File:2 conceptions of square root of 3 through tilings.svg
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Summary
[edit]Description2 conceptions of square root of 3 through tilings.svg |
English:
A tiling can cover an Euclidean plane entirely with an inifinite number of elements, which are either equilateral triangles edge‑to‑edge, each halved by an height, or squares of two different sizes, of which the ratio of dimensions equals the length ratio of an height of equilateral triangle to half its side. One or the other infinite tiling grounds a calculation of heights of a triangle with three sides of length 2, either via a proportion from the first tiling with congruent right triangles, or via a Pythagorean relation from the second tiling, called “Pythagorean tiling”. Conclusion of the two reasonings: the positive quantity the square of which is 3, more briefly it is the square root of 3: Our eyes capture forms and ideas in a tiling, an hexagon may appear that is regular, or an enlargement that multiplies lengths by two and areas by four, or lengths by three and areas by nine. Once the idea was clarified, the following question crossed certain minds: when the coefficient of areas is not a perfect square, what is the magnification scale exactly? The challenge of absolute rationality will lead over the centuries to the invention of so‑called “real” numbers, the set of which represented by ℝ contains irrational numbers Français : Un pavage peut couvrir un plan Euclidien en entier par une infinité d’éléments, qui sont soit des triangles équilatéraux bord à bord, chacun partagé en deux par une hauteur, soit des carrés de deux tailles différentes, dont le rapport des dimensions est celui d’une hauteur de triangle équilatéral à une moitié de son côté. L’un ou l’autre pavage infini est à l’origine du calcul de des hauteurs d’un triangle ayant trois côtés de longueur 2, soit via une proportion à partir du premier pavage par des triangles rectangles isométriques, soit via une relation de Pythagore à partir du second pavage, dit “de Pythagore”. Conclusion des deux raisonnements : la grandeur positive dont le carré est 3, plus brièvement c’est la racine carrée de 3 : Nos yeux captent des formes et des idées dans un pavage, un hexagone peut nous apparaître qui est régulier, ou un agrandissement qui multiplie les longueurs par deux et les aires par quatre, ou les longueurs par trois et les aires par neuf. Une fois l’idée éclaircie, la question suivante s’imposa à certains esprits : quand le coefficient de multiplication des aires n’est pas un carré parfait, que vaut exactement l’échelle d’agrandissement ? Le défi d’une rigueur absolue aboutira au fil des siècles à l’invention des nombres dits “réels”, dont l’ensemble désigné par ℝ contient des nombres irrationnels . Par exemple l’assertion “ est un nombre réel”, témoigne d’une réelle imagination collective pour qui ne manque pas d’air, ou n’est pas trop menacé par une montée des eaux, un contrôle eau frontière, etc. |
Date | |
Source | Own work |
Author | Arthur Baelde |
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 | 16:01, 24 February 2022 | 1,056 × 594 (9 KB) | Arthur Baelde (talk | contribs) | everywhere the same font‑family for a letter "a" that represents an area of 4 | |
09:19, 21 February 2022 | 1,056 × 594 (9 KB) | Arthur Baelde (talk | contribs) | to improve the repetitive pattern of dashed lines that delimit two adjacent squares of different sizes | ||
14:37, 20 February 2022 | 1,312 × 738 (9 KB) | Arthur Baelde (talk | contribs) | Uploaded own work with UploadWizard |
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