Tetradecahedron: Difference between revisions
Content deleted Content added
Punctuation corrections required by WP:MOS. |
Rm dubious tag with dubious rationale. Quoted text does not falsify this claim and many polyhedra meeting this description are described in the article text. |
||
| (36 intermediate revisions by 26 users not shown) | |||
| Line 1: | Line 1: | ||
{{short description|Polyhedron with 14 faces}} |
|||
[[image:Space-filling tetrakaidecahedron.png|thumb|right|240px|A tetradecahedron with |
[[image:Space-filling tetrakaidecahedron.png|thumb|right|240px|A tetradecahedron with D<sub>2d</sub>-symmetry, existing in the [[Weaire–Phelan structure]]]] |
||
A '''tetradecahedron''' is a [[polyhedron]] with 14 [[Face (geometry)|faces]]. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with [[regular polygon]] faces. |
A '''tetradecahedron''' is a [[polyhedron]] with 14 [[Face (geometry)|faces]]. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with [[regular polygon]] faces. |
||
A tetradecahedron is sometimes called a '''tetrakaidecahedron'''.<ref>{{MathWorld | id=Tetradecahedron | title=Tetradecahedron | access-date={{TODAY}}}}</ref><ref>{{Cite web |url=http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t099.htm |title=Tetradecahedron |access-date=29 October 2007 |archive-date=18 July 2011 |archive-url=https://web.archive.org/web/20110718131418/http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t099.htm |url-status=dead }}</ref> No difference in meaning is ascribed.<ref>{{MathWorld | id=Tetrakaidecahedron | title=Tetrakaidecahedron | access-date={{TODAY}}}}</ref><ref>{{Cite web |url=http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t117.htm |title=Tetrakaidecahedron |access-date=29 October 2007 |archive-date=28 September 2011 |archive-url=https://web.archive.org/web/20110928131654/http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t117.htm |url-status=dead }}</ref> The Greek word ''[[Kai (conjunction)|kai]]'' means 'and'. There is evidence that mammalian [[epidermis|epidermal]] cells are shaped like flattened tetrakaidecahedra, an idea first suggested by [[Lord Kelvin]].<ref>{{Cite journal|doi = 10.7554/eLife.19593|title = Epidermal cell turnover across tight junctions based on Kelvin's tetrakaidecahedron cell shape|year = 2016|last1 = Yokouchi|first1 = Mariko|last2 = Atsugi|first2 = Toru|last3 = Logtestijn|first3 = Mark van|last4 = Tanaka|first4 = Reiko J.|last5 = Kajimura|first5 = Mayumi|last6 = Suematsu|first6 = Makoto|last7 = Furuse|first7 = Mikio|last8 = Amagai|first8 = Masayuki|last9 = Kubo|first9 = Akiharu|journal = eLife|volume = 5|pmid = 27894419|pmc = 5127639 | doi-access=free }}</ref> The polyhedron can also be found in soap bubbles and in [[Sintering|sintered ceramics]], due to its ability to [[Honeycomb (geometry)|tesselate]] in 3D space.<ref>{{Cite web |date=2020-07-26 |title=Most space Filling Structure in the World! – Tetradecahedron |url=https://ardentmetallurgist.wordpress.com/2020/07/26/tetradecahedron/ |access-date=2022-11-15 |website=Ardent Metallurgist |language=en}}</ref><ref>{{Cite journal |last1=Wey |first1=Ming-Yen |last2=Tseng |first2=Hui-Hsin |last3=Chiang |first3=Chian-kai |date=2014-03-01 |title=Improving the mechanical strength and gas separation performance of CMS membranes by simply sintering treatment of α-Al2O3 support |url=https://www.sciencedirect.com/science/article/pii/S0376738813009356 |journal=Journal of Membrane Science |language=en |volume=453 |pages=603–613 |doi=10.1016/j.memsci.2013.11.039 |issn=0376-7388}}</ref> |
|||
