great dodecahedron

One way to construct a great dodecahedron is by faceting the regular icosahedron. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices.{{r|inchbald}} For each vertex of the icosahedron, the five neighboring vertices become those of a regular pentagon face of the great dodecahedron. The resulting shape has a pentagram as its vertex figure,{{r|pugh|barnes}} so its Schläfli symbol is $\backslash \{5,5/2\backslash \}$.{{r|kappraff}}

The great dodecahedron may also be interpreted as the second stellation of dodecahedron. The construction started from a regular dodecahedron by attaching 12 pentagonal pyramids onto each of its faces, known as the first stellation. The second stellation appears when 30 wedges are attached to it.{{r|cromwell}}

Given a great dodecahedron with edge length $E$,

$$\backslash text\{Inradius\}\; =\; \backslash frac\{\backslash sqrt\{50+10\backslash sqrt\{5\}\}\}\{20\}\backslash ,E$$

$$\backslash text\{Midradius\}\; =\; \backslash frac\{1+\backslash sqrt\{5\}\}\{4\}\backslash ,E$$

$$\backslash text\{Circumradius\}\; =\; \backslash frac\{\backslash sqrt\{10+2\backslash sqrt\{5\}\}\}\{4\}\backslash ,E$$

$$\backslash text\{Surface\; Area\}\; =\; 15\backslash sqrt\{5-2\backslash sqrt\{5\}\}\backslash ,E^2$$

$$\backslash text\{Volume\}\; =\; \backslash frac\{5\backslash sqrt\{5\}-5\}\{4\}\backslash ,E^3$$

{{multiple image

| image1 = Perspectiva Corporum Regularium 22c.jpg

| caption1 = Great dodecahedron in Perspectiva Corporum Regularium

| image2 = Alexander's Star.jpg

| caption2 = Alexander's Star in solved state

| total_width = 300

}}

Historically, the great dodecahedron is one of two solids discovered by Louis Poinsot in 1810, with some people named it after him, Poinsot solid. As for the background, Poinsot rediscovered two other solids that were already discovered by Johannes Kepler—the small stellated dodecahedron and the great stellated dodecahedron.{{r|barnes}} However, the great dodecahedron appeared in the 1568 Perspectiva Corporum Regularium by Wenzel Jamnitzer, although its drawing is somewhat similar.{{r|ss}}

The great dodecahedron appeared in popular culture and toys. An example is Alexander's Star puzzle, a twisty puzzle that is based on a great dodecahedron.{{cite magazine|url=https://archive.org/details/games-32-1982-October/page/n57/mode/2up|title=Alexander's star|magazine=Games|issue=32|date=October 1982|page=56}}

{{multiple image

| image1 = Compound of great dodecahedron and small stellated dodecahedron.png

| caption1 = Great dodecahedron shown solid, surrounding stellated dodecahedron only as wireframe

| image2 = Small stellated dodecahedron truncations.gif

| caption2 = Animated truncation sequence from {5/2, 5} to {5, 5/2}

| total_width = 300

}}

{{anchor|Compound}}The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ({{mset|{{sfrac|10|4}}}} "decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes.

A truncation process applied to the great dodecahedron produces a series of nonconvex uniform polyhedra. Truncating edges down to points produces the dodecadodecahedron as a rectified great dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the small stellated dodecahedron.

It shares the same edge arrangement as the convex regular icosahedron; the compound with both is the small complex icosidodecahedron.

{{Reflist|refs=

{{cite book

| last = Barnes | first = John

| year = 2012

| title = Gems of Geometry

| edition = 2nd

| url = https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA46

| page = 46

| publisher = Springer

| doi = 10.1007/978-3-642-30964-9

| isbn = 978-3-642-30964-9

}}

{{cite book

| last = Cromwell | first = Peter

| year = 1997

| title = Polyhedra

| url = https://books.google.com/books?id=OJowej1QWpoC&pg=PA265

| page = 265

| publisher = Cambridge University Press

| isbn = 978-0-521-66405-9

}}

{{cite journal

| last = Inchbald | first = Guy

| year = 2006

| title = Facetting Diagrams

| journal = The Mathematical Gazette

| volume = 90 | issue = 518 | pages = 253–261

| doi = 10.1017/S0025557200179653

| jstor = 40378613

}}

{{cite book

| last = Kappraff | first = Jay

| title = Connections: The Geometric Bridge Between Art and Science

| date = 2001

| url = https://books.google.com/books?id=twF7pOYXSTcC&pg=PA476

| page = 476

| publisher = John Wiley & Sons

| isbn = 978-981-02-4586-3

}}

{{cite book

| last = Pugh | first = Anthony

| year = 1976

| title = Polyhedra: A Visual Approach

| url = https://books.google.com/books?id=IDDxpYQTR7kC&pg=PA85

| page = 85

| publisher = University of California Press

| isbn = 978-0-520-03056-5

}}

{{cite book

| last1 = Scriba | first1 = Christoph

| last2 = Schreiber | first2 = Peter

| year = 2015

| title = 5000 Years of Geometry: Mathematics in History and Culture

| url = https://books.google.com/books?id=6Kp9CAAAQBAJ&pg=PA305

| page = 305

| publisher = Springer

| doi = 10.1007/978-3-0348-0898-9

| isbn = 978-3-0348-0898-9

}}

}}

• {{mathworld2 | urlname = GreatDodecahedron| title =Great dodecahedron | urlname2 = UniformPolyhedron | title2 = Uniform polyhedron}}
• {{mathworld | urlname = DodecahedronStellations| title =Three dodecahedron stellations}}
• [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
• [https://web.archive.org/web/20070921221730/http://bulatov.org/metal/dodecahedron_7.html Metal sculpture of Great Dodecahedron]

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Category:Kepler–Poinsot polyhedra

Category:Regular polyhedra

Category:Polyhedral stellation

Category:Toroidal polyhedra