Degenerate Triangles: Questions & Poincaré Conjecture

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SUMMARY

The discussion centers on the concept of degenerate triangles in relation to circles and the Poincaré conjecture. It is established that a circle cannot be so large that connecting three points results in a degenerate triangle; they can only approach this condition. The Planck length is clarified as irrelevant in mathematics, and points in mathematical theory are conceptual and do not occupy physical space. The conversation raises questions about the implications of these concepts in physics and the use of mathematics as a descriptive tool.

PREREQUISITES
  • Understanding of degenerate triangles in geometry
  • Familiarity with the Poincaré conjecture in topology
  • Basic knowledge of the Planck length in physics
  • Conceptual grasp of mathematical points and their properties
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  • Explore the implications of the Poincaré conjecture in modern topology
  • Research the role of the Planck length in quantum physics
  • Study the properties and definitions of degenerate triangles in geometry
  • Investigate how mathematics serves as a descriptive tool in theoretical physics
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Jim Lundquist
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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.
 
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Jim Lundquist said:
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.
 
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Jim Lundquist said:
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.
 

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