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| | The author's title is Christy Brookins. The favorite pastime for him and his kids is to play lacross and he would by no means give it up. Some time in the past he selected to live in North Carolina and he doesn't plan on altering it. He is an info officer.<br><br>Here is my blog post - phone psychic [[http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ Highly recommended Resource site]] |
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| !bgcolor=#e7dcc3 colspan=2|Final stellation of the icosahedron
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| | colspan="2" align="center" valign="top" |[[File:Complete icosahedron ortho stella.png|150px]][[File:Complete icosahedron ortho2 stella.png|150px]]<BR>Two symmetric [[orthographic projection]]s
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| |bgcolor=#e7dcc3|[[Symmetry group]]||[[Icosahedral symmetry|icosahedral]] (''I''<sub>''h''</sub>)
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| |bgcolor=#e7dcc3|Type||[[Stellated icosahedron]], 8th of 59
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| |bgcolor=#e7dcc3|Symbols||Du Val '''H'''<BR>[[Wenninger]]: W<sub>42</sub>
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| |bgcolor=#e7dcc3|[[Euler characteristic|Elements]]<BR>(As a star polyhedron)||''F'' = 20, ''E'' = 90<BR>''V'' = 60 (''χ'' = −10)
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| |bgcolor=#e7dcc3|[[Euler characteristic|Elements]]<BR>(As a simple polyhedron)||''F'' = 180, ''E'' = 270,<BR>''V'' = 92 (''χ'' = 2)
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| |bgcolor=#e7dcc3|Properties<BR>(As a star polyhedron)||[[Vertex-transitive]], [[face-transitive]]
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| |colspan=2|
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| {|
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| ![[Stellation diagram]]!![[Stellation]] core||[[Convex hull]]
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| |- valign=top align=center
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| |[[File:Seventeenth stellation of icosahedron facets.png|100px]]
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| |[[File:Icosahedron.png|100px]]<BR>[[Stellations of icosahedron|Icosahedron]]
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| |[[File:Complete icosahedron convex hull.png|100px]]<BR>[[truncated icosahedron]]
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| |}
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| |}
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| In [[geometry]], the '''complete''' or '''final stellation of the icosahedron'''<ref>Coxeter et al. (1938), pp 30–31</ref><ref>Wenninger, ''Polyhedron Models'', p. 65.</ref> is the outermost [[stellation]] of the [[icosahedron]], and is "complete" and "final" because it includes all of the cells in the icosahedron's [[stellation diagram]].
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| It is also called the '''echidnahedron'''. This [[polyhedron]] is the seventeenth [[stellation]] of the [[icosahedron]], and given as [[List of Wenninger polyhedron models#Stellations of icosahedron|Wenninger model index 42]].
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| As a geometrical figure, it has two interpretations, described below:
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| * As an [[regular polyhedron|irregular]] [[#As_a_star_polyhedron|star (self-intersecting) polyhedron]] with 20 identical self-intersecting [[enneagram (geometry)|enneagrammic]] faces, 90 edges, 60 vertices.
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| * As a [[#As_a_simple_polyhedron|simple polyhedron]] with 180 triangular faces (60 isosceles, 120 scalene), 270 edges, and 92 vertices. This interpretation is useful for [[polyhedron model]] building.
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| [[Johannes Kepler]] researched stellations that create regular star polyhedra (the [[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra]]) in 1619, but the complete icosahedron, with irregular faces, was first studied in 1900 by [[Max Brückner]].
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| ==History==
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| [[File:Kepler-Poinsot solids.svg|480px]]
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| * 1619: In ''[[Harmonices Mundi]]'', [[Johannes Kepler]] first applied the stellation process, recognizing the [[small stellated dodecahedron]] and [[great stellated dodecahedron]] as regular polyhedra.<ref>{{MathWorld|urlname = Kepler-PoinsotSolid|title = Kepler-Poinsot Solid}}</ref>
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| * 1809: [[Louis Poinsot]] rediscovered Kepler's polyhedra and two more, the [[great icosahedron]] and [[great dodecahedron]] as regular star polyhedra, now called the [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]].<ref>Louis Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, pp. 16–48, 1810.</ref>
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| * 1812: [[Augustin-Louis Cauchy]] made a further enumeration of star polyhedra, proving there are only 4 regular star polyhedra.<ref name=Cromwell>Cromwell (1999) (p. 259)</ref>
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| [[File:Bruckner Taf XI Fig 14.JPG|thumb|right|Brückner's model (Taf. XI, Fig. 14, 1900)<ref name=mb>Brückner, Max (1900)</ref>]]
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| * 1900: Max Brückner extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the ''complete stellation''.<ref name=mb />
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| * 1924: A.H. Wheeler in 1924 published a list of 20 stellation forms (22 including reflective copies), also including the ''complete stellation''.<ref>Wheeler (1924)</ref>
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| * 1938: In their 1938 book ''[[The Fifty Nine Icosahedra]]'', [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], [[Patrick du Val|P. Du Val]], H. T. Flather and J. F. Petrie stated a set of stellation rules for the regular icosahedron and gave a systematic enumeration of the fifty-nine stellations which conform to those rules. The complete stellation is referenced as the eighth in the book.
