June 2019, ScientificAmerican.com 17LEIF RISTROPHApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York UniversityBIOMECHANICSFlight
Simulator
Finding the ideal wing shape
through evolutionHumans have long drawn inspiration
from bird wings to design mechanical
ones—and now a team of mathematicians
has taken this biomimicry to a new level.
By 3-D-printing a variety of wing shapes,
racing them in a laboratory and feeding
the data into an algorithm that simulates
evolution, the researchers found that a
teardrop-shaped wing is fastest for both
flapping flight and swimming.
This is the first time such a combined
process has been used to find an optimal
wing shape for fast flight, says Leif Ristroph,
a mathematician at New York University’s
Courant Institute of Mathematical Sciences
and senior author of the new study.
Specific aspects of the teardrop shape
help to make the optimal wing faster than
its competitors, Ristroph says. These
include its front-to-back asymmetry (when
viewed from the side), characterized by
a rounded front, forward placement of its
thickest point and a slender, trailing tail.
The razor-thin back edge resembles that
of a bird wing, which typically narrows to a
single feather. The finding suggests birds’
wings have evolved to be as thin as possi-
ble, the researchers write in the study,
which was published in January in the Pro-
ceedings of the Royal Society A.Ristroph and his collaborators 3-D-print-
ed a first “generation” of 10 plastic wings.
They attached each wing to a motor-driv-
en horizontal rod that caused it to flap up
and down in water. They measured its
swimming speed and extrapolated its fly-
ing speed. They tested a variety of shapes,
including ones based on conventional air-
plane wings, flattened spheres and a pea-
nutlike structure, Ristroph says.
The researchers fed the wing speed
data into the evolutionary algorithm, which
produced a second generation of eight
“daughter” wings. Faster wing shapes were
more likely to be passed on to the next
generation, but the algorithm also allowed
“mutations” that could yield new shapes.
The two fastest wings from the first gener-
ation were also added to the second. The
process of 3-D printing and laboratory rac-
ing was then repeated with the second
generation of 10 wings. Altogether the
researchers created 15 generations of
wings. The fastest wing—the teardrop
shape—evolved in the 11th generation and
persisted in the following ones. The algo-
rithm’s attempts to improve this shape in
subsequent generations yielded ones that
were too slender to 3-D-print.
The study “is tremendously interest-
ing,” says Geoffrey Spedding, an aero-
space engineer at the University of South-
ern California, who was not involved with
the work. He notes that the optimal wing
is “more like a fish fin,” which makes it bet-
ter suited for swimming or propelling ob -
jects forward than for generating lift, as in
airplane flight. — Rachel CrowellNonideal wing shape generates vortices ( visualized here with red dye ) at its leading edge that
interfere with trailing-edge vortices ( green dye ).© 2019 Scientific AmericanLEIF RISTROPHApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York UniversityBIOMECHANICSFlight
Simulator
Finding the ideal wing shape
through evolutionHumans have long drawn inspiration
from bird wings to design mechanical
ones—and now a team of mathematicians
has taken this biomimicry to a new level.
By 3-D-printing a variety of wing shapes,
racing them in a laboratory and feeding
the data into an algorithm that simulates
evolution, the researchers found that a
teardrop-shaped wing is fastest for both
flapping flight and swimming.
This is the first time such a combined
process has been used to find an optimal
wing shape for fast flight, says Leif Ristroph,
a mathematician at New York University’s
Courant Institute of Mathematical Sciences
and senior author of the new study.
Specific aspects of the teardrop shape
help to make the optimal wing faster than
its competitors, Ristroph says. These
include its front-to-back asymmetry (when
viewed from the side), characterized by
a rounded front, forward placement of its
thickest point and a slender, trailing tail.
The razor-thin back edge resembles that
of a bird wing, which typically narrows to a
single feather. The finding suggests birds’
wings have evolved to be as thin as possi-
ble, the researchers write in the study,
which was published in January in the Pro-
ceedings of the Royal Society A.Ristroph and his collaborators 3-D-print-
ed a first “generation” of 10 plastic wings.
They attached each wing to a motor-driv-
en horizontal rod that caused it to flap up
and down in water. They measured its
swimming speed and extrapolated its fly-
ing speed. They tested a variety of shapes,
including ones based on conventional air-
plane wings, flattened spheres and a pea-
nutlike structure, Ristroph says.
The researchers fed the wing speed
data into the evolutionary algorithm, which
produced a second generation of eight
“daughter” wings. Faster wing shapes were
more likely to be passed on to the next
generation, but the algorithm also allowed
“mutations” that could yield new shapes.
The two fastest wings from the first gener-
ation were also added to the second. The
process of 3-D printing and laboratory rac-
ing was then repeated with the second
generation of 10 wings. Altogether the
researchers created 15 generations of
wings. The fastest wing—the teardrop
shape—evolved in the 11th generation and
persisted in the following ones. The algo-
rithm’s attempts to improve this shape in
subsequent generations yielded ones that
were too slender to 3-D-print.
The study “is tremendously interest-
ing,” says Geoffrey Spedding, an aero-
space engineer at the University of South-
ern California, who was not involved with
the work. He notes that the optimal wing
is “more like a fish fin,” which makes it bet-
ter suited for swimming or propelling ob -
jects forward than for generating lift, as in
airplane flight. — Rachel CrowellNonideal wing shape generates vortices ( visualized here with red dye ) at its leading edge that
interfere with trailing-edge vortices ( green dye ).sad0619Adva3p.indd 17 4/17/19 6:19 PM
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