
Overturned polygons: shapes with less than two sides - galfarragem
http://chalkdustmagazine.com/blog/overturned-polygons/
======
akkartik
The entire system is really cool!

Polygons with a negative number of sides:
[http://chalkdustmagazine.com/blog/thinking-outside-
box](http://chalkdustmagazine.com/blog/thinking-outside-box)

Polygons with a fractional number of sides:
[http://chalkdustmagazine.com/blog/fractional-
polygons](http://chalkdustmagazine.com/blog/fractional-polygons) (hint: the
pentacle you may summon a demon with is a fractional-sided polygon)

Polygons with -2 to 2 sides: [http://chalkdustmagazine.com/blog/overturned-
polygons](http://chalkdustmagazine.com/blog/overturned-polygons)

I think the analysis of [-2..2] (including the boundaries, 2-sided 'polygons'
that he calls 'digons') is questionable. Polygons consist of straight lines,
so bending lines and introducing loops seems unjustifiable until you can tell
me how to distinguish a bent line from a curve. Until then, polygons with -2
to 2 sides are just lines, at least the parts we can see in our mathematical
universe. I can understand the temptation to fill in the number line, but
perhaps we should just accept that some of the shapes are singularities,
hidden within a mathematical event horizon.

~~~
empath75
The loops are just notation. They still have sharp angles.

~~~
akkartik
I guess I'm not persuaded by the whole thread through the articles of angles
and polygons having direction, clockwise vs counter-clockwise and whatnot. If
the loops are just sharp angles, then every polygon within [-2, 2] is
identical to some polygon outside [-2, 2].

This disconnect is perhaps most obvious at a couple of ends of the spectrum.

a) The digon (2-sided polygon). Euclidean geometry assumes lines are atomic
and have no inner structure. So a digon would be just a line segment.

b) The monagon (1-sided polygon). If the loops are just points then this is
just a line (extending to infinity).

Perhaps polygons made of directional lines (rays but also distinguishing loops
from non-loops) are a distinct, internally consistent non-euclidean geometry.

~~~
Sharlin
It is often useful to have polygon edges be directional, that is, identify
them with either of the two equal-but-opposite vectors between two vertices.
In 3D computer graphics you usually want to have either "clockwise" or
"anticlockwise" polygons, but not a mixture. This is because the handedness of
a polygon determines which way the surface normal points. In addition to
shading this is used to determine whether a polygon is a "backface" (that is,
faces away from the camera and need not be rendered assuming it's a part of a
solid shape).

Re: the digon, as the article mentions, some digons have nonzero area in
elliptic geometry (such as on the surface of a sphere). Yes, it's non-
Euclidean, but not too unfamiliar.

~~~
akkartik
Yeah, I'm familiar with basic computer graphics. It's perfectly reasonable to
adopt a _convention_ about handedness as a way to distinguish inside from
outside. But that's all it is. What seems real to me is the inside-outside
distinction, not the handedness. When you make a loop, inside/outside doesn't
change relative to the no-loop case. So it's not clear what a loop _is_. Is it
just a convention for distinguishing between polygons that are otherwise
identical? That would be fine, but it wasn't clear that that's what the author
thinks.

~~~
ithkuil
The inside/out distinction is the whole point. The author of the article was
indeed reasoning whether the antipolygon is the hole left from the cutout of
the corresponding positive polygon or even the rest of the space outside of
the cutout

~~~
akkartik
Are you disagreeing with me? I'm not sure where, since I've said:

 _" What seems real to me is the inside-outside distinction..."_

 _"...the analysis of [-2..2] is questionable."_ [So the rest of the negative
number line swapping inside and outside is not questionable.]

------
TapamN
I remember playing with LOGO (MicroWorlds, specifically) in 2nd or 3rd grade,
and I was given code that would draw a n-sided polygon. I got the idea to draw
polygons with weird numbers of sides (like a 0-sized polygon, or a polygon
with a non-integer number of sides) and at one point I got a circle made with
a dashed line (I'm pretty sure this was the 0-sided polygon).

It was (and still is) super confusing since it had no pen up commands at all,
so everything should have been connected. I guess it couldn't handle the weird
angles and distances it was being asked to draw.

------
wlesieutre
On a related note, I discovered yesterday that AutoCAD won't let you draw a
closed "polyline" object with two sides.

One of the features of the polyline object is that you can turn edges into
arcs when you need a curve, and I was trying to make a "D" shape. Had to add a
third vertex in the middle of the flat side before it would let me.

~~~
athom
That's weird. I remember learning AutoCAD back in the release 10 and 11 days.
They used to have this "donut" command that would draw a 2D donut shape using
a polyline with thickness, and just two points joined by arcs.

Now I think of it, I think I remember: it used two points, with arcs instead
of line segments, and "closed" to get the second arc. I'm not sure if you
could force the closing segment back to a straight one, but that could have
probably got your "D" shape, if so.

~~~
wlesieutre
I tried it again today and it worked, so there must have been some other
reason it was failing. Maybe one of my lines was out of plane? But then I'm
not sure why adding a midpoint on the straight edge fixed it. I must have
accidentally fixed some other problem at the same time.

Here's another odd note though, if you have a polyline with a single arc and
use properties to set it to closed, it adds a straight edge and you get a D
shape. If you use PEDIT CLOSE it adds another arc to close it instead, and it
tries to continue tangent at the end, completing a whole circle.

------
foobarbecue
Shapes with _fewer_ than two sides.

~~~
yesenadam
TL;DR Nuh-uh.

But that's not how most people talk, is it? Maybe in your circles, but not in
mine. Is that a useful grammatical point to insist on? Why? "Less" is shorter
and easier to say, everyone knows what it means. I guess that why it's
universally used nowadays.

ps Often, maybe most of the time that I see "its" on here it's spelt wrongly,
and it seems very soon that - e.g. "The dog wagged it's tail" \- will be
correct. I even frequently see plurals made with apostrophes, even on HN, and
fear that may be an acceptable plural form one day.. But language isn't a
fixed thing, as some people seem to think. However everyone speaks/writes, is
correct. Where else did - or could - the first grammar books come from but
writing down rules abstracted from how people actually spoke/wrote?

~~~
foobarbecue
If people like me didn't resist change, slogF ripsni() s'nork.

~~~
yesenadam
Hahaha thanks for the laugh. Hmm well, Spanish has the RAE, (Royal Spanish
Academy) which I at first thought a silly idea - an academy to control
spelling. But over the years (of learning Spanish) I came to appreciate them.
They did most of their work in the 18th and 19th C, changing Spanish spelling
to be much less like French (e.g. getting rid of 'gn' in words) and reverting
a lot of words to their 'proper' Latin consonants (like making sure 'b' and
'v' weren't swapped).

There's this lovely story - _How an eight-year-old boy invented the new word
'petaloso'_ \- thanks to RAE's Italian equivalent.

[https://www.bbc.com/news/blogs-
trending-35653871](https://www.bbc.com/news/blogs-trending-35653871)