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| *1974: In [[Magnus Wenninger|Wenninger]]'s 1974 book ''[[List of Wenninger polyhedron models|Polyhedron Models]]'', the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W<sub>42</sub>.
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| *1995: Andrew Hume named it in his [[Netlib]] polyhedral database as the '''echidnahedron'''<ref>The name ''echidnahedron'' may be credited to Andrew Hume, [http://netlib.sandia.gov/master/index.html developer] of the [[netlib]] [http://netlib.sandia.gov/polyhedra/index.html polyhedron database]:<br/>"... and some odd solids including the echidnahedron (my name; its actually the final stellation of the icosahedron)." [http://groups.google.com/group/geometry.research/browse_thread/thread/83fef9c1c0df12df/569b136b9a82f709?hl=en&lnk=st&q=Echidnahedron#569b136b9a82f709 geometry.research; "polyhedra database"; August 30, 1995, 12:00 am.]<br/></ref> (the [[echidna]], or spiny anteater, is a small [[mammal]] that is covered with coarse [[hair]] and [[Spine (zoology)|spines]]).
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| ==Interpretations==
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| [[File:Icosahedron stellation diagram center.png|thumb|left|Stellation diagram of the icosahedron with numbered cells. The complete icosahedron is formed from all the cells in the stellation, but only the outermost regions, labelled "13" in the diagram, are visible.]]
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| === As a stellation ===
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| {{main|The Fifty Nine Icosahedra}}
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| The [[stellation]] of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. ''The Fifty Nine Icosahedra'' enumerates the stellations of the regular [[icosahedron]], according to a set of rules put forward by [[J. C. P. Miller]], including the '''complete stellation'''. The Du Val symbol of the complete stellation is '''H''', because it includes all cells in the stellation diagram up to and including the outermost "h" layer.<ref name=Cromwell />
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| ===As a simple polyhedron===
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| {| class=wikitable width=250 align=right
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| |[[File:Complete icosahedron net stella.png|250px]]<BR>A [[polyhedral model]] can be constructed by 12 sets of faces, each folded into a group of five pyramids.
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| |}
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| As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an [[Euler characteristic]] of 2.<ref>[http://polyhedra.org/poly/show/141/echidnahedron Echidnahedron] at polyhedra.org</ref>
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| The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio<ref name=mw>{{MathWorld | urlname=Echidnahedron | title=Echidnahedron}}</ref>
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| :<math>\sqrt {\frac {3}{2} \left (3 + \sqrt{5} \right ) } \, : \,
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| \sqrt {\frac {1}{2} \left (25 + 11\sqrt{5} \right ) } \, : \,
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| \sqrt {\frac {1}{2} \left (97 + 43\sqrt{5} \right ) } \, .</math>
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| {| class=wikitable width=400
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| |+ Convex hulls of each sphere of vertices
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| !Inner
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| !Middle
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| !Outer
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| !All three
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| |-
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| !20 vertices
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| !12 vertices
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| !60 vertices
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| !92 vertices
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| |- valign=top
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| |[[File:Dodecahedron.png|100px]]<BR>[[Dodecahedron]]
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| |[[File:Icosahedron.png|100px]]<BR>[[Icosahedron]]
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| |[[File:Complete icosahedron convex hull.png|100px]]<BR>Nonuniform<BR>[[truncated icosahedron]]
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| |[[File:Complete icosahedron ortho stella.png|100px]]<BR>Complete icosahedron
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| |}
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| When regarded as a three-dimensional solid object with edge lengths ''a'', φ''a'', φ<sup>2</sup>''a'' and φ<sup>2</sup>''a''√2 (where φ is the [[golden ratio]]) the complete icosahedron has surface area<ref name=mw />
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| :<math>S=\frac{1}{20}(13211 + \sqrt{174306161})a^2\, ,</math>
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| and volume<ref name=mw />
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| :<math>V=(210+90\sqrt{5})a^3\, .</math>
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| === As a star polyhedron ===
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| {| class=wikitable align=right width=300
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| |- valign=top
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| |[[File:Echidnahedron with enneagram face.png|150px]]<BR>Twenty (9/4) polygon faces (one face is drawn yellow with 9 vertices labeled.)
