If our test for whether a regular polygon can tesselate with itself is whether the degrees of an individual interior angle can divide 360 to yield an integer (some examples of these integers are 6 [for equilateral triangles], 4 [for squares], and 3 [for regular hexagons]), why do circles (with interior angles each of 180) not tesselate?
My thinking was that 180 goes into 360 two times, so we should be able to put two circles at a point and have them tesselate.
Does this have something to do with three points being necessary to define a plane, and our formula suggesting that a mere two circles could work?
Is there something about curves that makes them very different from line segments joined together? I would find this intriguing, since we can get to a circle by thinking of it as a regular polygon with n sides as n goes to infinity (I think).*
I have googled this and not found explanations for why the above algorithm (Let n be number of sides. If 360/(((n-2)*180)/n) is an integer, a regular polygon of this number of sides tessalates. [Or, simplified, we could use whether 360/(180-360/n) is an integer.]) delivers an incorrect result in this case.
This (https://study.com/academy/answer/will-a-semi-circle-tessellate.html) may contain an answer, but it's behind a paywall.
Is there some sense in which sides and interior angles break down as concepts when n goes to infinity?
Is there some issue of 180-360/n as n goes to infinity actually being slightly larger than 180?
*What are the rules with regard to considering curved shapes as versions of lines (or planes) stuck together in a particular way, with their lengths approaching 0 and their number approaching infinity? I don't only mean calculus (which I take to be about finding gradients and areas or volumes under curves), but also this type of thing, where tessellation and angles or other geometric things are involved.
That condition is necessary (because the total angle around the vertices must cover a full turn) but not sufficient. (Being a polygon might be required.)
Tessellation with a flat interior angle can be considered to be achieved by a "bigon", i.e. two half-lines delimiting a half-plane.
For more about tessellations: https://en.wikipedia.org/wiki/Wallpaper_group
Tessellations are straight edge graphs. Two tiles share an edge and three share a vertex. Hence a reasonable condition of existence is $n\ge3$, corresponding to interior angles $\ge120°$.