View source for Great icosahedron

{{Short description|Kepler-Poinsot polyhedron with 20 faces}}
{{Reg polyhedra db|Reg star polyhedron stat table|gI}}
[[File:Great icosahedron.stl|thumb|3D model of a great icosahedron]]

In [[geometry]], the '''great icosahedron''' is one of four [[Kepler–Poinsot polyhedra]] ([[nonconvex]] [[List of regular polytopes#Non-convex 2|regular polyhedra]]), with [[Schläfli symbol]] {{math|{3,{{frac|5|2}}} }} and [[Coxeter-Dynkin diagram]] of {{CDD|node_1|3|node|5|rat|d2|node}}. It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a [[pentagram]]mic sequence.

The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the {{math|(''n''–1)}}-dimensional [[simplex]] faces of the core {{mvar|n}}-polytope (equilateral triangles for the great icosahedron, and [[line segment]]s for the pentagram) until the figure regains regular faces. The [[grand 600-cell]] can be seen as its four-dimensional analogue using the same process.

==Construction ==
The edge length of a great icosahedron is <math>\frac{7+3\sqrt{5}}{2}</math> times that of the original icosahedron.

== Images ==

{| class="wikitable" width=560
!Transparent model
![[density (polytope)|Density]]
![[Stellation diagram]]
![[Net (polyhedron)|Net]]
|- valign=top
|[[File:GreatIcosahedron.jpg|140px]]<BR>A transparent model of the great icosahedron (See also [[Media:GreatIcosahedron.gif|Animation]])
|[[File:Great icosahedron cutplane.png|140px]]<BR>It has a density of 7, as shown in this cross-section.
|[[File:Great icosahedron stellation facets.svg|140px]]<BR>It is a [[stellation]] of the icosahedron, counted by Wenninger as model [W41] and the [[List of Wenninger polyhedron models#Stellations of icosahedron|16th of 17 stellations of the icosahedron]] and 7th of 59 stellations by [[Donald Coxeter|Coxeter]].
|{{nowrap|[[File:Great icosahedron net.png|140px]] × 12}}<BR>Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines.
|}

{| class="wikitable" width=180
|+ [[Spherical tiling]]
|[[File:Great icosahedron tiling.svg|180px]]<BR>This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)
|}

{{-}}

== Formulas ==

For a great icosahedron with edge length E (the edge of its dodecahedron core),

<math display=block>\text{Inradius} = \frac{\text{E}(3\sqrt{3}-\sqrt{15})}{4}</math>

<math display=block>\text{Midradius} = \frac{\text{E}(\sqrt{5}-1)}{4}</math>

<math display=block>\text{Circumradius} = \frac{\text{E}\sqrt{2(5-\sqrt{5})}}{4}</math>

<math display=block>\text{Surface Area} = 3\sqrt{3}(5+4\sqrt{5})\text{E}^2</math>

<math display=block>\text{Volume} = {\tfrac{25+9\sqrt{5}}{4}}\text{E}^3</math>

== As a snub ==
The ''great icosahedron'' can be constructed as a uniform [[Snub (geometry)|snub]], with different colored faces and only [[tetrahedral symmetry]]: {{CDD|node_h|3x|rat|d2|node_h|3x|rat|d2|node_h}}. This construction can be called a ''retrosnub tetrahedron'' or ''retrosnub tetratetrahedron'',<ref>{{KlitzingPolytopes|../incmats/gike.htm|uniform polyhedra|Great icosahedron}}</ref> similar to the [[snub tetrahedron]] symmetry of the [[icosahedron]], as a partial faceting of the [[truncated octahedron]] (or ''[[omnitruncation|omnitruncated]] tetrahedron''): {{CDD|node_h|3|node_h|3|node_h}}. It can also be constructed with 2 colors of triangles and [[pyritohedral symmetry]] as, {{CDD|node_h|3x|rat|d2|node_h|4|node}} or {{CDD|node_h|3x|rat|d2|node_h|4|rat|3x|node}}, and is called a ''retrosnub octahedron''.

{| class=wikitable
!Tetrahedral
!Pyritohedral
|-
|[[File:Retrosnub tetrahedron.png|200px]]
|[[File:Pyritohedral great icosahedron.png|200px]]
|-
!{{CDD|node_h|3x|rat|d2|node_h|3x|rat|d2|node_h}}
!{{CDD|node_h|3x|rat|d2|node_h|4|node}}
|}

== Related polyhedra ==
[[File:Great stellated dodecahedron truncations.gif|thumb|right|240px|Animated truncation sequence from {5/2, 3} to {3, 5/2}]]
It shares the same [[vertex arrangement]] as the regular convex [[icosahedron]]. It also shares the same [[edge arrangement]] as the [[small stellated dodecahedron]].

A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the [[great icosidodecahedron]] as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the [[great stellated dodecahedron]].

The [[Truncation (geometry)|truncated]] ''great stellated dodecahedron'' is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two [[great dodecahedron|great dodecahedra]] inscribed within and sharing the edges of the icosahedron.

{| class="wikitable" width=500
!Name
![[Great stellated dodecahedron|Great<BR>stellated<BR>dodecahedron]]
![[Truncated great stellated dodecahedron]]
![[Great icosidodecahedron|Great<BR>icosidodecahedron]]
![[Truncated great icosahedron|Truncated<BR>great<BR>icosahedron]]
!Great<BR>icosahedron
|- align=center
![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
|{{CDD|node|3|node|5|rat|d2|node_1}}
|{{CDD|node|3|node_1|5|rat|d2|node_1}}
|{{CDD|node|3|node_1|5|rat|d2|node}}
|{{CDD|node_1|3|node_1|5|rat|d2|node}}
|{{CDD|node_1|3|node|5|rat|d2|node}}
|-
!Picture
|[[File:Great stellated dodecahedron.png|100px]]
|[[File:Icosahedron.png|100px]]
|[[File:Great icosidodecahedron.png|100px]]
|[[File:Great truncated icosahedron.png|100px]]
|[[File:Great icosahedron.png|100px]]
|}

==References==
{{reflist}}
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger  | title=Polyhedron Models | publisher=[[Cambridge University Press]] | year=1974 | isbn=0-521-09859-9 }}
*{{Cite book | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Du Val | first2=P. | last3=Flather | first3=H. T. | last4=Petrie | first4=J. F. | title=The fifty-nine icosahedra | publisher=Tarquin | edition=3rd | isbn=978-1-899618-32-3 |mr=676126 | year=1999 }} (1st Edn University of Toronto (1938))
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, 3.6 6.2 ''Stellating the Platonic solids'', pp.&nbsp;96–104

== External links ==
* {{mathworld2 | urlname = GreatIcosahedron| title =Great icosahedron | urlname2 = UniformPolyhedron | title2 = Uniform polyhedron}}
** {{mathworld | urlname = IcosahedronStellations| title = Fifteen stellations of the icosahedron}}
* [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]

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{{Icosahedron stellations}}

[[Category:Kepler–Poinsot polyhedra]]
[[Category:Regular polyhedra]]
[[Category:Polyhedral stellation]]
[[Category:Deltahedra]]