# The Four-Vertex Theorem

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This shows the result of the four-vertex theorem: A simple closed curve has at least four vertices. You can transform the closed curve by dragging the locator. If the caustic extends beyond the window, you can reduce its size.

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Contributed by: Takaharu Tsukada (January 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Manfredo P. do Carmo, *Differential Geometry of Curves and Surfaces*, Prentice Hall, 1976.

Ian R. Porteous, *Geometric Differentiation: For the Intelligence of Curves and Surfaces*, Cambridge University Press, 1994.

## Permanent Citation

"The Four-Vertex Theorem"

http://demonstrations.wolfram.com/TheFourVertexTheorem/

Wolfram Demonstrations Project

Published: January 5 2012