A point is created at the origin and labeled O. A prime number p is picked and 2 points are created at (p,0) and (-p,0). The points are labeled B and C. Using OB and OC as the bottoms, two 5 sided regular polygons are created. They are blue in the diagram. The orange area represents the overlap between the two polygons.
What is the value of the Area of polygon OBGHI divided by the area of the orange polygon?
We will use geogebra to get the analytical answer.
1) Place a slider on the screen, it will be called a and range it from -10 to 10.
2) create points (0,0),(a,0) and (-a,0).
3) Rename A to O.
4) Move the slider until a = 5 which is a prime.
5) Use the regular polygon tool and click C and then O and enter 5 as the number of sides. A five sided regular polygon will be created.
6) Use the regular polygon tool again and click O and then B and enter 5 as the number of sides. A second five sided regular polygon will be created.
7) Use the intersection tool and click ED and HI to find the intersection point J.
8) Use the polygon tool and click on A,D,J,I and back to A to create poly3. Right click it and color it orange.
9) Enter poly1/poly3 in the input bar and set rounding to 15 significant digits.
Copy I, you should get l = 4.73606797749979. Now we would like to turn that into an exact answer if we can.
Take that over to Wolfram and enter "Closed form 4.73606797749979." You will get:
Or use your own M and enter:
RootApproximant[ 4.73606797749979] // FullSimplify
which is correct. Now move the slider around to convince yourself that for any AO and BO that number is a constant.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.