File:2 conceptions of square root of 5 through tilings.svg
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Description2 conceptions of square root of 5 through tilings.svg |
English:
All triangular elements of tilings are congruent to the initial right triangle, placed in the top left‑hand corner. Its perpendicular sides measure 1 and 2. the length of its hypotenuse. On the right, the wallpaper as background is a tiling by two kinds of squares. Their dimensions are the same lengths 1 and 2. Such a periodic tiling, called “Pythagorean tiling”, is a possible base to prove the Pythagorean theorem by comparing areas. On the right, a pattern of the background could be endlessly repeated. At the bottom, one of them is filled by five elements of tilings: four right triangles around a square. Each side of this tiling and each of its four angles is the sum of the two acute angles of the initial triangle, as marked at one vertex. 90 degrees is the sum of two acute angles of any right triangle, so we have a rhombus‑shaped pattern that has a right angle. It is a square, and the area of any minimal pattern is squared. See a few patterns on another image. This tiling of five elements has an area of 5, because each element has an area equal to the area unit: The sum of two angles equal to 90 degrees appears also at the lower vertex of the triangular tiling, in the bottom left‑hand corner. Thus each angle of this large triangle equals the corresponding angle of the initial triangle. The two triangles are similar, and the side lengths of one of them are proportional to the side lengths of the other one. Since the product of extremes equals the product of means in the following proportion:
So the triangular tiling is times as large as the initial triangle, the hypotenuse of which measures Français : Tous les éléments de pavage triangulaires sont isométriques au triangle rectangle initial, placé dans le coin en haut à gauche. Ses côtés perpendiculaires mesurent 1 et 2. la longueur de son hypoténuse. À droite, le motif périodique en arrière‑plan est un pavage par deux sortes de carrés. Leurs dimensions sont les mêmes longueurs 1 et 2. Un tel pavage périodique, appelé “pavage de Pythagore”, est une base possible de preuve du théorème de Pythagore, par comparaison d’aires. À droite, un motif répétitif de l’arrière‑plan pourrait se répéter indéfiniment. En bas, l’un d’eux est rempli de cinq éléments de pavage : quatre triangles rectangles autour d’un carré. Chaque côté du pavage et chacun de ses quatre angles est la somme de deux angles aigus du triangle initial, comme indiqué à un sommet. 90 degrés est la somme de deux angles aigus de n’importe quel triangle rectangle, de sorte que notre motif répétitif en forme de losange possède un angle droit. C’est un carré, et l’aire d’un motif répétitif minimal est le carré de Voir quelques motifs répétitifs sur une autre image. L’aire du pavage de cinq éléments est 5, car chaque élément du pavage a une aire égale à l’unité : La somme de deux angles égale à 90 degrés apparait aussi au sommet inférieur du pavage triangulaire, dans le coin en bas à gauche. Chaque angle de ce grand triangle est donc égal à l’angle correspondant du triangle initial. Les deux triangles sont semblables, et les longueurs des côtés de l’un sont proportionnelles aux longueurs des côtés de l’autre. Puisque le produit des extrêmes est égal au produit des moyens dans l’égalité suivante :
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Source | Own work |
Author | Arthur Baelde |
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SVG development InfoField | This /Baelde was created with a text editor. |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 12:16, 13 May 2022 | 1,056 × 594 (6 KB) | Arthur Baelde (talk | contribs) | Fewer double arrows and other improvements | |
09:23, 4 January 2022 | 1,056 × 594 (7 KB) | Arthur Baelde (talk | contribs) | Uploaded own work with UploadWizard |
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Short title | 2 conceptions of √5 through tilings |
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Width | 1056 |
Height | 594 |