The branch with the furthest reach

School of Engineering and Applied Sciences, and Department of Physics, Harvard University Cambridge, MA 02138, USA

^a lm@seas.harvard.edu

Received: 16 June 2011
Accepted: 17 November 2011

Abstract

How should a given amount of material be moulded into a cantilevered beam clamped at one end, so that it will have the furthest horizontal reach? Here, we formulate and solve this variational problem for the optimal variation of the cross-section area of a heavy cantilevered beam with a given volume V, Young's modulus E, and density ρ, subject to gravity g. We find that the cross-sectional area should vary according a universal profile that is independent of material parameters, with both the length and maximum reach-out distance of the branch that scale as (EV/ρg)^1/4, with a universal self-similar shape at the tip with the area of cross-section a∼s³, s being the distance from the tip, consistent with earlier observations of tree branches, but with a different local interpretation than given before. A simple experimental realization of our optimal beam shows that our result compares favorably with that of our observations. Our results for the optimal design of slender structures with the longest reach are valid for cross-sections of arbitrary shape that can be solid or hollow and thus relevant for a range of natural and engineered systems.