What is carpenter's theorem?

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What is carpenter's theorem?

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Asked 6/19/2012 9:16:15 AM

Updated 9/29/2024 1:47:45 PM

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User: What is carpenter's theorem?

Weegy: In 2000, we solved the following three problems positively: Convexifying polygons: Given a simple polygon in the plane (whose edges are considered rigid bars and whose vertices are considered hinges), [ is there a continuous motion of the polygon that preserves the lengths of the edges and never causes edges to cross, and results in the polygon being convex? Carpenter's rule conjecture: Given an open polygonal chain (also called a polygonal path or polygonal arc) in the plane, is there a continuous motion with the same properties that results in the arc becoming straight? A generalization: Given a collection of polygonal arcs and simple polygons in the plane, none of which intersect each other, such that no polygon contains another arc or polygon, there is a motion that preserves all of the edge lengths and never crosses any edges, and results in the arcs becoming straight and the polygons being convex. Furthermore: If desired, an arc or polygon can be contained inside another polygon, but this arc or polygon cannot be guaranteed to be straightened or convexified (as this is not always possible). The motion is expansive: the distance between every pair of vertices only increases. The motion is piecewise-differentiable. The motion preserves any symmetries present in the initial configuration. The configuration space of a polygonal arc or simple polygon, modulo isometries, is contractible. The proof ends up being quite simple with the right ideas (in particular the notion of increasing pairwise distances) and tools (in particular the theory of rigidity) in hand. ]
Expert answered|mrspainter22|Points 130|

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Asked 6/19/2012 9:16:15 AM

Updated 9/29/2024 1:47:45 PM

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Carpenter's theorem states that if a triangle has a given perimeter, the triangle with the largest area is an equilateral triangle.

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