File:Academ scale ratio square root of 5.svg
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English: A tiling fills the pentagon ABCST with twelve isosceles triangles of two different kinds. Both of them fills the triangle ABC, one of each kind: BKA and AKC. The ten others of the tiling fills STA, five of each kind. Five of one kind are congruent to AKC, they have two angles equal to 36°. The five others are congruent to BKA, they have two angles equal to 72°. Thus, the area of STA is five times that of ABC. A similarity of which the the scale ratio is √5 and angle 180° transforms ABC into STA, transforming two sides of ABC drawn in red into two sides of STA drawn in green.
The ratios of lengths AC/AK and AK/BK are equal to the golden ratio φ. The scale ratio √5 is: SA/AC = ST/AB = (φ + 2)/φ. Therefore, φ = (1 + √5)/2. Français : Douze triangles isocèles de deux sortes différentes constituent un pavage du pentagone ABCST. Deux d’entre eux pavent le triangle ABC, un de chaque sorte : BKA et AKC. Les dix autres du pavage remplissent STA, cinq de chaque sorte. Cinq d’une sorte sont isométriques (superposables) à AKC, ils ont deux angles égaux à 36°. Les cinq autres sont isométriques à BKA, ils ont deux angles égaux à 72°. L’aire de STA est donc cinq fois celle de ABC. Une similitude de rapport √5 et d’angle 180° transforme ABC en STA. Elle transforme deux côtés de ABC tracés en rouge en deux côtés de STA tracés en vert. Les rapports de longueurs AC/AK et AK/BK sont égaux au nombre d’or φ. L’échelle √5 est : SA/AC = ST/AB = (φ + 2)/φ. Donc, φ = (1 + √5)/2. |
| Date | |
| Source | Own work |
| Author | Yves Baelde |
| Other versions | Published in France in “Souci d’exactitude” (Yves Baelde), Bulletin de l’APMEP n° 401, december 1995 (p.876). |
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
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| current | 10:01, 23 May 2010 | | 500 × 400 (3 KB) | Baelde (talk | contribs) | {{Information |Description={{en|1=A tiling fills the pentagon ABCST with twelve isosceles triangles of two different kinds. Both of them fills the triangle ABC, one of each kind: BKA and AKC. The ten others of the tiling fills STA, five of each kind. Five |
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