Neat! Is it based on the T-graphs in Kenyon & Sheffield's "Dimers, tilings and trees"?
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Replying to @blockspins
It was indeed inspired by Kenyon's work, but my construction is as follows: -Relax a network of zero rest length springs(blue) -Construct the Maxwell reciprocal figure(red) -Form a new quad grid(green) using the above 2 networks as the diagonals...pic.twitter.com/ivmCYgs99O
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Replying to @KangarooPhysics @blockspins
What's a Maxwell reciprocal figure? And am I right in guessing that, in relaxing the springs, you make each vertex equal to the average of its neighbors?
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Replying to @akivaw @blockspins
It's known by various names, but essentially the forces on each node correspond to a closed polygon in the reciprocal, for structures in equilibrium (this is the key idea in graphic statics) https://homepages.abdn.ac.uk/j.s.reid/pages/Maxwell/Legacy/MaxRecip.html … https://en.wikipedia.org/wiki/Cremona_diagram …
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..and yes-relaxing a network of zero rest length springs is equivalent to iteratively moving each point to the average of its neighbours.
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Replying to @KangarooPhysics @blockspins
What happens if you translate the red figure (leaving the blue where it is)?
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Replying to @akivaw @KangarooPhysics
That's a good question. In that vein let me also ask how you choose the overall scale of the reciprocal diagram?
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Replying to @blockspins @akivaw
Both interesting questions. Translating the red relative to the blue results in the squares being translated by half that vector. Also shown here - the green quads can become concave or even self intersecting, but the Varignon parallelograms still stay square.pic.twitter.com/yqxkaOpDPt
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As for scaling - in the ones shown above, the lengths of the reciprocal are all equal to their corresponding edges in the spring mesh. If we scale the reciprocal it changes all the squares into rectangles with the same aspect ratio. Here's 1:2pic.twitter.com/EjmSajFa4s
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...and here are just the Varignon parallelograms as the relative scale varies. In the 2 limits the rectangles become the lines of the spring mesh and its reciprocal figure. This is the same alternative form of graphic statics I wrote about here: https://twitter.com/KangarooPhysics/status/1044325364779102209 …pic.twitter.com/OSEgazeRsr
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This is brilliant, thank you!
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