國立清華大學 數學系

Abstract: Given a positive integer n, an n-Calabi triangle is a non-equilateral triangle in which the largest regular n-polygon can be positioned in different ways (with one side of the polygon lying on different sides of the triangle). For n=4, Calabi found that the n-Calabi triangle (for squares) is isosceles with the ratio of the base to either leg being 1.55138752454..., which is a root of a cubic equation. Jerrard and Wetzel found the 3-Calabi triangle. In both n=3 and 4, the n-Calabi triangles are unique (up to similarity). In this talk we show that there are many n-Calabi triangles for n greater than 4 and give a necessary and sufficient condition for an isosceles triangle to be an n-Calabi triangle. We will also discuss some related results and unsolved questions in general covering problems.