page_title: Winding number
section_title: Polygons
hierarchical_section_title: Winding number / Alternative definitions / Polygons
caption_attribution_description: Star polygon 9/4
context_page_description: In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.
  Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics, including string theory.
context_section_description: In polygons, the winding number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q.
caption: The boundary of the regular Enneagram {9/4} winds around its centre 4 times, so it has a density of 4.
exif: {}