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.