Comment by akkartik | Hacker News Reader

Comment by akkartik | Hacker News Reader

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.

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)

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.

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.

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.]

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