This is in continuation of Two circles
In a 8.5x11 sheet of paper I drew two non-overlapping ellipses of equal area, with major axis of each equalling twice the minor axis - both completely inside the paper, of course.
What's the largest portion of the paper I could cover with these ellipses?
What would be the answer if I drew THREE equal ellipses?
I have been trying to picture what the optimum configuration for the ellipses would be in the 2 and 3 ellipse cases.
For two ellipses the figure has 180 degree rotational symmetry. Both ellipses are parallel and each is tangent to both long sides and one short side each. The ellipses are tangent to each other in the center of the rectangle.
For three ellipses the figure has mirror symmetry over the line dividing the rectangle into two 8.5 by 5.5 rectangles.
The first ellipse is tangent to the long side at the midpoint and has its major axis parallel to the long side.
The other two ellipses are reflections of each other. They are tangent to each other at the symmetry line. They are also both tangent to the long side opposite the first ellipse. They are also each tangent to the first ellipse and one short side.
The math gets messy rather quickly for two ellipses, I did not even try to analytically structure the three ellipse case.