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.
- What is the area of ABD?
- What is the function that describes BD?
- 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.