Editing Dodecahedron (section)

{{Short description|Polyhedron with 12 faces}}
{{About-distinguish|the three-dimensional shape|Roman dodecahedron}}

{| class="wikitable floatright" width=320
|+ Common dodecahedra
|- style="text-align:center;"
!colspan=4|I{{sub|h}}, order 120
|- style="text-align:center; vertical-align:bottom;"
|[[Regular dodecahedron|Regular]]
|[[Small stellated dodecahedron|Small stellated]]
|[[Great dodecahedron|Great]]
|[[Great stellated dodecahedron|Great stellated]]
|- style="text-align:center; vertical-align:bottom;"
|[[File:Dodecahedron.png|80px]]
|[[File:Small stellated dodecahedron.png|80px]]
|[[File:Great dodecahedron.png|80px]]
|[[File:Great stellated dodecahedron.png|80px]]
|-
!T{{sub|h}}, order 24
!T, order 12
!O{{sub|h}}, order 48
!Johnson (J{{sub|84}})
|- style="text-align:center; vertical-align:bottom;"
|[[Pyritohedron]]
|[[Tetartoid]]
|[[Rhombic dodecahedron|Rhombic]]
|[[Triangular dodecahedron|Triangular]]
|- style="text-align:center; vertical-align:bottom;"
|[[File:Pyritohedron.png|x80px]]
|[[File:Tetartoid.png|x80px]]
|[[File:Rhombicdodecahedron.jpg|x80px]]
|[[File:snub_disphenoid.png|x80px]]
|- align=center
!colspan=2|D{{sub|4h}}, order 16
!colspan=2|D{{sub|3h}}, order 12
|- align=center
|[[rhombo-hexagonal dodecahedron|Rhombo-hexagonal]]
|[[Rhombic dodecahedron#Parallelohedron|Rhombo-square]]
|[[Trapezo-rhombic dodecahedron|Trapezo-rhombic]]
|[[Triangular-rhombic dodecahedron|Rhombo-triangular]]
|- align=center
|[[File:Rhombo-hexagonal dodecahedron.png|80px]]
|[[File:Squared rhombic dodecahedron.png|80px]]
|[[File:Trapezo-rhombic dodecahedron.png|80px]]
|[[File:Triangular square dodecahedron.png|80px]]
|}

In [[geometry]], a '''dodecahedron''' ({{etymology|grc|''{{Wikt-lang|grc|δωδεκάεδρον}}'' ({{grc-transl|δωδεκάεδρον}})|}}; {{etymology||''{{Wikt-lang|grc|δώδεκα}}'' ({{grc-transl|δώδεκα}})|twelve||''{{Wikt-lang|grc|ἕδρα}}'' ({{grc-transl|ἕδρα}})|base, seat, face}}) or '''duodecahedron'''<ref>1908 Chambers's Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary</ref> is any [[polyhedron]]  with twelve flat faces. The most familiar dodecahedron is the [[regular dodecahedron]] with regular pentagons as faces, which is a [[Platonic solid]]. There are also three [[Kepler–Poinsot polyhedron|regular star dodecahedra]], which are constructed as [[stellation]]s of the convex form. All of these have [[icosahedral symmetry]], order 120.

Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular:
The [[#Pyritohedron|pyritohedron]], a common crystal form in [[pyrite]], has [[pyritohedral symmetry]], while the [[#Tetartoid|tetartoid]] has [[tetrahedral symmetry]].

The [[rhombic dodecahedron]] can be seen as a limiting case of the pyritohedron, and it has [[octahedral symmetry]]. The [[elongated dodecahedron]] and [[trapezo-rhombic dodecahedron]] variations, along with the rhombic dodecahedra, are [[space-filling polyhedra|space-filling]]. There are numerous [[#Other dodecahedra|other dodecahedra]].

While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.<ref name="cross">{{Cite journal |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.2020.1712564 |title=Platonic Solids and High Genus Covers of Lattice Surfaces |doi=10.1080/10586458.2020.1712564 |journal=[[Experimental Mathematics]] |date=May 27, 2020 |language=en-US |last1=Athreya |first1=Jayadev S. |last2=Aulicino |first2=David |last3=Hooper |first3=W. Patrick|volume=31 |issue=3 |pages=847–877 |arxiv=1811.04131 |s2cid=119318080 }}</ref>