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## #1 2022-12-04 21:01:25

mathdrop
Member
Registered: 2022-03-07
Posts: 73

### Egg formula

The longitudinal cut of an eggshell.
After everything below I hold the opinion, that it is an asymmetric ellipse.

First I digged into Wikipedia.

The cause of the shape of eggs according to an article, linked there, about Mary Caswell Stoddard and her team:

"It begins when an unfertilized egg cell is added to a globule of yolk, and sent down a bird’s oviduct... ...On its travels, it is fertilized by sperm, surrounded by white, and coated in two membranes. The membranes are pumped with fluid like a balloon being inflated, and finally surrounded by a shell. Counter-intuitively, it’s not the shell that matters most, but the membranes. If you dissolve the shell in acid, the naked egg will still retain its original shape."

They took the Baker-Formula and matched bird eggs with it (EggxTractor).

There are different construction methods.
The Wikipedia article also links a paper titled the mathematics of egg shape of Yutaka Nishiyama.
After explaining the Descartes approach, he goes on with the Cassini approach, which is more or less a shortcut of the former.
This could probably be read before taking on the Baker-Formula:

y=t(1+a)^(1/1+a)*(1−a)^(a/1+a)

t=(equatorial diameter)^(-1) ?
a=ln(heightright)/ln(heightleft)

There is another equation which
I struggle to let go of just yet.
It's the Blaschke one, I found on mathtinkering:

x^1.5-l^0.5x+y^2=0
y = ± (sqrt(x)*sqrt(sqrt(a)-sqrt(x)))

Since 2021 this formula of Narushin (see there for variable descriptions) was hyped: