A Paradoxical Dissection

نوشته شده در موضوع تصاویر پارادوکس در 15 ژانویه 2015

Is it probable to disintegrate an 8×8 block and file a pieces to form a 13×5 rectangle? Common clarity indicates that this can't be done, since a area of a block is 64, while a area of a rectangle is 65. However, this animation seems to uncover that it is possible. What is going on?



de22f u mad math imgur A Paradoxical Dissection





Since a rectangle is incomparable than a square, a pieces should not fit together precisely, though there should be a gap. In a prior image, a opening was lonesome adult by creation really tiny adjustments to a shapes. When a shapes are drawn accurately, a opening is revealed.

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You competence be wondering how we can trust that this sketch is accurate, since a prior sketch also looked flattering convincing. This is where math comes to a rescue! Notice that a slope of a hypotenuse of a immature triangle is 3/8, though a slope of a longest side of a blue trapezoid is 2/5. Since 3/8 is reduction than 2/5, a slight opening exists between a immature triangle and a blue trapezoid.

Incidentally, a fact that 3/8 is really tighten to 2/5 is not a coincidence, though is formed on a properties of Fibonacci numbers. Note that 2, 3, 5, and 8 are uninterrupted Fibonacci numbers. One can emanate a identical enigmatic ratiocination by regulating any 4 uninterrupted Fibonacci numbers.

But a slow doubt still remains. The opening looks flattering tiny — how do we know that it has area 1? There are many ways to determine this, though one of a many engaging ways is to use Pick’s Theorem. Pick’s postulate states that if a polygon is drawn on a grid such that all vertices are during grid points (points with integer coordinates), afterwards a area is equal to i + b/2 − 1, where i is a series of interior grid points and b is a series of grid points on a boundary. In this case, there are 4 grid points on a boundary: (0,0), (8,3), (13,5), and (5,2). There are no grid points in a interior, so a area is 0 + 4/2 − 1 = 1.

This ratiocination is apparently due to Sam Loyd. The missing block puzzle is a famous various that is formed on dissecting a right triangle instead of a square.

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Article source: http://mathblag.wordpress.com/2011/08/28/a-paradoxical-dissection/

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