View source for Pentagrammic prism

{{Multiple issues|
{{Notability|date=May 2020}}
{{More citations needed|date=May 2020}}
}}
{{Prism polyhedra db|Prism star polyhedron stat table|P5d2}}
In [[geometry]], the '''pentagrammic prism''' is one of an infinite set of nonconvex [[Prism (geometry)|prisms]] formed by square sides and two regular [[star polygon]] caps, in this case two [[pentagram]]s.

It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex [[pentagonal prism]].

It is the 78th model in the [[list of uniform polyhedra]], as the first representative of [[Prismatic uniform polyhedron|uniform star prisms]], along with the [[pentagrammic antiprism]], which is the 79th model.

==Geometry==
It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U<sub>78</sub> as a [[uniform polyhedron]].<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/78.html|title=78: pentagrammic prism|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref>

The pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior, depending on how the interior is defined. One definition of the interior is the set of points from which a [[ray (geometry)|ray]] crosses the boundary an odd number of times; this makes the central pentagon exterior, as every ray beginning within it crosses two edges.

== Gallery ==
{| class=wikitable width=400
|[[Image:Pentagram_prism.png|120px]]<BR>An alternative representation with hollow centers to the pentagrams.
|[[File:Pentagrammic prism.stl|200px]]<BR>3D model of a (uniform) pentagrammic prism
|}
{{clear}}
== Pentagrammic dipyramid==
{{Infobox face-uniform polyhedron |
  name=Pentagrammic dipyramid |
  Image_File=Pentagram_Dipyramid.png |
  Polyhedron_Type=[[Star bipyramid]]|
  Schläfli = {} + {5/2} |
  Coxeter={{CDD|node_f1|2x|node_f1|5|rat|2x|node}} |
  Face_List=10 [[triangle]]s|
  Edge_Count=15 |
  Vertex_Count=7 |
  Symmetry_Group=''D''<sub>5h</sub>, [5,2], (*225), order 20|
  Rotation_Group  = ''D''<sub>''5''</sub>, [5,2]<sup>+</sup>, (225), order 10|
  Face_Type=V4.4.5|
  Dual=pentagrammic prism|
  Property_List=[[face-transitive]], ([[deltahedron]])
}}
In [[geometry]], the '''pentagrammic [[dipyramid]]''' (or '''bipyramid''') is first of the infinite set of [[face-transitive]] star dipyramids containing [[star polygon]] arrangement of edges.  It has 10 intersecting [[isosceles triangle]] faces. It is topologically identical to the [[pentagonal dipyramid]].

Each star dipyramid is the [[dual polyhedron|dual]] of a star polygon based [[Prism (geometry)|uniform prism]]. 

{| class=wikitable
|[[File:Pentagrammic bipyramid (dual uniform).stl|320px]]<BR>3D model of a dual uniform pentagrammic dipyramid
|[[File:Pentagrammic bipyramid (regular faces).stl|320px]]<BR>3D model of a pentagrammic dipyramid with regular faces
|}

=== Related polyhedra ===

There are two pentagrammic [[trapezohedron|trapezohedra]] (or deltohedra), being dual to the [[pentagrammic antiprism]] and [[pentagrammic crossed antiprism]] respectively, each having intersecting [[Kite (geometry)|kite]]-shaped faces (convex or concave), and a total of 12 vertices:
{| class=wikitable
!{{mset|{{frac|5|2}}}} trapezohedron
!{{mset|{{frac|5|3}}}} trapezohedron
|-
|[[File:5-2 deltohedron.png|240px]]
|[[File:5-3 deltohedron.png|240px]]
|}

== References ==
{{Reflist}}

== External links ==
* {{mathworld | urlname = PentagrammicPrism| title = Pentagrammic prism}}
* {{mathworld | urlname = PentagrammicDipyramid | title = Pentagrammic dipyramid }}
*{{mathworld |urlname=PentagrammicDeltohedron |title=Pentagrammic deltohedron}}
*{{mathworld |urlname=PentagrammicConcaveDeltohedron |title=Pentagrammic concave deltohedron}}
*http://www.mathconsult.ch/showroom/unipoly/78.html
*http://bulatov.org/polyhedra/uniform/u03.html
*[http://www.software3d.com/Prism52.php Paper model of pentagrammic prism]
*https://web.archive.org/web/20050313234702/http://www.math.technion.ac.il/~rl/kaleido/data/03.html
*https://web.archive.org/web/20060211140715/http://www.ac-noumea.nc/maths/amc/polyhedr/no_conv5_.htm
*[https://web.archive.org/web/20060516191627/http://www.korthalsaltes.com/pentagrammic_prism.htm Paper Model (net) Pentagrammic Prism]

[[Category:Uniform polyhedra]]
[[Category:Prismatoid polyhedra]]

{{Polyhedron-stub}}