Part 1:
Regular hexagon ABCDEF is surrounded by two parabolas. One contains A,B,C,D and the other contains D,E,F,A. What is the ratio between the area of the intersection of the parabola's interiors to the area of the hexagon?
Part 2:
Regular hexagon ABCDEF is surrounded by two parabolas. One contains A,B,C,D and the other contains C,D,E,F. What is the ratio between the area of the intersection of the parabola's interiors to the area of the hexagon?
Part 3:
Regular hexagon ABCDEF is surrounded by two parabolas. One contains A,B,C,D and the other contains B,C,D,E. What is the ratio between the area of the intersection of the parabola's interiors to the area of the hexagon?
The solutions are:
Part 1: (2nd parabola rotated 180 degrees), ratio = 32/27
Part 2: (2nd parabola rotated 120 degrees), ratio = 35/27
Part 3: (2nd parabola rotated 60 degrees), ratio = 59/27
The links above show all the calculations. One can see that the
integrals had to be done carefully to find the rational ratios
above. (Approximations from simulations could not definitively
assure the ratios are indeed rational fractions, although they
did help to check the results.
Some inventive integration methods were devised, especially for
Case 2, the most difficult case. The integration difficulty
originated from the fact that the rotated parabolas are mathematically
complex and not necessarily functions at all.
See: http://www.math-principles.com/2014/11/rotation-of-parabola-2.html
It is surprising that the two irrational areas, when ratioed,
yield a rational number. The most amazing part of the result is that
the denominators of each ratio are the same, as if pointing to
some deeper connection between the hexagon area (irrational)
and the positioning of each parabola on the vertices of the
hexagon, that produced that irrational overlap area.
Is there any literature on this provocative result?
In the following post: "More hugging" I explore whether
these rational rations will pop up if the problem is done
with other polygons than a hexagon
Edited on July 31, 2023, 1:41 am
Summary of results
