While Mai and Joey were preparing for the party, Tea Gardner and Yugi Muto, both lovers, hung out at the church, where they stared at a stain-glassed Reuleaux pentagon window, filled with pink and red colors. The conversation started:
Tea: Wow! What a beautiful rose! Isn't it beautiful, Yugi?
Yugi: Yes, it is!
Tea: I have a math problem for you to try out. Suppose we start with five unit circles, where each center is intersected by two circular arcs. Then, this makes a Reuleaux pentagon.
Yugi: That is easy to remember...
Tea: But there's more! Draw another Reuleaux pentagon inside, where each vertex intersects the midpoint of the circular arcs. Repeat this infinitely, so we have much like the stain-glassed one. Here is the following:
[After minutes of perfect-sketching]
Here is the rose with alternating red and white petals. Your goal is to compute the area of all red petals.
Yugi: Touche! What a long geometry problem you asked here! Is there a specific formula to generalize the area of the red petals after finite number of iterations?
Tea: If I give that away, then you will know the answer to this fun problem! Solve this correctly, and we will enjoy a nice party! :)
What is the area of all red petals in the second diagram, where the diagonal of the largest Reuleaux pentagon is 1? If your area is , input as your answer.
A Reuleaux polygon is a curvilinear polygon formed by an odd number of circular arcs. Like a Reuleaux triangle, the apex points of a Reuleaux polygon are centers of identical circles.
For the main problem, determine the area of a Reuleaux pentagon.
Bonus: Generalize this for any odd number of vertices and circular arcs. You should notice that as the number of edges approaches , a Reuleaux polygon becomes close to a circle of area , where is the radius of the large circle.
This is the end of the fourth chapter of the story. Check chapter directory if you are interested: