Elongated square pyramid: Difference between revisions

Elongated square pyramid
Elongated square pyramid
Type	Johnson; J7 – J8 – J9
Faces	4 triangles; 1+4 squares
Edges	16
Vertices	9
Vertex configuration	; ;
Symmetry group
Dihedral angle (degrees)	triangle-to-triangle: 109.47°; square-to-square: 90°; triangle-to-square: 144.74°;
Properties	convex, composite
Net

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In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.

Construction

The elongated square bipyramid is a composite, since it can constructed by attaching two equilateral square pyramids onto the faces of a cube that are opposite each other, a process known as elongation.^[1]^[2] This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight equilateral triangles and four squares as their faces.^[3] A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as $J_{15}$ , the fifteenth Johnson solid.^[4]

Properties

Given that $a$ is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is $(1/{\sqrt {2}})a$ . Therefore, the height of an elongated square bipyramid is:^[5] $a+{\frac {1}{\sqrt {2}}}a=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a.$ Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:^[3] $\left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2}.$ Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:^[3] $\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.$

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group $C_{4v}$ of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:^[6]

The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, $\arccos(-1/3)\approx 109.47^{\circ }$ ,
The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, $\pi /2=90^{\circ }$ ,
The dihedral angle of an equilateral square pyramid between square and triangle is $\arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }$ . Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is $\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.$

References

^ Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
^ 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.
^ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. S2CID 220150682.
^ Sapiña, R. "Area and volume of the Johnson solid $J_{8}$ ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.
^ 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.

External links

Weisstein, Eric W., "Johnson solid" ("Elongated square pyramid") at MathWorld.

[timofeenko-2010-1] Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.

[rajwade-2] 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-3] 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.

[uehara-4] Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. S2CID 220150682.

[pye-5] Sapiña, R. "Area and volume of the Johnson solid $J_{8}$ ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.

[johnson-6] 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.

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[2]

[3]

[4]

[5]

[6]

