Merging the Two Sets of Integer Sided Triangles

  One can reconcile the two sets of triangles by allowing λ to have rational values which are half integers in this case. We can still index the points with integers by using μ=2λ.

Both sets of points will then map onto the same curve.

We can also map all these points onto a unit circle by letting x=a/c and y=b/c which are rational numbers p/q with p and q integers.

The series of points with rational coordinates on the unit circle are not unique. Starting with point on the circle with rational coordinates near the horizontal axis we can use the trigonometric relations for the sum of two angles to step along the circle and get another point with rational coordinates since the rational numbers are closed under the arithmetic operations of addition, subtraction, multiplication and division (excluding zero).

For Pythagoras the proportions of the sides of the triangles and the calculation of lengths and areas would have been his main concern. In ancient Egypt angles were determined by right triangles and was expressed as a seked. The best the Pythagoreans appear to have done in terms of angles were the standard angles of an arc, 30°, 45°, 60° and 90°, and their coordinates.