Lagrange's Four-Square Theorem Seen Using Polygons and Lines

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Any natural number can be represented as the sum of the squares of four non-negative integers. For most numbers there are multiple representations. In this Demonstration, the four integers (not squared) may be viewed using three different options.

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1. For each set of four numbers , a polygon is displayed with vertices , , , and .

2. Each set is sorted from smallest to largest. The first two numbers and the last two numbers are viewed as points so that the representation becomes a line segment. Where there are multiple representations, a polygon is formed by connecting each line segment to the next line segment in the list.

3. This is similar to option 2 except that the coordinates of the second point are reversed.

In all three options, each polygon is randomly colored to create a unique portrait for each natural number. Mouseover the polygons to see the sets of four non-negative integers.

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Contributed by: Noel Patson (July 2013)
Open content licensed under CC BY-NC-SA

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Noel Patson

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