Problem of the Week #25: Monday June 26th, 2023 We never had a topic for June, so I thought we could go back and visit geometry again for the last week of June. As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Let’s consider figures constructed by the following process: Starting with a square, choose two points on the perimeter of the square to make a line, and then move both end points incrementally in the same direction (clockwise or counterclockwise) and the same distance along the perimeter of the square to make subsequent lines. For example, the figure below is constructed by starting with the top two corners of the square, and each subsequent line moves the end points clockwise by 1/10 of the side length of the square.
Notice that there is a white region in the middle. This region is an approximation of a curved figure, and this approximation will improve as we decrease the incremental distance for the end points at each step.
a). Describe the shape of this curved figure approximated by the process above.
b). Calculate the area of this curved figure approximated by the process above.
c). Choose a starting line (two starting end points) that would decrease the area of this curved figure.
d). What is the minimum area of the curved figure?
e). Choose a starting line (two starting end points) that would result in a curved figure that has its four vertices exactly on the perimeter of the square.
f). What is the area of the curved figure in e)?
g). What is the maximum area of the curved figure?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.