Elongated triangular orthobicupola: Difference between revisions

Elongated triangular orthobicupola
Elongated triangular orthobicupola
Type	Johnson; J34 – J35 – J36
Faces	8 triangles; 12 squares
Edges	36
Vertices	18
Vertex configuration
Symmetry group
Properties	convex
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In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

Construction

The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces.^[1] This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid $J_{35}$ .^[3]

Properties

An elongated triangular orthobicupola with a given edge length $a$ has a surface area, by adding the area of all regular faces:^[2] $\left(12+2{\sqrt {3}}\right)a^{2}\approx 15.464a^{2}.$ Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up:^[2] $\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.$

It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group $D_{3h}$ of order 12. Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon $120^{\circ }=2\pi /3$ , and that between its base and square face is $\pi /2=90^{\circ }$ . The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately $70.5^{\circ }$ , that between each square and the hexagon is $54.7^{\circ }$ , and that between square and triangle is $125.3^{\circ }$ . The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:^[4] ${\begin{aligned}{\frac {\pi }{2}}+70.5^{\circ }&\approx 160.5^{\circ },\\{\frac {\pi }{2}}+54.7^{\circ }&\approx 144.7^{\circ }.\end{aligned}}$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^[5]

References

^ Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
^ ^a ^b ^c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
^ Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.
^ "J35 honeycomb".

External links

Weisstein, Eric W., "Elongated triangular orthobicupola" ("Johnson solid") at MathWorld.

[rajwade-1] Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.

[berman-2] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[francis-3] Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.

[johnson-4] Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.

[5] "J35 honeycomb".

