Where Can I Find a Proof for the Four Vertex Theorem?

In summary, the Four Vertex Theorem is a mathematical theorem that states any simple closed curve can be divided into at least four distinct convex polygons. It was first discovered by Russian mathematician Georgy Voronoy in 1908 and later rediscovered by American mathematician George David Birkhoff in 1917. This theorem has significant implications in geometry, topology, computer graphics, and robotics. It has been proven and is considered a fundamental theorem in mathematics. However, there are exceptions to the theorem, such as shapes with holes or self-intersections, and it only applies to two-dimensional shapes in a plane.
  • #1
adityatatu
15
0
Can somebody please give me an online reference for a good proof of 'The Four Vertex Theorem'? :confused:
 
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  • #2
I haven't read it in detail yet, but this looks like what you're after:

http://ada.math.uga.edu/teaching/math4250/Html/Four-vertex.htm [Broken]
 
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  • #3


There are several resources available online for a proof of the Four Vertex Theorem. One possible source is the MathWorld website, which offers a detailed explanation and proof of the theorem using the concept of convex polygons. Another option is the Wolfram Demonstrations Project, which includes interactive demonstrations of the theorem and its proof. Additionally, many math forums and communities such as Math Stack Exchange and Quora have discussions and explanations of the theorem with references to various proofs. It is always recommended to consult multiple sources and verify the validity of the proof before accepting it.
 

What is the Four Vertex Theorem?

The Four Vertex Theorem is a mathematical theorem that states that any simple closed curve in a plane can be decomposed into at least four distinct convex polygons. This means that any shape with a smooth boundary can be divided into four or more smaller shapes with straight edges.

Who discovered the Four Vertex Theorem?

The Four Vertex Theorem was first discovered by Russian mathematician Georgy Voronoy in 1908. However, it was not widely recognized until it was independently rediscovered by American mathematician George David Birkhoff in 1917.

What is the significance of the Four Vertex Theorem?

The Four Vertex Theorem has significant implications in the fields of geometry and topology. It provides a way to decompose complex shapes into simpler ones, making it useful in computer graphics and image processing. It also has applications in robotics, as it can be used to determine the best path for a robot to navigate around obstacles.

Is the Four Vertex Theorem a proven theorem?

Yes, the Four Vertex Theorem has been proven and is considered a fundamental theorem in mathematics. It has been rigorously tested and verified by numerous mathematicians, and its validity has been established using various mathematical techniques.

Are there any exceptions to the Four Vertex Theorem?

Yes, there are a few exceptions to the Four Vertex Theorem. The theorem does not hold for shapes with holes or for shapes that have self-intersections. In these cases, the number of convex polygons needed to decompose the shape may be more than four. Additionally, the theorem only applies to two-dimensional shapes in a plane and does not hold for three-dimensional objects.

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