Tiling Pythagorean Proof (Roger Nelsen’s “Proof Without Words II”) for a 15‑year‑old — plus Al‑Nayrizi (c. 900 AD) and a Desmos build

Goal: See and verify the Pythagorean theorem a² + b² = c² by a simple tiling (rearrangement) idea you can draw and animate in Desmos. We'll explain the visual proof (Roger Nelsen’s "proof without words II" style), note the similar historical proof attributed to Al‑Nayrizi (circa 900 AD), and give step‑by‑step Desmos instructions so you can explore with sliders.

1. The visual idea (tiling)

Start with a large square whose side length is a + b. Inside that square place four congruent right triangles, each with legs of length a and b and hypotenuse c, one in each corner. If you place the triangles with their legs along the sides of the big square, their hypotenuses meet and leave a small central square whose side length is c.

Look at the area of the big square in two ways:

(1) Area of the big square directly: (a + b)^2
(2) Area as sum of the four triangles + the little central square:
    4 * (1/2 * a * b) + c^2 = 2ab + c^2

Equate them:
(a + b)^2 = 2ab + c^2
a^2 + 2ab + b^2 = 2ab + c^2
=> a^2 + b^2 = c^2

That equality is exactly the Pythagorean theorem. This is the essence of a "proof without words": the picture plus the area decomposition gives the result.

2. Historical note — Al‑Nayrizi (circa 900 AD)

Al‑Nayrizi (sometimes spelled Annairizi in older sources) was an Islamic mathematician from around 900 AD who gave geometric demonstrations that are variants of dissection/rearrangement proofs. His style—breaking shapes into parts and comparing areas—is in the same family of arguments as the tiling proof above. Many medieval and ancient mathematicians produced geometric area arguments for the Pythagorean relation, and the rearrangement idea (placing congruent right triangles inside a larger square) is a natural and powerful way to see the theorem.

3. Build this in Desmos (step‑by‑step)

Open Desmos (www.desmos.com/calculator) and follow these steps. Use sliders for a and b so you can change their lengths and watch the picture update.

1. Create sliders: type a = 3 and b = 2 (Desmos will make sliders). Set sensible slider ranges (for example 0.5 to 6) so the picture stays visible.
2. Define the hypotenuse length (just for reference): c = sqrt(a^2 + b^2). Desmos will show the calculated c value.
3. Draw the big square (bottom‑left corner at the origin):

   polygon((0,0),(a+b,0),(a+b,a+b),(0,a+b))

   This draws the outside square of side a+b.
4. Draw the four right triangles as filled polygons so the leftover central shape becomes visually obvious:

   polygon((0,0),(a,0),(0,b))
   polygon((a+b,0),(a,0),(a+b,b))
   polygon((a+b,a+b),(a+b,b),(a,a+b))
   polygon((0,a+b),(0,b),(a,a+b))
       

   Each polygon() command creates one triangle. Color or style them in Desmos to make the picture clear (Desmos lets you change color and opacity).
5. Optional: draw the four hypotenuses explicitly (they appear as triangle edges). If you want to mark the central square, you can draw the lines connecting the two endpoints of each hypotenuse (they are already edges of triangles). The central square becomes visually clear as the region not covered by the triangles.
6. Label areas (use Desmos text tool or create numeric expressions):

   BigSquareArea = (a+b)^2
   TrianglesArea = 4*(1/2*a*b) = 2ab
   CenterArea = c^2

   Compare BigSquareArea and TrianglesArea + CenterArea numerically on the side panel to see they match for all slider values.
7. Play: move the sliders a and b. The figure rearranges automatically and you can see the central leftover region always has area c^2 even though a and b change.

4. Why this convinces you (quick recap)

• The picture is not a numerical trick — it’s geometry: the same big square is partitioned in two different ways (once as a single square, once as 4 triangles + central square).
• Because those two partitions are the same whole, their areas are equal. The algebra that follows from that area equality is exactly a^2 + b^2 = c^2.

5. Extra experiments

• Try integer a and b (e.g., a=3, b=4) and check that c becomes 5 (classic 3–4–5 right triangle).
• Animate a or b in Desmos (click the little play button on a slider) to see the shape change and to convince yourself the area identity holds always.
• Try to create a second tiling that arranges the four triangles to leave two squares (one of area a^2 and one of area b^2) — doing that shows another view of why a^2 + b^2 = c^2.

If you want, I can produce a ready‑to‑paste Desmos expression list or a share link (I can write exact polygon lines and label formulas) so you can copy/paste it and see the figure immediately. Tell me if you want the Desmos file text or a direct link.