Ladder
Funicular — the ladder
2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
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The shape that carries itself, and the arch that is its reflection
Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is.
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The polygon that finds the shape
A hanging string under five loads has no smooth curve in it — it has five vertices and six straight segments, and every slope in it is a running sum divided by one number.