File:A Pythagorean tiling View 2.svg

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English: In order to prove the  Pythagorean theorem,  such a tessellation and its pattern grid can be associated to a set of puzzle pieces, 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  a  and  b:  the leg lengths of the given triangle.  On this drawing, every square element of the tiling, any tile has a slope equal to the ratio of sizes:  a / b  =  tan 30°.  Thus a square pattern is indefinitely repeated horizontally and vertically:  see   <pattern id="pg"  in the source code.

See another page for more informations.
 
Français : Afin de prouver le  théorème de Pythagore,  un tel pavage quadrillé par ses motifs répétés peut être associé à un jeu de pièces de puzzle, comme ci-dessous ou dans une autre page en français.

Ce pavage classique est créé à partir d’un triangle rectangle donné.  Un plan euclidien est entièrement couvert d’une infinité de carrés, dont les dimensions sont  a  et  b :  les longueurs des côtés de l’angle droit du triangle donné.  Dans ce dessin, tout élément carré du pavage, n’importe quel carreau a une pente égale au rapport des dimensions :  a / b  =  tan 30°.  Ainsi un motif carré est répété à l’infini horizontalement et verticalement :  voir   <pattern id="pg"  dans le code source.

Voir une autre page pour plus d’informations.
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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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current07:00, 14 October 2012Thumbnail for version as of 07:00, 14 October 2012600 × 600 (672 bytes)Baelde (talk | contribs){{Information |Description ={{en|1=Is evoked a tiling of an Euclidean plane by an infinity of squares of two sizes. Here the ratio of sizes is √{{overline|3}}...

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