Problem of the Week #8: Monday Feb. 27th, 2023 Last one in this geometry streak. This one seems to be more on the conceptual side. 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. **I've just noticed this week that the Terms of Service page was not working correctly. It seems to be fixed for now. Sorry for those who tried to access it and was told the page didn't exist.
a). Consider a sphere. What range of shapes can we obtain by slicing it with a plane and looking at the cross section? Sketch all possible shapes and indicate the range of possible sizes.
b). Now do the same for a right cylinder and a right cone.
c). Consider a regular tetrahedron. What range of shapes can we obtain by slicing it with a plane and looking at the cross section? Sketch all possible shapes and indicate the ranges of possible angles and side lengths.
d). Now do the same for a cube and a regular octahedron.
e). What range of cross sectional shapes can we obtain if we allow the three dimensional shapes to be irregular, for example, an ellipsoid, an oblique cylinder or cone, an irregular tetrahedron or octahedron, a cuboid, etc.?
f). Share your own problem inspired by these considerations.
g). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.