Degenerate Triangles: Questions & Poincaré Conjecture

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Do such questions cause problems using math as a descriptive tool in physics, or are there fudge factors?There are no fudge factors in math or physics. These questions may lead to further exploration and understanding. The Poincaré conjecture does not directly relate to these questions.
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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.
 

FAQ: Degenerate Triangles: Questions & Poincaré Conjecture

What is a degenerate triangle?

A degenerate triangle is a triangle that has one or more of its sides with a length of zero, making it essentially a line or a point. It can also refer to a triangle with all three vertices lying on the same line.

How do degenerate triangles relate to the Poincaré conjecture?

The Poincaré conjecture is a famous unsolved problem in mathematics that states that every simply connected, closed 3-dimensional manifold is topologically equivalent to a 3-dimensional sphere. It is closely related to degenerate triangles because these triangles play a crucial role in the proof of the conjecture.

What is the significance of the Poincaré conjecture?

The Poincaré conjecture has been called one of the most important and difficult problems in mathematics. Its proof would have a major impact in many areas of mathematics, including topology, geometry, and algebraic geometry. It is also important because it is one of the seven Millennium Prize Problems, with a prize of $1 million for its solution.

How have degenerate triangles been used in attempts to prove the Poincaré conjecture?

One of the key steps in the proof of the Poincaré conjecture is to show that any 3-manifold can be dissected into pieces that can be reassembled into a 3-sphere. This dissection involves breaking down the 3-manifold into smaller pieces, which often include degenerate triangles. By understanding the properties of these triangles, mathematicians have been able to make progress towards proving the conjecture.

Has the Poincaré conjecture been proven?

Yes, the Poincaré conjecture was proven by Russian mathematician Grigori Perelman in 2002. His proof was confirmed by other mathematicians in 2006 and was recognized as one of the greatest mathematical achievements of the 21st century. As a result, Perelman was awarded the Fields Medal in 2006 and the Millennium Prize in 2010.

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