File:A tiling in order to prove the Pythagorean theorem.svg

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English:
Such periodic tilings can be a mnemonic to depict or prove the Pythagorean theorem with jigsaw puzzles, as shown below.

This classical tiling is created from a given right triangle, an Euclidean plane is entirely covered with an infinity of squares, the sizes of which are the leg lengths of the given triangle. On this drawing, every square element of the tiling has a slope equal to the ratio of lengths:  a / b  =  tan θ.   Thus a square pattern in dashed red is indefinitely repeated horizontally and vertically:  see   <pattern id="pg"  in the source code.  On this image, the square elements of the tiling have a ratio of sizes equal to square root of three or its multiplicative inverse, mathematically written:
1 / √3  =  √3 / 3  =  tan 30°.

Assume that the given triangle is not isosceles, so every pattern in dashed red is divided into five polygonal surfaces: three similar triangles and two quadrilaterals. If any piece of a jigsaw puzzle is congruent to one of these five polygons, the total area of five pieces of the five kinds is constant, whatever their assemblage. This total area is the area of a square pattern in dashed red: the squared hypotenuse length, or the area of another assemblage, forming two square elements of the tiling:  a 2 + b 2.  Hence this identity:  a 2 + b 2 = c 2.  The visual demonstration of the Pythagorean theorem will be complete with a second image of such a tiling, and a third image about the particular case of an isosceles right triangle, where a pattern in dashed red is divided into four half elements of the tiling.
 

Français :
De tels pavages périodiques peuvent être un moyen mnémotechnique pour illustrer ou prouver le théorème de Pythagore avec des puzzles, comme indiqué ci-dessous.

Ce pavage classique est créé à partir d’un triangle rectangle donné, un plan Euclidien est entièrement couvert par une infinité de carrés, dont les tailles sont les longueurs des côtés de l’angle droit du triangle donné. Dans ce dessin, chaque élément carré du pavage a une pente égale au rapport des longueurs :  a / b  =  tan θ.   Ainsi un motif carré en rouge pointillé est indéfiniment répété horizontalement et verticalement :  voir   <pattern id="pg"  dans le code source.  Dans cette image, les éléments carrés du pavage ont un rapport de tailles égal à racine carrée de trois ou son inverse, écrit mathématiquement:
1 / √3  =  √3 / 3  =  tan 30°.

Supposons que le triangle donné ne soit pas isocèle, alors chaque motif en rouge pointillé est partagé en cinq surfaces polygonales : trois triangles semblables et deux quadrilatères. Si n’importe quelle pièce d’un puzzle est superposable à l’un de ces cinq polygones, l’aire totale de cinq pièces des cinq sortes est constante, quel que soit leur assemblage. Cette aire totale est l’aire d’un motif carré en rouge pointillé : la longueur de l’hypoténuse au carré, ou l’aire d’un autre assemblage, formant deux éléments carrés du pavage :  a 2 + b 2.  D’où cette identité:  a 2 + b 2 = c 2.  La démonstration visuelle du théorème de Pythagore sera complète avec une deuxième image d’un tel pavage, et une troisième image sur le cas particulier d’un triangle rectangle isocèle, où un motif en pointillé rouge est partagé en quatre moitiés d’éléments du pavage.
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Author Baelde
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 Pythagorean theorem 

   A right triangle is given, from which a periodic tiling is created, from which puzzle pieces are constructed.

On three previous images, the hypotenuses of copies of the given triangle are in dashed red.  On left, a periodic square in dashed red takes another position relative to the tiling:  its center is the one of a small tile.  And one of the puzzle pieces is square, its size is the one of a small tile.  The four other puzzle pieces have stripes. They can form together a large tile, and they are congruent, because of a rotation a quarter turn around the center of any tile that leaves unchanged the tiling and the grid in dashed red.  Therefore the area of a large tile equals four times the area of a striped piece.  In case where the initial triangle is isosceles, the midpoint of any segment in dashed red is a common vertex of four tiles with equal sizes:  a =  b,  and each striped piece is still a quarter of a tile, it is an isosceles triangle.  Whatever the shape of the initial triangle, the two assemblages of the five puzzle pieces have equal areas:
 a 2 + b 2  =  c 2.    Hence  the  Pythagorean  theorem.

 Periodic tilings by squares 

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