Degenerate Triangles: Questions & Poincaré Conjecture

[Jim Lundquist]

Please forgive me...I am not a mathematician, but I have a couple questions that have been puzzling me. In theory, can a circle be so large that connecting 3 points on that circle result in a degenerate triangle? If the length of a straight line drawn between two points on a circle is Planck distance, how can another point fall between those 2 points? Do such questions cause problems using math as a descriptive tool in physics, or are there fudge factors? One more thing...does the Poincaré conjecture factor into these questions in any way? Thank you.

 

[mfb]

In theory, can a circle be so large that connecting 3 points on that circle result in a degenerate triangle?

No, it can only get very close.

The Planck length has no relevance in mathematics, and even in physics it is not the "smallest step size" or anything like that.

 

[YoungPhysicist]

If the length of a straight line drawn between two points on a circle is Planck distance, how can another point fall between those 2 points?

Points in math takes up no space. It is just a concept.