Tetrated dodecahedron: Difference between revisions
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{{short description|Near-miss Johnson solid with 28 faces}} |
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{{Infobox polyhedron |
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| image = Tetrated dodecahedron.svg |
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|align=center colspan=2|[[Image:Tetrated Dodecahedron.gif|240px|Tetrated dodecahedron]] |
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| euler = |
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| faces = 4 [[equilateral triangle]]s<br>12 [[isosceles triangle]]s<br>12 [[pentagon]]s |
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| edges = 54 |
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| vertices = 28 |
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|bgcolor=#e7dcc3|Faces||4+12 [[triangle]]s<br>12 [[pentagon]]s |
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| vertex_config = 4 ({{math|5.5.5}})<BR>12 ({{math|3.5.3.5}})<BR>12 ({{math|3.3.5.5}}) |
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|- |
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| schläfli = |
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|bgcolor=#e7dcc3|Edges||54 |
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| wythoff = |
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| conway = |
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|bgcolor=#e7dcc3|Vertices||28 |
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| coxeter = |
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| symmetry = {{mvar|T<sub>d</sub>}} |
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|bgcolor=#e7dcc3|[[Vertex configuration]]||4 (5.5.5)<BR>12 (3.5.3.5)<BR>12 (3.3.5.5) |
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| rotation_group = |
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|- |
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| surface_area = |
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|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||T<sub>d</sub> |
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| volume = |
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| angle = |
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|bgcolor=#e7dcc3|Properties||convex |
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| dual = |
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| properties = [[convex polytope|convex]] |
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| vertex_figure = |
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[[File:Tetrated dodecahedron.stl|thumb|3D model of a tetrated dodecahedron]] |
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[[File:Physical model of tetrated dodecahedron.png|thumb|Model built with [[polydron]]]] |
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| ⚫ | In [[geometry]], the '''tetrated dodecahedron''' is a [[near-miss Johnson solid]]. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003, and named, by Robert Austin.<ref>[http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/Tetrated%20Dodecahedra.html Tetrated dodecahedra]</ref> |
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It has 28 [[Face (geometry)|faces]]: twelve regular [[pentagon]]s arranged in four panels of three pentagons each, four [[equilateral triangle]]s (shown in blue), and six pairs of [[isosceles triangle]]s (shown in yellow). All edges of the tetrated [[dodecahedron]] have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This [[polyhedron]] has [[tetrahedral symmetry]]. |
It has 28 [[Face (geometry)|faces]]: twelve regular [[pentagon]]s arranged in four panels of three pentagons each, four [[equilateral triangle]]s (shown in blue), and six pairs of [[isosceles triangle]]s (shown in yellow). All edges of the tetrated [[dodecahedron]] have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This [[polyhedron]] has [[tetrahedral symmetry]]. |
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Topologically, as a near-miss Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always |
Topologically, as a near-miss Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always equilateral, while the pentagons and the other triangles only have reflection symmetry. |
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== Net == |
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The 12 pentagons and 16 triangles are colored in this net by their locations within the tetrahedral symmetry. |
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== Related polyhedra == |
== Related polyhedra == |
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Latest revision as of 16:08, 30 November 2023
| Tetrated dodecahedron | |
|---|---|
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| Type | Near-miss Johnson solid |
| Faces | 4 equilateral triangles 12 isosceles triangles 12 pentagons |
| Edges | 54 |
| Vertices | 28 |
| Vertex configuration | 4 (5.5.5) 12 (3.5.3.5) 12 (3.3.5.5) |
| Symmetry group | Td |
| Properties | convex |
| Net | |
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In geometry, the tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003, and named, by Robert Austin.[1]
It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.
Topologically, as a near-miss Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always equilateral, while the pentagons and the other triangles only have reflection symmetry.
Related polyhedra
[edit]| Dodecahedron (Platonic solid) |
Icosidodecahedron (Archimedean solid) |
Pentagonal orthobirotunda (Johnson solid) |
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