Overturned polygons: shapes with less than two sides(chalkdustmagazine.com)
chalkdustmagazine.com
Overturned polygons: shapes with less than two sides
http://chalkdustmagazine.com/blog/overturned-polygons/
22 comments
Shapes with fewer than two sides.
I used to own a domain monohedron.TLD - but the bad math joke wasn't really appreciated, ever :/
But now we are allowed continuously many sides..
I am wrong and foxes and superflyguy are abosolutely right! Should be fewer in this case!
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?
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?
If people like me didn't resist change, slogF ripsni() s'nork.
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
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
Read the article again and explain why you think the headline is wrong. (Hint: this morning my walk to work was less than three miles.)
I did not think of that! I concede the point.
The entire system is really cool!
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.
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.
From what the articles are saying it seems like they're either making an assumption of or are working in some non-Euclidean space, and the visuals are merely Euclidean projections or representations, which are necessarily curved.
(Warning: likely butchering the terminology, am not a mathematician)
(Warning: likely butchering the terminology, am not a mathematician)
You might also like Knuth's (3/2)-ary Trees: https://www.youtube.com/watch?v=P4AaGQIo0HY
The loops are just notation. They still have sharp angles.
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.
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.
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.
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.
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.
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
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.]
"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.]
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.
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.
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.
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.
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.
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.
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.
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.