The image shows two sets of puzzle pieces that depict the Pythagorean theorem.  Two surface areas are equal for any right triangle:  a 2 + b 2 = c 2,  it is the conclusion of the theorem in traditional notations.
If a given right triangle is not isosceles, we can construct five puzzle pieces like on the image left.  In this case,  where  a ≠ b,  a “Pythagorean tiling”  is a possible medium to construct the puzzle pieces:  a periodic tiling of the Euclidean plane by squares of two different sizes, where any square by any side abuts on exactly one square of another size.  The tiling is manifestly periodic when a pattern grid is overlaid on the tiling.  By shifting this grid with no change of orientation relative to the tiling, we get other sets of puzzle pieces for a same right triangle, as shown on other images.
On the image left, one of the puzzle pieces is a triangle equal to the given triangle, of which the right angle is symbolized by two little perpendicular segments.  The letter “a” is chosen to denote the smallest side length of this triangle.  Two squares are shaped by the puzzle pieces above the line in dashed blue.  Their total surface area is  a 2 + b 2.  Below this line in dashed blue, the same pieces form one square.  Its dimension equals the hypotenuse length of the initial triangle, denoted by c.  The five pieces have a constant area, whatever their assemblage.  Hence the conclusion of the theorem.

In case of isosceles right triangle, when  a = b,  all puzzle pieces are equal to the given triangle, and the smallest triangular piece disappears.  The hypotenuse length is here denoted by d,  that is the diagonal length of a square of size a.  And a square of dimension d  have two diagonals measuring  2 a.  Seeing that a diagonal of a square is larger than its sides, the lengths  (a,  d,  2 a )  are ordered by increasing order, as written on the top right of the image.  In other words, 1 and 2 are the integers approximating square root of two:  a number known since ancient times to be irrational.
Own work
Baelde