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| |[[File:Enneagram 9-4 icosahedral.svg|150px]]<BR>[[Isogonal_figure#k-isogonal_figures|2-isogonal]] (9/4) faces
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| |}
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| The complete stellation can also be seen as a self-intersecting [[star polyhedron]] having 20 faces corresponding to the 20 faces of the underlying icosahedron. Each face is an irregular 9/4 [[star polygon]], or [[Enneagram (geometry)|enneagram]].<ref name=Cromwell /> Since three faces meet at each vertex it has 20 × 9 / 3 = 60 vertices (these are the outermost layer of visible vertices and form the tips of the "spines") and 20 × 9 / 2 = 90 edges (each edge of the star polyhedron includes and connects two of the 180 visible edges).
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| When regarded as a star icosahedron, the complete stellation is a [[noble polyhedron]], because it is both [[isohedral]] (face-transitive) and [[isogonal figure|isogonal]] (vertex-transitive).
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| If its faces were regular [[Enneagram (geometry)|enneagrams]], it could be a regular polyhedron with Schläfli symbol {9/4,3}.
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| ==See also==
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| *[[Kepler–Poinsot polyhedron]]
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| *[[List of Wenninger polyhedron models]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Brückner, Max (1900). [http://books.google.com/books?id=dERmNV8lxt4C ''Vielecke und Vielflache: Theorie und Geschichte'']. Leipzig: B.G. Treubner. ISBN 978-1-4181-6590-1. {{de icon}} [http://www.worldcat.org/oclc/25080888?tab=details WorldCat] English: ''Polygons and Polyhedra: Theory and History''. Photographs of models: [http://books.google.com/books?id=dERmNV8lxt4C&printsec=titlepage&source=gbs_summary_r&cad=0#PRA1-PR5,M1 Tafel VIII (Plate VIII)], etc. [http://bulatov.org/polyhedra/bruckner1900/index.html High res. scans.]
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| * A. H. Wheeler, ''Certain forms of the icosahedron and a method for deriving and designating higher polyhedra'', Proc. Internat. Math. Congress, Toronto, 1924, Vol. 1, pp 701–708
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| * {{Citation | 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))
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| *[[Wenninger, Magnus J.]], [http://books.google.com/books?id=N8lX2T-4njIC&lpg=PP1&dq=Polyhedron%20models&pg=PP1#v=onepage&q=&f=false ''Polyhedron models'']; Cambridge University Press, 1st Edn (1983), Ppbk (2003). ISBN 978-0-521-09859-5. (Model 42, p 65, ''Final stellation of the icosahedron'')
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| *{{cite book
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| | last = Cromwell
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| | first = Peter R.
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| | title = Polyhedra
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| | publisher = Cambridge University Press
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| | year = 1997
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| | isbn = 0-521-66405-5
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| | url=http://books.google.com/books?id=OJowej1QWpoC&lpg=PP1&dq=Polyhedra&pg=PP1#v=onepage&q=&f=false}}
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| *Jenkins, Gerald, and Magdalen Bear. ''The Final Stellation of the Icosahedron: An Advanced Mathematical Model to Cut Out and Glue Together''. Norfolk, England: Tarquin Publications, 1985. ISBN 978-0-906212-48-6.
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| ==External links==
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| * [http://www.prospero78.freeserve.co.uk/icosa/Echidnahedron1.doc With instructions for constructing a model of the echidnahedron] ([[Microsoft Word|.doc]]) by Ralph Jones
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| * [http://www.steelpillow.com/polyhedra/icosa/stelfacet/StelFacet.htm Towards stellating the icosahedron and faceting the dodecahedron] by Guy Inchbald
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| *{{MathWorld | urlname=IcosahedronStellations | title=Fifty nine icosahedron stellations}}
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| **{{MathWorld | urlname=Echidnahedron | title=Echidnahedron}}
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| *[http://mathserver.sdu.edu.cn/mathency/math/i/i011.htm Stellations of the icosahedron]
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| *[http://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-index.html 59 Stellations of the Icosahedron]
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| *[[VRML]] model: http://www.georgehart.com/virtual-polyhedra/vrml/echidnahedron.wrl
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| *[[Netlib]]: [http://netlib.sandia.gov/polyhedra Polyhedron database, model 141]
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| {{Icosahedron stellations}}
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| {{good article}}
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| {{DEFAULTSORT:Complete Icosahedron}}
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| [[Category:Polyhedral stellation]]
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| [[Category:Polyhedra]]
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