Lagrange's Four-Square Theorem Seen Using Polygons and Lines
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.[more]
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.[less]
"Lagrange's Four-Square Theorem Seen Using Polygons and Lines"
Wolfram Demonstrations Project
Published: July 11 2013