Megascaling Rule: Conservation of
Tensile Cross-Section across Fractal Dimensions
Leonardo noted that tree structure usually follows a simple conservation
rule, that equivalent cross-sectional area is maintained at every
branching scale. Thus tree-trunk cross-section equals branch cross-section
equals twig cross-section. This is attributed to the tree's need for
optimal strength and extension with minimal mass.
This rule approximates optimal megscaling of soft kites. Starting with the
main rope(s) of a kite as the the thickest "trunk," a continual branching
progression of scaling jumps occurs all the way down to the thread or
membrane thickness. The diffuse kinetic energy of the wind "fans-in" by
degrees to the main ropes to match the concentrated kinetic energy of the
anchor, resulting in maximal static force handling. A branching factor of
2 is the natural minimum, with the ideal number of branching steps
determined by the ratio of the wind load on a twig to the total load on
This is how Mothra loadpaths are generally sized, with each major loadpath
branching as smoothly as practical. It is rigger's art to choose COTS
loadpath ropes to form a nearly ideal sequence, and especially to design
the junctions to maintain full strength, avoiding knots or heavy expensive
shackles with splices and soft-shackle methods.
Unlike trees, with severe scaling limits imposed by cantilever tower
structure, branching tensile networks seem to mostly avoid normal scaling
laws. Two factors in particular help; that the cross-sectional strength of
a rope grows at the square of bluff-body drag, and that the energy of the
airmass processed by a wing grows at the square of membrane area.
These advantages and the ready methods help predict soft-kite megascaling
feasibility, even though more study remains to find the technological
limits. Many subtleties to the fractal branching system are noted in
trees- Leaves fail first to unload wind peaks. The progression of flexible
compliance and statistical load averaging ensures that the trunk rarely
fails before its branches.
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