View source for Augmented truncated tetrahedron

{{Short description|65th Johnson solid}}
{{Infobox polyhedron
 | image = Augmented truncated tetrahedron.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[augmented tridiminished icosahedron|''J''{{sub|64}}]] – '''''J''{{sub|65}}''' – [[augmented truncated cube|''J''{{sub|66}}]]}}
 | faces = 8 [[triangle]]s <br> 3 [[Square (geometry)|square]]s <br> 3 [[hexagon]]s
 | edges = 27
 | vertices = 15
 | symmetry = {{math|[[Point groups in three dimensions|''C''{{sub|3v}}]]}}
 | vertex_config = {{math|2x3(3.6{{sup|2}})<br>3(3.4.3.4)<br>6(3.4.3.6)}}
 | properties = [[Convex polytope|convex]]
 | net = Johnson solid 65 net.png
}}
[[File:J65 augmented truncated tetrahedron.stl|thumb|3D model of an augmented truncated tetrahedron]]

In [[geometry]], the '''augmented truncated tetrahedron''' is a polyhedron constructed by attaching a [[triangular cupola]] onto an [[truncated tetrahedron]]. It is an example of a [[Johnson solid]].

== Construction ==
The augmented truncated tetrahedron is constructed from a [[truncated tetrahedron]] by attaching a [[triangular cupola]].{{r|rajwade}} This cupola covers one of the truncated tetrahedron's four hexagonal faces, so that the resulting polyhedron's faces are eight [[equilateral triangle]]s, three [[square]]s, and three [[regular hexagon]]s.{{r|berman}} Since it has the property of [[Convex set|convexity]] and has [[regular polygon]]al faces, the augmented truncated tetrahedron is a [[Johnson solid]], denoted as the sixty-fifth Johnson solid <math> J_{65} </math>.{{r|francis}}

== Properties ==
The surface area of an augmented truncated tetrahedron is:{{r|berman}}
<math display="block"> \frac{6 + 13 \sqrt{3}}{2}a^2 \approx 14.258a^2, </math>
the sum of the areas of its faces. Its volume can be calculated by slicing it off into both truncated tetrahedron and triangular cupola, and adding their volume:{{r|berman}}
<math display="block"> \frac{11 \sqrt{2}}{4}a^3 \approx 3.889a^3. </math>

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the triangular cupola, the [[pyramidal symmetry]] <math> C_{3 \mathrm{v}} </math>. Its [[dihedral angle]]s can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following:{{r|johnson}}
* its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°;
* its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
* its dihedral angle between triangle and square is as in the triangular cupola's angle: 125.3°
* its dihedral angle between triangle and square, on the edge where the triangular cupola and truncated tetrahedron are attached, is the sum of both triangular cupola's square-hexagon angle and the truncated tetrahedron's triangle-hexagon angle: approximately 164.17°; and
* its dihedral angle between triangle and hexagon, on the edge where triangular cupola and truncated tetrahedron are attached, is the sum of the dihedral angle of a triangular cupola and truncated tetrahedron between that: approximately 141.3°;

== 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 = AugmentedTruncatedTetrahedron | title = Augmented truncated tetrahedron }}

{{Johnson solids navigator}}

[[Category:Johnson solids]]


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