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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 l

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 8y ago
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 8y ago
  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 8y ago
    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 8y ago
      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.]

Sharlin 8y ago

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 8y ago
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 8y ago
  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 8y ago
    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.]