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
 {{Short description|Johnson solid with 20 faces}}
 {{Infobox polyhedron
-|image=elongated triangular orthobicupola.png
+ | image = Elongated triangular orthobicupola.png
-|type=[[Johnson solid|Johnson]]<br>{{math|[[pentagonal orthobirotunda|''J''{{sub|34}}]] –  '''''J''{{sub|35}}''' – [[elongated triangular gyrobicupola|''J''{{sub|36}}]]}}
+ | type = [[Johnson solid|Johnson]]<br>{{math|[[pentagonal orthobirotunda|''J''{{sub|34}}]] –  '''''J''{{sub|35}}''' – [[elongated triangular gyrobicupola|''J''{{sub|36}}]]}}
-|faces=2+6 [[triangle]]s<br>2x3+6 [[Square (geometry)|square]]s
+ | faces = 8 [[triangle]]s<br>12 [[Square (geometry)|square]]s
-|edges=36
+ | edges = 36
-|vertices=18
+ | vertices = 18
-|symmetry={{math|''D''{{sub|3h}}}}
+ | symmetry = <math> D_{3h} </math>
-|vertex_config={{math|6(3.4.3.4)<br>12(3.4<sup>3</sup>)}}
+ | vertex_config = <math> \begin{align}
+ &6 \times (3 \times 4 \times 3 \times 4) + \\
-|dual=-
+ &12 \times (3 \times 4^3)
-|properties=[[convex set|convex]]
+\end{align} </math>
-|net=Johnson solid 35 net.png
+ | properties = [[convex set|convex]]
+ | net = Johnson solid 35 net.png
 }}
-In [[geometry]], the '''elongated triangular orthobicupola''' or '''cantellated triangular prism''' is one of the [[Johnson solid]]s  ({{math|''J''{{sub|35}}}}). As the name suggests, it can be constructed by elongating a [[triangular orthobicupola]] ({{math|''J''{{sub|27}}}}) by inserting a [[hexagon]]al [[prism (geometry)|prism]] between its two halves. The resulting solid is superficially similar to the [[rhombicuboctahedron]] (one of the [[Archimedean solid]]s), with the difference that it has threefold [[rotational symmetry]] about its axis instead of fourfold symmetry.
+In [[geometry]], the '''elongated triangular orthobicupola''' is a polyhedron constructed by attaching two regular [[triangular cupola]] into the base of a regular [[hexagonal prism]]. It is an example of [[Johnson solid]].
+== Construction ==
-{{Johnson solid}}
+The elongated triangular orthobicupola can be constructed from a [[hexagonal prism]] by attaching two regular [[triangular cupola]]e onto its base, covering its hexagonal faces.{{r|rajwade}} This construction process known as [[Elongation (geometry)|elongation]], giving the resulting polyhedron has 8 [[equilateral triangle]]s and 12 squares.{{r|berman}} A [[Convex set|convex]] polyhedron in which all faces are [[Regular polygon|regular]] is [[Johnson solid]], and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid <math> J_{35} </math>.{{r|francis}}
-== Volume ==
+== Properties ==
+An elongated triangular orthobicupola with a given edge length <math> a </math> has a surface area, by adding the area of all regular faces:{{r|berman}}
-The volume of ''J''<sub>35</sub> can be calculated as follows:
+<math display="block"> \left(12 + 2\sqrt{3}\right)a^2 \approx 15.464a^2. </math>
+Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up:{{r|berman}}
+<math display="block"> \left(\frac{5\sqrt{2}}{3} + \frac{3\sqrt{3}}{2}\right)a^3 \approx 4.955a^3. </math>
+It has the same [[Point groups in three dimensions|three-dimensional symmetry groups]] as the [[triangular orthobicupola]], the dihedral group <math> D_{3h} </math> of order 12. Its [[dihedral angle]] can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the [[internal angle]] of a regular hexagon <math> 120^\circ = 2\pi/3</math>, and that between its base and square face is <math> \pi/2 = 90^\circ </math>. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately <math> 70.5^\circ </math>, that between each square and the hexagon is <math> 54.7^\circ </math>, and that between square and triangle is <math> 125.3^\circ </math>. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:{{r|johnson}}
-''J''<sub>35</sub> consists of 2 cupolae and hexagonal prism.
+<math display="block"> \begin{align}
+ \frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\
+ \frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ.
+\end{align} </math>
+==Related polyhedra and honeycombs==
-The two cupolae makes 1 cuboctahedron = 8 tetrahedra + 6 half-octahedra.
-octahedron = 4 tetrahedra, so total we have 20 tetrahedra.
+The elongated triangular orthobicupola forms space-filling [[Honeycomb (geometry)|honeycomb]]s with [[Tetrahedron|tetrahedra]] and [[square pyramid]]s.<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J35.html|title = J35 honeycomb}}</ref><br>
-What is the volume of a tetrahedron?
-Construct a tetrahedron having vertices in common with alternate vertices of a
-cube (of side <math>\textstyle\frac{1}{\sqrt{2}}</math>, if tetrahedron has unit edges). The 4 triangular
-pyramids left if the tetrahedron is removed from the cube form half an
-octahedron = 2 tetrahedra. So
+== References ==
-:<math>V_\mathrm{tetrahedron} = \frac{1}{3} V_\mathrm{cube} = \frac{1}{3} \frac{1}{{\sqrt{2}}^3} = \frac{\sqrt{2}}{12}</math>
+{{reflist|refs=
+<ref name="berman">{{cite journal
-The hexagonal prism is more straightforward. The hexagon has area <math>\textstyle 6 \frac{\sqrt{3}}{4}</math>, so
+ | 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
-:<math>V_\mathrm{prism} = \frac{3 \sqrt{3}}{2}</math>
+ | 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
-Finally
+ | 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
-:<math>V_{J_{35}} = 20 V_\mathrm{tetrahedron} + V_\mathrm{prism} =
+ | last = Rajwade | first = A. R.
-\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}</math>
+ | 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>
+}}
-numerical value:
-:<math>V_{J_{35}} = 4.9550988153084743549606507192748</math>
-==Related polyhedra and honeycombs==
-The elongated triangular orthobicupola forms space-filling [[Honeycomb (geometry)|honeycomb]]s with [[Tetrahedron|tetrahedra]] and [[square pyramid]]s.<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J35.html|title = J35 honeycomb}}</ref><br>
-==References==
-{{Reflist}}
 ==External links==
 * {{MathWorld2|title2=Johnson solid|urlname2=JohnsonSolid| urlname=ElongatedTriangularOrthobicupola | title=Elongated triangular orthobicupola}}
 {{Johnson solids navigator}}
-{{Polyhedron-stub}}
 [[Category:Johnson solids]]

v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)