Talk:Isoperimetric ratio

Question[edit]

Can two regions, each enclosed by a simple closed curve not similar to the other, have the same isoperimetric ratio? I think that at least for polygons the answer is "no", but I'm not sure and I could not find any text discussing this question... DakarMath (talk) 05:34, 10 November 2015 (UTC)[reply]

Of course yes. The isoperimetric ratio of triangles varies continuously from small (for equilateral triangles) to arbitrarily large (for instance both for isosceles triangles with very sharp apex angles, or with very obtuse apex angles. For any particular value greater than the value for an equilateral triangle, there exist two isosceles triangles with that value (one with a sharp apex and one with a flat apex) along with infinitely many other scalene triangles. —David Eppstein (talk) 06:15, 10 November 2015 (UTC)[reply]

Very simple to understand example! Thank you. DakarMath (talk) 06:25, 10 November 2015 (UTC)[reply]