Five peas in a pod; five beans in a circle

[fivemack]

set parametric
e=(sqrt(5)-1)/4
plot sin(t),cos(t)*e,e*e*sin(t)+(1-e*e),e*e*cos(t),e*sin(t),e*cos(t),(0.5+0.5*e-e*e)+(0.5-0.5*e-e*e)*sin(t),(0.5-0.5*e-e*e)*cos(t),(-1+e*e)+(e*e*sin(t)),e*e*cos(t),-(0.5+0.5*e-e*e)-(0.5-0.5*e-e*e)*sin(t),(0.5-0.5*e-e*e)*cos(t)

e=0.46
v=0.4073
u=0.673
phi=0.55
cp=cos(phi)
sp=sin(phi)
plot sin(t),cos(t),sin(t),e*cos(t),v-cp*e*sin(t)+sp*e*e*cos(t),u+sp*e*sin(t)+cp*e*e*cos(t),-v+cp*e*sin(t)-sp*e*e*cos(t),u+sp*e*sin(t)+cp*e*e*cos(t),v-cp*e*sin(t)+sp*e*e*cos(t),-u-sp*e*sin(t)-cp*e*e*cos(t),-v+cp*e*sin(t)-sp*e*e*cos(t),-u-sp*e*sin(t)-cp*e*e*cos(t)




The smallest peas match the radius-of-curvature of the ends of the ellipse; the smaller beans touch their mirror-images and the central ellipse and match the radius-of-curvature of the outer circle. Those constraints define e.

Comments

[hairyears.livejournal.com]

I recognise one of those: it is the most efficient configuration for a dirigible airship.

Things not as widely known as they should be: airships aren't the shape you think they are - we see the outer skin, and think that's full of hydrogen or helium, but the actual gasbags are always spherical and held in a rigid (or semi-rigid) structure with an aerodynamic fuselage; and that the low-drag profile isn't about speed, it's about reducing the effect of crosswinds so that lightweight engines can keep the 'dirigible' pointed in the direction it's supposed to go.