Overturned polygons: shapes with less than two sides
chalkdustmagazine.com
chalkdustmagazine.com
Polygons with a negative number of sides: http://chalkdustmagazine.com/blog/thinking-outside-box
Polygons with a fractional number of sides: 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
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.
(Warning: likely butchering the terminology, am not a mathematician)
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.
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.
"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.]
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.
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.
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.
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.
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?
There's this lovely story - How an eight-year-old boy invented the new word 'petaloso' - thanks to RAE's Italian equivalent.