English: It is possible to associate such tilings with some proofs of the Pythagorean theorem, 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. How many methodical arrangements of colours for all tiles, it is a mathematical problem.
Français : Il est possible d’associer de tels pavages à certaines preuves du théorème de Pythagore, 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. Combien de dispostions méthodiques de couleurs pour tous les carreaux, voilà un problème mathématique.
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.
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{{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 [[w:Square root of 3|is square...