree (which doesn't make much sense because you can't
define a proper square on a spherical surface) and steradian (which is again based on an irrational number).

I agree in principle, but I think rather than calling it a
circle, you should just call it tau (for the rest of this
anno, please replace
all [tau] with lowercase greek symbol tau as shown in link
since it appears there's no way to display that here). 90
degrees
is
[tau]/4 as
mentioned in the tau manifesto, but it would be equally
valid it us this in a metric type notation as 250 m[tau]
(rather
than 250 mcir as you suggest).

I admit that this is maybe not quite right,since [tau] is
a
dimensionless number like pi. So saying 0.25
[tau]
= 90
degrees isn't right. It should really be 0.25 [tau] radians,
but
perhaps the radians could simply be implied, especially
when used with the milli prefix. Therefore 90 degrees =
0.25 [tau] radians = 250 m[tau].

I waded into that link a ways. Ow. literal pain somewhere just behind my right eyeball. Fascinating, but I think I'll check it out later when I'm not bucking a cold. (+)

When working on embedded systems, I often divide the circle into 256, which makes the calculations a lot easier. You can store an angle in an 8-bit number and it wraps automatically. I don't think there is a name for this unit though.

Actually, hang on. A lot of problems are caused by pi being
irrational. However, if we simply define pi to be 1, and
then redefine the integers to be multiples of pi, then
everything gets easier.

You might think this would cause problems with counting, or
with basic arithmetic, but no. For instance, "2" now
becomes 2*pi/pi, and the pi's cancel meaning that 2 is still
2. But now pi squared = 1 = pi. The area of a circle
becomes pi*r^2, which is simply r^2. Et cetera.

It turns out that both the circle and the sphere (as units of
angle and solid angle) already exist. The "turn" is a unit of
angle (1 turn = 360°), and the "spat" is a unit of solid angle
(1
spat = solid angle of a sphere from its centre).

Moreovermore, [mixtela] will be diluted to learn that
1/256th of a turn has a name - the "binary radian" or "brad".
256 brads = 360°.

<considers making "reinventing the wheel" pun; decides
against>

I disagree because this is not duodecimal. However, I have
some sympathy with a binary/octal/hexadecimal approach.
I've seen a Peters Projection map with decimal angular
units on it though, so I'm not sure it's unbaked.

From my profile (not that I'm saying you should have
looked there first (though I
guess you really should have (but this gives me an
opportunity to explain it in a
more visible place, and a reminder to proofread and edit
my explanation, so I'm
happy)) but just because I don't want to rewrite it and I
feel I explained it pretty
well there (apart from the just-discovered need for
proofreading and editing)):

// I keep a list in Evernote of ideas to post here.
Whenever I post an idea here, I
add at the bottom of the idea body a line of the format
"X/Y [Z]", where X is how
many ideas from my list I've posted (including the current
idea), Y is the total
number of ideas on my list on the day of posting, and Z is
either the date I thought
up that idea or the date I added that idea to my list. (I
started recording these
dates on my list on 2017-10-02.) This is to facilitate
analysis of my progress in
posting my ideas, by myself or anyone else who cares to
(probably nobody). "n/a"
means I posted the idea here without putting it on my list
first—this is usually the
same day or close to it. //