File:AcademViews PlatonicDodecahedron RegularDecagon.svg
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[edit]DescriptionAcademViews PlatonicDodecahedron RegularDecagon.svg |
English: Elevation view and top view of a platonic dodecahedron, with notations and equalities. The faces of the solid have the same colors in this image and this other one. Thanks to the colors of the faces we recognize the points of the figure, and we can still denote points by letters, despite the absence of letters. φ is the golden ratio. The dodecahedron is centrally symmetric, its center is Ω. Its edge length is a, and (φ a ) or d is the diagonal length of its faces. (φ d ) or (d + a ) is the distance between two opposite edges.
Two opposite faces are horizontal. The intersection of the plane of the bottom face with the circumscribed sphere of the solid is the circumscribed circle of this face, dashed depicted in green in the top view. The common axis of the horizontal faces is depicted like Ω when seen from above, and the outline of the solid is a convex regular decagon. When the solid rotates one fifth of a turn around this vertical axis, it returns to its initial orientation whatever the sense of rotation. Infinitely many horizontal cross sections are convex regular pentagons, the ten vertices of the two largest are vertices of the solid. Any horizontal cross section that is not pentagonal is a convex decagon, which has the vertices of two congruent regular pentagons with the same center. Such a decagon is regular if and only if the plane of the cross section passes through Ω, the center of the solid. Ω is the center of this regular decagon with white sides, of which the vertices are the midpoints of ten slanted edges of the dodecahedron. Français : Vue en élévation et vue de dessus d’un dodécaèdre de Platon, avec des notations et des égalités. Les faces du solide ont les mêmes couleurs dans cette image et dans cette autre. Grâce aux couleurs des faces on reconnaît les points de la figure, et on peut quand même désigner des points par des lettres, malgré l’absence de lettres. φ est le nombre d’or. Le centre Ω du dodécaèdre est son centre de symétrie. La longueur de ses arêtes est a, et (φ a ) ou d est la longueur d’une diagonale d’une face. (φ d ) ou (d + a ) est la distance entre deux arêtes opposées. Deux faces opposées sont horizontales. L’intersection du plan de la face inférieure avec la sphère circonscrite au solide est le cercle circonscrit à cette face, représenté en pointillé vert dans la vue de dessus. L’axe commun des faces horizontales est représenté comme Ω en vue de dessus, et le contour du solide est un décagone régulier convexe. Quand le solide tourne d’un cinquième de tour autour de cet axe vertical, il revient à son orientation initiale quel que soit le sens de rotation. Un nombre infini de sections horizontales sont des pentagones réguliers convexes, les dix sommets des deux plus grands sont des sommets du solide. Toute section horizontale qui n’est pas pentagonale est un décagone convexe, qui a les sommets de deux pentagones réguliers isométriques et concentriques. Un tel décagone est régulier si et seulement le plan de section passe par Ω, le centre du solide. Ω est le centre de ce décagone régulier aux côtés blancs, dont les sommets sont les milieux de dix arêtes obliques du dodécaèdre. |
Date | |
Source | Own work |
Author | Yves Baelde |
SVG development InfoField | This /Baelde was created with a text editor. |
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[edit]Arthur Baelde, the copyright holder of this work, hereby publishes it under the following license:
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 01:17, 8 October 2010 | 720 × 960 (4 KB) | Baelde (talk | contribs) | {{Information |Description={{en|1=Elevation view and top view of a platonic dodecahedron, with notations and equalities. The faces of the solid have the same colors in this image and |
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