53,033 research outputs found
On the Square Peg Problem and some Relatives
The Square Peg Problem asks whether every continuous simple closed planar
curve contains the four vertices of a square. This paper proves this for the
largest so far known class of curves.
Furthermore we solve an analogous Triangular Peg Problem affirmatively, state
topological intuition why the Rectangular Peg Problem should hold true, and
give a fruitful existence lemma of edge-regular polygons on curves. Finally, we
show that the problem of finding a regular octahedron on embedded spheres in
R^3 has a "topological counter-example", that is, a certain test map with
boundary condition exists.Comment: 15 pages, 14 figure
- …