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Encyclopedia > Truncated icosahedron
Truncated icosahedron
Truncated icosahedron
(Click here for rotating model)
Type Archimedean solid
Elements F=32, E=90, V=60 (χ=2)
Faces by sides 12{5}+20{6}
Schläfli symbol t{3,5}
Wythoff symbol 2 5 | 3
Coxeter-Dynkin
Symmetry Ih
References U25, C27, W9
Properties Semiregular convex

Colored faces
Truncated icosahedron
5.6.6
(Vertex figure)

Pentakis dodecahedron
(dual polyhedron)

Net

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. Download high resolution version (878x874, 80 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated icosahedron, made by me using POV-Ray, see :image:poly. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... H.S.M. Coxeter. ... This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ... Image File history File links Download high-resolution version (1000x1000, 238 KB)Image for Truncated icosahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Truncated_icosahedron_vertfig. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Download high resolution version (844x843, 106 KB)Pentakis dodecahedron, made by me using POV-Ray, see image:poly. ... A pentakis dodecahedron is a catalan solid which looks a bit like an overinflated dodecahedron. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... Image File history File links Download high resolution version (437x635, 52 KB)Flattened polyhedron, source is image:makepoly. ... Categories: Polyhedra | Stub ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... Look up pentagon in Wiktionary, the free dictionary. ... A regular hexagon. ...

Contents

Construction

This polyhedron can be constructed from an icosahedron with the 12 vertices truncated (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges. An icosahedron noun (plural: -drons, -dra ) is a polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... A truncated cube - faces double in sides, and vertices replaced by new faces. ...


Icosahedron

Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Canonical coordinates

Canonical coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3φ), (±1,±3φ,0), (±3φ,0,±1) and the orthogonal cuboids (±2,±(1+2φ),±φ), (±(1+2φ),±φ,±2), (±φ,±2,±(1+2φ)) along with the orthogonal cuboids (±1,±(2+φ),±2φ), (±(2+φ),±2φ,±1), (±2φ,±1,±(2+φ)), where φ = (1+√5)/2 is the golden mean. Using φ2 = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9φ + 10. The edges have length 2. In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In anatomy, the cuboid bone is a bone in the foot. ... // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ...


Geometric relations

The truncated icosahedron easily verifies the Euler characteristic: It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

32 + 60 − 90 = 2.

With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons). A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ...


Applications

Compared to a football (soccer ball).

A football (soccer) ball is an example of a truncated icosahedron found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but is more spherical due to the pressure of the air inside and the elasticity of the ball. Image File history File links Trunc-icosa. ... Image File history File links Trunc-icosa. ... A football is a ball used to play one of the different sports known as football. ... Football (soccer) - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A football is a ball used to play one of the different sports known as football. ...


This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs (Richard Rhodes. Dark Sun: The Making of the Hydrogen Bomb, ISBN 0-684-82414-0. Touchstone Books, 1996., p. 195). The gadget, partially assembled on the shot tower for the Trinity test. ... A post-war Fat Man model. ... The mushroom cloud of the atomic bombing of Nagasaki, Japan, 1945, rose some 18 km (11 mi) above the epicenter. ...


The truncated icosahedron can also be described as a model of the Buckminsterfullerene (C60) molecule. The diameter of the football and the Buckminsterfullerene molecule are 22 cm and ca. 1 nm, respectively, hence the size ratio is 200,000,000 : 1. The fullerenes, discovered in 1985 by researchers at Rice University, are a family of carbon allotropes named after Richard Buckminster Fuller and are sometimes called buckyballs. ... A nanometre (American spelling: nanometer, symbol nm) is a unit of length in the metric system, equal to one thousand-millionth of a metre, which is the current SI base unit of length. ...


Truncated icosahedra in the arts

A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli illustrating The Divine Proportion. Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ...


See also

Look up Truncated icosahedron in
Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (from wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Spinning truncated icosahedron, made by me using POV-Ray, see :image:poly. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... An icosahedron noun (plural: -drons, -dra ) is a polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... The truncated dodecahedron is an Archimedean solid. ... The truncated rhombic triacontahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the twelve vertices where five faces meet at their acute corners. ... The first hyperbolic soccerball model The hyperbolic soccerball is a tiling of a surface frequently used as a manipulative for studying the properties of hyperbolic geometry. ... The fullerenes, discovered in 1985 by researchers at Rice University, are a family of carbon allotropes named after Richard Buckminster Fuller and are sometimes called buckyballs. ...

References

  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)

This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

External links


 
 

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