Polygons¶
Recall (doc/postdoc). Not a first course.
Job: piecewise-linear closed curves; regular \(n\)-gon \(\Leftrightarrow\) cyclic \(+\) equilateral; interior angle formulas.
Core¶
\(n\)-gon: \(n\) vertices, \(n\) sides, simple if non-self-intersecting.
| Quantity (convex) | Formula |
|---|---|
| Interior angle sum | \((n-2)\pi\) |
| Each regular interior | \(\dfrac{(n-2)\pi}{n}\) |
| Exterior sum | \(2\pi\) (one full turn) |
| Diagonals | \(\binom{n}{2}-n=n(n-3)/2\) |
Regular: all sides equal and all angles equal \(\Leftrightarrow\) inscribed in circle and equilateral (char \(0\) plane).
Tessellation: regular tilings of plane for \(n=3,4,6\) only.
Used later for¶
- Circles: cyclic polygons, Ptolemy
- Complex analysis: roots of unity vertices
- Algebraic topology: polygonal presentations of surfaces
- Optimization: polytopes, LP feasible sets
Check¶
Does interior angle sum \((n-2)\pi\) hold for non-convex simple polygons?
Yes for simple planar polygons (triangulate); fails for complex self-intersecting or spherical polygons (excess).