Shearing a Polygon into a Triangle of Equal Area
What Is This About?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Shearing a Polygon into a Triangle of Equal Area

The applet suggests a construction [Eves] of a triangle with the area equal to that of a given convex polygon. According to the Wallace-Bolyai-Gerwien Theorem, any two polygons of equal area are equidecomposable: it is possible to cut one into polygonal pieces that can be rearranged to form the other polygon. Thus we already know how to solve the quadrature problem for (or squaring of) a convex polygon. The applet shows a different, yet no less constructible, way of achieving the same goal.

In any polygon, a triple of successive vertices, say, a, b, c, form a ear if Δabc lies inside the polygon. In a convex polygon, any three successive vertices constitute an ear. Let's consider three such vertices. Two of the three vertices (a and c) are joined by a diagonal of the polygon and the ear is completed by two of its sides meeting at the third vertex (b). Note that for any location of b on the line parallel to ac, the ear has the same area. For, all such triangles have the same base (ac) and the same altitude (the fixed distance between the line and ac.) Going from one such triangle to another is induced by a shearing transformation with the base line fixed. (Observe in passing that every simple polygon with at least four vertices has at least two ears.)

For two positions of b, b will lie on the extensions of the sides of the polygon incident to a and c. By choosing any of the two and discarding one of the vertices (a or c) we obtain a convex polygon of the same area as the given one, but with fewer vertices. The process may continue until only three vertices remain. No further reduction in the number of vertices becomes possible.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?
 

References

  1. H. Eves, An Introduction to the History of Mathematics, Brooks Cole; 6 edition (January 2, 1990)

Quadrature: A Child's Play

What Is Shear Transform?

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Copyright © 1996-2017 Alexander Bogomolny

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