What this is about

This is a visual ("proof without words") argument for the Pythagorean theorem using a tiling idea that you can imagine printed repeatedly across a medieval textile. It shows that the area made from the two smaller squares (a^2 and b^2) equals the area of the big square (c^2), by rearranging congruent pieces.

Materials and the plan

No fancy tools needed: pencil, ruler, and colored pencils (optional). We will draw one tile made of right triangles and small squares, arrange pieces inside two different big squares, and compare areas. The core idea: same set of pieces fills both shapes, so the areas are equal.

Step-by-step drawing (one tile / one arrangement)

1. Start with a right triangle with legs of lengths a and b and hypotenuse c. Draw one such triangle. (Example: a=3, b=4, c=5 is a nice integer example.)
2. Make four copies of that triangle. Each triangle has area (1/2)ab, so four of them total area 2ab.
3. Now draw a square of side length (a+b). Inside it, place the four right triangles so their right angles sit at the four corners of the big (a+b)-by-(a+b) square and their hypotenuses form a tilted central square. The triangles fit exactly; there will be a central empty square whose side length is c (the hypotenuse). This is the classic arrangement used in many visual Pythagorean proofs.
   • The big square's area is (a+b)^2.
   • That equals the area of the 4 triangles plus the area of the central (c-by-c) square, so (a+b)^2 = 4*(1/2 ab) + c^2 = 2ab + c^2.
   • Rearrange: c^2 = (a+b)^2 - 2ab = a^2 + 2ab + b^2 - 2ab = a^2 + b^2, proving the theorem.
4. Roger Nelson's 'Proof Without Words II' uses a very similar tiling/arrangement idea but emphasizes a tile that is repeated (tiled) over the plane. Imagine making one tile that contains the four triangles plus small complementary squares, and then repeating it across fabric: the way the pieces pack demonstrates the same area counting across the pattern.
   • One useful way to picture his version: split the (a+b)-square into a-by-a and b-by-b corner squares plus rectangles built from the triangles. When you rearrange those corner squares and triangles, they can be reassembled into a single c-by-c square. The tile's pieces are preserved, so repeating the tile makes the textile show many local instances of the same identity.

How to draw Nelson's tile concretely (practical)

1. Draw a square of side a+b. Mark a point on the left side a units up from the bottom, and on the bottom side a units right from the left corner. Connect those marks to create a right triangle of legs a and b. Do the symmetric marks around the square so you get the four right triangles placed as described above.
2. Color the four triangles one color and the central rotated square another. Now draw the same tile shifted so the triangles match edges—this creates a lattice like a medieval repeated motif. The fact that the triangles and small squares match up in the tiling is what gives the impression of a woven pattern where each motif contains the area-equivalence.

Why this is convincing (short explanation)

Both arrangements (the big (a+b)-square with four triangles inside, and the decomposition into an a-by-a square and a b-by-b square plus triangles) use the same exact pieces: four identical right triangles plus one square piece. If those pieces fill a big square of side (a+b) and also can be arranged to show a square of side c plus the triangles, their areas must match. Algebraic bookkeeping of areas gives c^2 = a^2 + b^2.

Connection to medieval textiles

Medieval textiles and floor tilings often use repeating motifs built from simple shapes (triangles, squares, rhombi). If you turn the tile from the proof into a repeating pattern, each repeat visually encodes the same area relationships. The repeated pattern reinforces the idea: local moves and rearrangements (like how a craftsman might cut and sew pieces of cloth) preserve area, so global area identities (like a^2 + b^2 = c^2) hold across the whole fabric. Historically, craftspeople worked with geometry intuitively when designing repeating patterns, so a tiling proof nicely connects algebraic truths to patterns you could weave or embroider.

Try it yourself (exercise)

1. Use a=3 and b=4. Draw a 7-by-7 square and place four 3-4-5 triangles at the corners. Verify the central square has side 5 by measuring or using the Pythagorean relation.
2. Cut out paper triangles and assemble: make two shapes—one that visibly shows a 5-by-5 central square inside a 7-by-7 square, and another arrangement that isolates a 3-by-3 and a 4-by-4 square. You will see the same pieces fill both, so 9 + 16 = 25.

Short summary

Roger Nelson's 'Proof Without Words II' is a tiling-style visual proof: the same set of triangles and square pieces can be arranged to show a big square of side c or two smaller squares of sides a and b. Repeating that tile across a textile makes a medieval-looking pattern and emphasizes that area-preserving rearrangements prove the Pythagorean theorem: a^2 + b^2 = c^2.

If you'd like, I can give you a step-by-step printable diagram to cut out, or show coordinates so you can draw the tile perfectly with a ruler and compass.