To create an arch out of frictionless stones, that is, the elements feel compression only (except for the end stones anchored to the ground), we should use an inverse catenary, like the St. Louis Gateway Arch.
By analogy, does this mean the ideal shape of a dome is a catenary of revolution? Most domes seem to be built as sections of a sphere. In particular, are geodesic domes, which are sections of a sphere, actually optimal in minimizing stress on individual elements?
The catenary results from hanging a chain of uniform density. Consider the inverse problem. What varying linear density function results in the hung shape being a section of a circle (or another given shape)? Spherical dome?
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