Editing Augmented hexagonal prism

{{Short description|54th Johnson solid}}
{{Infobox polyhedron
 | image = augmented_hexagonal_prism.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[biaugmented pentagonal prism|''J''{{sub|53}}]] – '''''J''{{sub|54}}''' – [[parabiaugmented hexagonal prism|''J''{{sub|55}}]]}}
 | faces = 4 [[triangle]]s<br>5 [[square (geometry)|square]]s<br>2 [[hexagon]]s
 | edges = 22
 | vertices = 13
 | symmetry = {{math|''C''{{sub|2v}}}}
 |vertex_config = {{math|2x4(4{{sup|2}}.6)<br>1(3{{sup|4}})<br>4(3{{sup|2}}.4.6)}}
 | dual = monolaterotruncated hexagonal bipyramid
 | properties = [[Convex polytope|convex]]
 | net = Johnson solid 54 net.png
}}

In [[geometry]], the '''augmented hexagonal prism''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|54}}}}). As the name suggests, it can be constructed by augmenting a hexagonal [[prism (geometry)|prism]] by attaching a [[square pyramid]] ({{math|''J''{{sub|1}}}}) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a [[parabiaugmented hexagonal prism]] ({{math|''J''{{sub|55}}}}), a [[metabiaugmented hexagonal prism]] ({{math|''J''{{sub|56}}}}), or a [[triaugmented hexagonal prism]] ({{math|''J''{{sub|57}}}}).

== Construction ==
The augmented hexagonal prism is constructed by attaching one [[equilateral square pyramid]] onto the square face of a [[hexagonal prism]], a process known as [[Augmentation (geometry)|augmentation]].{{r|rajwade}} This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four [[equilateral triangle]]s, five [[square (geometry)|square]]s, and two regular [[hexagon]]s.{{r|berman}} A convex polyhedron in which all of the faces are regular is a Johnson solid, and the augmented hexagonal prism is among them, enumerated as <math> J_{54} </math>.{{r|francis}} Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the [[parabiaugmented hexagonal prism]] <math> J_{55} </math>, the [[metabiaugmented hexagonal prism]] <math> J_{56} </math>, and the [[triaugmented hexagonal prism]] <math> J_{57} </math>.{{r|rajwade}}

== Properties ==
An augmented hexagonal prism with edge length <math> a </math> has surface area{{r|berman}}
<math display="block"> \left(5 + 4\sqrt{3}\right)a^2 \approx 11.928a^2, </math>
the sum of two hexagons, four equilateral triangles, and five squares area. Its volume{{r|berman}}
<math display="block"> \frac{\sqrt{2} + 9\sqrt{3}}{2}a^3 \approx 2.834a^3, </math>
can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.{{r|berman}}

It has an [[axis of symmetry]] passing through the apex of a square pyramid and the centroid of a prism square face, rotated in a half and full-turn angle. Its [[dihedral angle]] can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following:{{r|johnson}}
* The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, <math> \arccos \left(-1/3\right) \approx 109.5^\circ </math>
* The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, <math> 2\pi/3 = 120^\circ </math>
* The dihedral angle of an augmented hexagonal prism between square-to-hexagon is the dihedral angle of a hexagonal prism between its base and its lateral face, <math> \pi/2 </math>
* The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is <math> \arctan \left(\sqrt{2}\right) \approx 54.75^\circ </math>. Therefore, the dihedral angle of an augmented hexagonal prism between square-to-triangle and between triangle-to-hexagon, on the edge in which the square pyramid and hexagonal prism are attached, are <math display="block"> \begin{align}
 \arctan \left(\sqrt{2}\right) + \frac{2\pi}{3} \approx 174.75^\circ, \\
 \arctan \left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.75^\circ.
\end{align}
</math>.

== References ==
{{reflist|refs=

<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}</ref>

<ref name="francis">{{cite journal
 | last = Francis | first = Darryl
 | title = Johnson solids & their acronyms
 | journal = Word Ways
 | date = August 2013
 | volume = 46 | issue = 3 | page = 177
 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | authorlink = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603| doi-access = free
}}</ref>

<ref name="rajwade">{{cite book
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | page = 84&ndash;89
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

}}

==External links==
* {{MathWorld2 | urlname2=JohnsonSolid | title2=Johnson Solid| urlname=AugmentedHexagonalPrism | title=Augmented hexagonal prism }}

[[Category:Johnson solids]]
{{Johnson solids navigator}}