|
Kisebb lapszögű szabályos testekkel együtt határolhat félig szabályos testeket. Ebben is hasonlít a hatszöghöz, ami szintén lehet félig szabályos test lapja. Végül határolhat négy dimenziós konkáv szabályos testeket.
<!--[[Kép:snub tetrahedron.png|bélyegkép|Ikozaéder as a snub tetraéder]]
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are [[invariant (mathematics)|invariant]] under the same [[rotation]]s as the [[tetrahedron]], and are somewhat analogous to the [[snub cube]] and [[snub dodecahedron]], including some forms which are [[chirality (mathematics)|chiral]] and some with T<sub>h</sub>-symmetry, i. e. have different planes of symmetry from the tetrahedron. The icosahedron has a large number of [[stellation]]s, including one of the [[Kepler-Poinsot polyhedra]] and some of the regular compounds, which could be discussed here.
The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°. Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive [[defect (geometry)|defect]] for folding in three dimensions, icosahedra cannot be used as the [[cell (geometry)|cells]] of a convex regular [[polychoron]] because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex [[polytope]] in ''n'' dimensions, at least three [[facet (mathematics)|facets]] must meet at a [[peak (geometry)|peak]] and leave a positive defect for folding in ''n''-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the [[snub 24-cell]]), just as hexagons can be used as faces in semi-regular polyhedra (for example the [[truncated icosahedron]]). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral [[120-cell]], one of the ten non-convex regular polychora.
An icosahedron can also be called a [[Gyroelongated dipyramid|gyroelongated pentagonal bipyramid]]. It can be decomposed into a [[gyroelongated pentagonal pyramid]] and a [[pentagonal pyramid]] or into a [[pentagonal antiprism]] and two equal [[pentagonal pyramid]]s.
The icosahedron can also be called a snub tetrahedron, as [[snub (geometry)|snubification]] of a regular tetrahedron gives a regular icosahedron. Alternatively, using the nomenclature for snub polyhedra that refers to a snub cube as a snub cuboctahedron (cuboctahedron = [[Rectification (geometry)|rectified]] cube) and a snub dodecahedron as a snub icosidodecahedron (icosidodecahedron = rectified dodecahedron), one may call the icosahedron the snub octahedron (octahedron = rectified tetrahedron).-->
=== Ikozaéder kontra dodekaéder ===
* Az ikozaéder szimmetria[[csoport]]ja izomorf az öt elemű halmazok páros permutációinak csoportjával. Ez a nem [[kommutatív]] csoport egyszerű, és az egyetlen nem triviális normálosztó az öt elemű halmaz permutációinak csoportjában (azaz az ötödfokú szimmetrikus csoportban). Mivel az általános ötödfokú egyenlet [[Galois-csoport]]ja izomorf az ötödfokú szimmetrikus csoportjával, azért az általános ötödfokú egyenlet nem oldható meg gyökjelekkel. Az Abel-Ruffini tétel bizonyítása ezt az egyszerű tényt használja, és Felix Klein könyvet írt, amiben az ikozaéder szimmetriáinak elméletét használta, hogy analitikus megoldást adjon az ötödfokú egyenletre.
<!--
The [[dice|die]] inside of a [[Magic 8-Ball]] that has printed on it 20 answers to yes-no questions is a regular icosahedron.
The icosahedron displayed in a functional form is seen in the [[Sol de la Flor]] light shade. The rosette formed by the overlapping pieces show a resemblance to the [[Frangipani]] flower.
-->
<!--==Lásd még==
* {{SZK IW}}[[:Kép:Icosahedron.gif|Spinning icosahedron]]
* [[Truncated icosahedron]]
* [[Icosahedral–hexagonal grids in weather prediction]]
== Jegyzetek ==
See [[Wikipedia:Footnotes]] for instructions.
{{források}}
-->
== Forrás ==
| 