Harry Hart's dissection of two similar circumscribed polygons to one is fully hingeable, rather than just wobbly-hingeable. The proof is evident upon a close examination of Figures 19.1 and 19.2. If we consider two consecutive quadrilaterals in Figure 19.2, and then the closest edge of each to the other, we see that the edges are parallel to each other. Of course, the relevant edges are also parallel to each other in the leftmost and rightmost polygons of Figure 19.1, since the two quadrilaterals are flush up against each other in those polygons. In fact, we see that every such pair of edges remain parallel when we take the right triangles of Figure 19.2 and rotate them at the same rate about their right angles. Rotation in a clockwise direction will give us the leftmost polygon in Figure 19.1, and rotation in a counterclockwise direction will give us the rightmost polygon. The rotations will keep the edges in each pair parallel. Thus the dissection is fully hingeable.
In a similar fashion, we argue that the dissection of two similar circumscribed polygons to two is also fully hingeable, this time referring to Figures 19.3 and 19.4.
Since a regular pentagon can be circumscribed, we can apply the dissection in Figure 19.1 to regular pentagons too. And indeed the two pentagons can also be congruent. In this case, it is possible to get a hinged dissection with ten pieces rather than eleven, as Gavin Theobald discovered in March 2007. Gavin took Collison's approach to dissecting two polygons to one and modified it. Instead of using Bradley's method for dissecting two triangles to one he used the Q-slide instead. The dissection turns out to be wobbly-hinged, but still, it's an improvement!
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Hinged Dissections: Swinging & Twisting
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Last updated January 19, 2015.