BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Allison Yip DTSTART:20261107T140500Z DTEND:20261107T153500Z DTSTAMP:20261109T050833Z UID:knotgraphgroup/176 DESCRIPTION:Title: Sedrakyan's Area Formula: A Breakthrough on a 2\,000-Year-Old Pro blem of Areas\nby Allison Yip as part of Knots\, graphs and groups\n\n \nAbstract\nThis work fills a 2\,000-year-old gap and establishes a struct ural “missing link” in polygon geometry. Heron’s formula dates back to approximately 60 AD\, while Bretschneider’s formula dates to 1842. Ye t no comparable unified formula existed for polygons with more than four s ides. This work shows that these two formulas\, which appeared fundamental ly different for centuries\, are particular cases of a single general form ula. The remarkable achievement is not merely that the formula uses only s ide lengths and certain distance lengths\, but that one unified formula co mputes the area of any convex n-gon for every n ≥ 3. This is the key nov elty\, addressing a problem previously considered impossible to resolve in a similarly general way for n ≥ 5. The formula thus overcomes the perce ived rigidity-related barrier to determining polygon areas for n ≥ 5. Th e area of a convex n-gon A₁A₂...Aₙ\, where n ≥ 3\, is given by\n\n S = (1/4) × √[4(A₁A₃)²(A₂B₂)² − (a₁² − a₂² + ⋯ + (−1)ⁿ⁻¹aₙ²)²]\,\n\nwhere AᵢAᵢ₊₁ = aᵢ\, with indices t aken modulo n\, and B₂ is the geometrically constructed point defined in the main results. The point B₂ can be viewed as a representative point associated with the figure’s area distribution\, analogous in spirit to the center of mass of a three-dimensional body. Potential applications inc lude distance geometry\, knot theory\, drone networks\, robotics (no GPS\, no coordinates\, only pairwise distances\, Area is mission-critical)\, an d other fields. For the full theorem\, geometric construction\, and proof\ , please visit: https://www.sedrakyans.org/sedrakyans-area-formula\n LOCATION:https://researchseminars.org/talk/knotgraphgroup/176/ END:VEVENT END:VCALENDAR