@@ Line 1: / Line 1: @@
+{{Short description|Polyhedron with cube and square pyramid}}
 {{Infobox polyhedron
-|image=elongated_square_pyramid.png
+ | image = elongated_square_pyramid.png
-|type=[[Johnson solid|Johnson]]<br>[[elongated triangular pyramid|J<sub>7</sub>]] - '''J<sub>8</sub>''' - [[elongated pentagonal pyramid|J<sub>9</sub>]]
+ | type = [[Johnson solid|Johnson]]<br>{{math|[[elongated triangular pyramid|''J''{{sub|7}}]] – '''''J''{{sub|8}}''' – [[elongated pentagonal pyramid|''J''{{sub|9}}]]}}
-|faces=4 [[triangle]]s<br>1+4 [[Square (geometry)|square]]s
+ | faces = 4 [[triangle]]s<br>1+4 [[Square (geometry)|square]]s
-|edges=16
+ | edges = 16
-|vertices=9
+ | vertices = 9
-|symmetry=[[cyclic symmetries|''C''<sub>4v</sub>]], [4], (*44)
+ | symmetry = <math> C_{4v} </math>
+ | angle = {{bulletlist
-|rotation_group=''C''<sub>4</sub>, [4]<sup>+</sup>, (44)
+ | triangle-to-triangle: 109.47°
-|vertex_config=4(4<sup>3</sup>)<br>1(3<sup>4</sup>)<br>4(3<sup>2</sup>.4<sup>2</sup>)
+ | square-to-square: 90°
-|dual=self
+ | triangle-to-square: 144.74°
-|properties=[[convex set|convex]]
+}}
-|net=Elongated_Square_Pyramid_Net.svg
+ | vertex_config = <math> 4 \times (4^3) </math><br><math> 1 \times (3^4) </math><br><math> 4 \times (3^2 \times 4^2) </math>
+ | properties = [[convex set|convex]], [[composite polyhedron|composite]]
+ | net = Elongated_Square_Pyramid_Net.svg
 }}
-In [[geometry]], the '''elongated square pyramid''' is one of the [[Johnson solid]]s (''J''<sub>8</sub>). As the name suggests, it can be constructed by elongating a [[square pyramid]] (''J''<sub>1</sub>) by attaching a [[cube (geometry)|cube]] to its square base. Like any elongated [[pyramid]], it is topologically (but not geometrically) self-[[dual polyhedron|dual]].
+In [[geometry]], the '''elongated square pyramid''' is a convex polyhedron constructed from a [[Cube (geometry)|cube]] by attaching an [[equilateral square pyramid]] onto one of its faces. It is an example of [[Johnson solid]].
+== Construction ==
-{{Johnson solid}}
+The elongated square bipyramid is a [[Composite polyhedron|composite]], since it can constructed by attaching two [[Equilateral square pyramid|equilateral square pyramids]] onto the faces of a [[Cube (geometry)|cube]] that are opposite each other, a process known as [[Elongation (geometry)|elongation]].{{r|timofeenko-2010|rajwade}} This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight [[Equilateral triangle|equilateral triangles]] and four squares as their faces.{{r|berman}} A convex polyhedron in which all of its faces are regular is a [[Johnson solid]], and the elongated square bipyramid is one of them, denoted as <math> J_{15} </math>, the fifteenth Johnson solid.{{r|uehara}}
-== Dual polyhedron ==
+== Properties ==
+Given that <math> a </math> is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is <math> (1/\sqrt{2})a </math>. Therefore, the height of an elongated square bipyramid is:{{r|pye}}
+<math display="block"> a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a.  </math>
+Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:{{r|berman}}
+<math display="block"> \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. </math>
+Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:{{r|berman}}
+<math display="block"> \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. </math>
-The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal.
+[[File:Pirámide cuadrada elongada.stl|thumb|3D model of a elongated square pyramid.]]
+The elongated square pyramid has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the equilateral square pyramid, the [[cyclic group]] <math> C_{4v} </math> of order eight. Its [[dihedral angle]] can be obtained by adding the angle of an equilateral square pyramid and a cube:{{r|johnson}}
-{| class=wikitable width=320
+* The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, <math> \arccos(-1/3) \approx 109.47^\circ </math>,
-|- valign=top
+* The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, <math> \pi/2 = 90^\circ </math>,
-!Dual elongated square pyramid
+* The dihedral angle of an equilateral square pyramid between square and triangle is <math> \arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>. Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is <math display="block"> \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. </math>
-!Net of dual
-|- valign=top
-|[[File:Dual elongated square pyramid.png|160px]]
-|[[File:Dual elongated square pyramid net.png|160px]]
-|}
-==Related polyhedra and honeycombs==
-The elongated square pyramid can form a [[tessellation of space]] with [[Tetrahedron|tetrahedra]],<ref>http://woodenpolyhedra.web.fc2.com/J8.html</ref> similar to a modified [[tetrahedral-octahedral honeycomb]].
 == See also==
@@ Line 37: / Line 40: @@
 ==References==
-{{Reflist}}
+{{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="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="pye">{{cite journal
+ | last = Sapiña | first = R.
+ | title = Area and volume of the Johnson solid <math> J_{8} </math>
+ | url = https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html
+ | issn = 2659-9899
+ | access-date = 2020-09-09
+ | language = es
+ | journal = Problemas y Ecuaciones
+}}</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
+ | publisher = Hindustan Book Agency
+ | page = 84&ndash;89
+ | isbn = 978-93-86279-06-4
+ | doi = 10.1007/978-93-86279-06-4
+}}</ref>
+<ref name="timofeenko-2010">{{cite journal
+ | last = Timofeenko | first = A. V.
+ | year = 2010
+ | title = Junction of Non-composite Polyhedra
+ | journal = St. Petersburg Mathematical Journal
+ | volume = 21 | issue = 3 | pages = 483–512
+ | doi = 10.1090/S1061-0022-10-01105-2
+ | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
+}}</ref>
+<ref name="uehara">{{cite book
+ | last = Uehara | first = Ryuhei
+ | year = 2020
+ | title = Introduction to Computational Origami: The World of New Computational Geometry
+ | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
+ | page = 62
+ | publisher = Springer
+ | isbn = 978-981-15-4470-5
+ | doi = 10.1007/978-981-15-4470-5
+ | s2cid = 220150682
+}}</ref>
+}}
 ==External links==
 * {{mathworld2 | urlname2 = ElongatedSquarePyramid  | title2 = Elongated square pyramid| urlname = JohnsonSolid | title = Johnson solid}}
+{{Johnson solids navigator}}
-{{Polyhedron-stub}}
+[[Category:Composite polyhedron]]
 [[Category:Johnson solids]]
 [[Category:Self-dual polyhedra]]
-[[Category:Pyramids and bipyramids]]
+[[Category:Pyramids (geometry)]]

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)