Fun Problem to Solve

Take a unit square ABCD in the coordinate plane: A(0,0), B(0,1), C(1,1), and D(1,0). Connect every point E on AD to every point F on AB such that AE=BF [as in fig. (1)]. The set of all these line segments will form a shape ABD, with BD being a continuous curve created by the infinite number of line segments EF generated through the described process. [fig. (2) and (2a)]. Repeat this process on all sides of the square to create 3 more congruent shapes which partially overlap [fig. (3)]. There will be a central white space remaining [fig. (3a)].

This construction leads to several interesting questions, none of which I’ve been able to answer.

  1. What is the area of ABD?
  2. What is the function that describes BD?
  3. What is the area of the central white space?

Enjoy! This is my first post here, so sorry for any formatting or rule breaking, but I think I’m good. Also, I don’t have a lot of practice in writing mathematics, so if the description above was confusing, the attached picture should be able to answer your questions (coordinates have been omitted).

Edit: As far as I know, I came up with this problem. This kind of curve is something I’ve doodled and seen doodled before, but I’ve never seen anyone ask any mathematical questions about it.

submitted by /u/lallumeur
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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs

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