One thing which fascinated me when I was young is that there's no elementary closed-form expression for the length of an ellipse, BUT there is one for the area of an ellipsoid of revolution (aka “spheroid”).
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Another way of looking at it : we need to write an infinite series if we want the length of an ellipse written with basic arithmetics. But that's also true for a circle !
We just hide this infinite series under the cute little label "π".
The length of an ellipse (half major axis a, excentricity e) is as follow.
The length of a circle has a "closed form" only after we define pi as being four times the value of the integral for e=0.
Also, for a long time, I was convinced that the length of an ellipse of semiaxes a,b had to be π·(a+b) (it's wrong!) because this works for a circle and it seemed that it “had to be” an elementary symmetric polynomial in a and b just like the area is π·a·b (this one is correct).
Déjà il y a plein de types d'hyper-ellipsoïdes selon le nombre de demi-axes qui sont égaux.
Je soupçonne, mais je n'ai pas vérifié, qu'en dimension IMPAIRE, si un unique axe est différent de tous les autres, alors il y a une formule pour la surface (enfin, l'hypersurface).
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We just hide this infinite series under the cute little label "π".
The length of a circle has a "closed form" only after we define pi as being four times the value of the integral for e=0.
Je soupçonne, mais je n'ai pas vérifié, qu'en dimension IMPAIRE, si un unique axe est différent de tous les autres, alors il y a une formule pour la surface (enfin, l'hypersurface).