Icosahedral symmetry: Difference between revisions
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{{Short description|Symmetry of a regular icosahedron}} | |||
[[File:Icosahedral_reflection_domains.png|Icosahedral symmetry reflection domains|thumb|right]] | |||
'''Icosahedral symmetry''' is a type of [[symmetry]] that is characteristic of the [[regular icosahedron]] and the [[regular dodecahedron]]. It is one of the [[Platonic solids]] and is denoted by the [[Schönflies notation]] \(I_h\), the [[Coxeter notation]] \(\{3,5\}\), and the [[orbifold notation]] \(\ast 532\). This symmetry group is of great interest in various fields, including [[geometry]], [[crystallography]], and [[virology]]. | |||
==Geometric Properties== | |||
Icosahedral symmetry is defined by the rotational symmetries of a regular icosahedron. The group of rotational symmetries is known as the [[icosahedral group]], which is a subgroup of the full icosahedral symmetry group that includes reflections. | |||
The icosahedral group has 60 rotational symmetries, which can be understood as the even permutations of the five diagonals of the icosahedron. These symmetries can be categorized into: | |||
* 1 identity element | |||
* 15 rotations through 72° or 288° about axes through opposite vertices | |||
* 20 rotations through 120° or 240° about axes through the midpoints of opposite edges | |||
* 24 rotations through 180° about axes through the centers of opposite faces | |||
[[File:Soccer_ball.svg|A soccer ball, an example of icosahedral symmetry in sports|thumb|left]] | |||
== | ==Applications== | ||
Icosahedral symmetry is not only a mathematical curiosity but also appears in nature and human-made objects. For example, many [[viruses]] have icosahedral symmetry, which allows them to form a stable and efficient structure with a minimal amount of genetic material. The [[capsid]] of these viruses is often composed of identical protein subunits arranged in an icosahedral pattern. | |||
In sports, the design of a [[soccer ball]] is based on a truncated icosahedron, which exhibits icosahedral symmetry. This design provides a near-spherical shape that is easy to manufacture and handle. | |||
== | ==Mathematical Representation== | ||
The mathematical representation of icosahedral symmetry involves group theory and [[Coxeter groups]]. The full icosahedral symmetry group, including reflections, is isomorphic to the symmetric group \(S_5\), which has 120 elements. The rotational subgroup is isomorphic to the alternating group \(A_5\), which has 60 elements. | |||
[[File:Sixteenth_stellation_of_icosahedron.png|The sixteenth stellation of the icosahedron, showing complex symmetry|thumb|right]] | |||
== | ==Stellations and Related Polyhedra== | ||
Icosahedral symmetry is also evident in the stellations of the icosahedron. A stellation is a process of extending the faces or edges of a polyhedron to form a new polyhedron. The icosahedron has 59 stellations, each exhibiting the same symmetry. | |||
The [[disdyakis triacontahedron]] is another polyhedron related to icosahedral symmetry. It is the dual of the [[truncated icosahedron]] and has 120 faces, each of which is a scalene triangle. | |||
[[ | [[File:Spherical_disdyakis_triacontahedron.png|Spherical disdyakis triacontahedron, illustrating complex icosahedral symmetry|thumb|left]] | ||
==Related Pages== | |||
* [[Platonic solid]] | |||
* [[Symmetry group]] | |||
* [[Coxeter group]] | |||
* [[Virus structure]] | |||
* [[Soccer ball]] | |||
File:Spherical_disdyakis_triacontahedron.png| | |||
[[Category:Symmetry]] | |||
[[Category:Polyhedra]] | |||
[[Category:Mathematics]] | |||
Latest revision as of 11:47, 23 March 2025
Symmetry of a regular icosahedron
Icosahedral symmetry is a type of symmetry that is characteristic of the regular icosahedron and the regular dodecahedron. It is one of the Platonic solids and is denoted by the Schönflies notation \(I_h\), the Coxeter notation \(\{3,5\}\), and the orbifold notation \(\ast 532\). This symmetry group is of great interest in various fields, including geometry, crystallography, and virology.
Geometric Properties[edit]
Icosahedral symmetry is defined by the rotational symmetries of a regular icosahedron. The group of rotational symmetries is known as the icosahedral group, which is a subgroup of the full icosahedral symmetry group that includes reflections.
The icosahedral group has 60 rotational symmetries, which can be understood as the even permutations of the five diagonals of the icosahedron. These symmetries can be categorized into:
- 1 identity element
- 15 rotations through 72° or 288° about axes through opposite vertices
- 20 rotations through 120° or 240° about axes through the midpoints of opposite edges
- 24 rotations through 180° about axes through the centers of opposite faces
Applications[edit]
Icosahedral symmetry is not only a mathematical curiosity but also appears in nature and human-made objects. For example, many viruses have icosahedral symmetry, which allows them to form a stable and efficient structure with a minimal amount of genetic material. The capsid of these viruses is often composed of identical protein subunits arranged in an icosahedral pattern.
In sports, the design of a soccer ball is based on a truncated icosahedron, which exhibits icosahedral symmetry. This design provides a near-spherical shape that is easy to manufacture and handle.
Mathematical Representation[edit]
The mathematical representation of icosahedral symmetry involves group theory and Coxeter groups. The full icosahedral symmetry group, including reflections, is isomorphic to the symmetric group \(S_5\), which has 120 elements. The rotational subgroup is isomorphic to the alternating group \(A_5\), which has 60 elements.
Stellations and Related Polyhedra[edit]
Icosahedral symmetry is also evident in the stellations of the icosahedron. A stellation is a process of extending the faces or edges of a polyhedron to form a new polyhedron. The icosahedron has 59 stellations, each exhibiting the same symmetry.
The disdyakis triacontahedron is another polyhedron related to icosahedral symmetry. It is the dual of the truncated icosahedron and has 120 faces, each of which is a scalene triangle.