A tetradecahedron is sometimes called a '''tetrakaidecahedron'''.<ref>[http://mathworld.wolfram.com/Tetradecahedron.html Tetradecahedron]</ref><ref>[http://web.archive.org/web/20110718131418/http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t099.htm Tetradecahedron]</ref> No difference in meaning is ascribed.<ref>[http://mathworld.wolfram.com/Tetrakaidecahedron.html Tetrakaidecahedron]</ref><ref>[http://web.archive.org/web/20110928131654/http://bbs.sachina.pku.edu.cn/stat/math_world/math/t/t117.htm Tetrakaidecahedron]</ref> The Greek word ''[[Kai (conjunction)|kai]]'' means 'and'. |
|||
==Convex== |
== Convex == |
||
There are 1,496,225,352 topologically distinct ''convex'' tetradecahedra, excluding mirror images, having at least 9 vertices.<ref>[http://www.numericana.com/data/polycount.htm Counting polyhedra]</ref> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) |
There are 1,496,225,352 topologically distinct ''convex'' tetradecahedra, excluding mirror images, having at least 9 vertices.<ref>[http://www.numericana.com/data/polycount.htm Counting polyhedra]</ref> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) |
||
== Examples== |
== Examples == |
||
An incomplete list of forms includes: |
An incomplete list of forms includes: |
||
Tetradecahedra having all [[regular polygon]]al faces (all exist in irregular-faced forms as well): |
Tetradecahedra having all [[regular polygon]]al faces (all exist in irregular-faced forms as well): |
||
*[[Archimedean solid]]s: |
* [[Archimedean solid]]s: |
||
**[[Cuboctahedron]] (8 [[triangles]], 6 [[Square (geometry)|squares]]) |
** [[Cuboctahedron]] (8 [[triangles]], 6 [[Square (geometry)|squares]]) |
||
**[[Truncated cube]] (8 triangles, 6 [[octagon]]s) |
** [[Truncated cube]] (8 triangles, 6 [[octagon]]s) |
||
**[[Truncated octahedron]] (6 squares, 8 [[hexagon]]s) |
** [[Truncated octahedron]] (6 squares, 8 [[hexagon]]s) |
||
*[[Prism (geometry)|Prisms]] and [[antiprism]]s: |
* [[Prism (geometry)|Prisms]] and [[antiprism]]s: |
||
**[[Dodecagonal prism]] (12 squares, 2 [[dodecagon]]s) |
** [[Dodecagonal prism]] (12 squares, 2 [[dodecagon]]s) |
||
**[[Hexagonal antiprism]] (12 triangles, 2 hexagons) |
** [[Hexagonal antiprism]] (12 triangles, 2 hexagons) |
||
*[[Johnson solid]]s: |
* [[Johnson solid]]s: |
||
**J<sub>18</sub>: [[Elongated triangular cupola]] (4 triangles, 9 squares, 1 hexagon) |
** J<sub>18</sub>: [[Elongated triangular cupola]] (4 triangles, 9 squares, 1 hexagon) |
||
**J<sub>27</sub>: [[Triangular orthobicupola]] (8 triangles, 6 squares) |
** J<sub>27</sub>: [[Triangular orthobicupola]] (8 triangles, 6 squares) |
||
**J<sub>51</sub>: [[Triaugmented triangular prism]] (14 triangles) |
** J<sub>51</sub>: [[Triaugmented triangular prism]] (14 triangles) |
||
**J<sub>55</sub>: [[Parabiaugmented hexagonal prism]] (8 triangles, 4 squares, 2 hexagons) |
** J<sub>55</sub>: [[Parabiaugmented hexagonal prism]] (8 triangles, 4 squares, 2 hexagons) |
||
**J<sub>56</sub>: [[Metabiaugmented hexagonal prism]] (8 triangles, 4 squares, 2 hexagons) |
** J<sub>56</sub>: [[Metabiaugmented hexagonal prism]] (8 triangles, 4 squares, 2 hexagons) |
||
**J<sub>65</sub>: [[Augmented truncated tetrahedron]] (8 triangles, 3 squares, 3 hexagons) |
** J<sub>65</sub>: [[Augmented truncated tetrahedron]] (8 triangles, 3 squares, 3 hexagons) |
||
**J<sub>86</sub>: [[Sphenocorona]] (12 triangles, 2 squares) |
** J<sub>86</sub>: [[Sphenocorona]] (12 triangles, 2 squares) |
||
**J<sub>91</sub>: [[Bilunabirotunda]] (8 triangles, 2 squares, 4 pentagons) |
** J<sub>91</sub>: [[Bilunabirotunda]] (8 triangles, 2 squares, 4 pentagons) |
||
Tetradecahedra having at least one irregular face: |
Tetradecahedra having at least one irregular face: |
||
*[[Heptagonal |
* [[Heptagonal bipyramid]] (14 triangles) (see ''[[Dipyramid]]'') |
||
*[[Heptagonal trapezohedron]] (14 [[Kite (geometry)|kites]]) (see [[Trapezohedron]]) |
* [[Heptagonal trapezohedron]] (14 [[Kite (geometry)|kites]]) (see ''[[Trapezohedron]]'') |
||
*[[Tridecagonal pyramid]] (13 triangles, 1 regular [[tridecagon]]) (see [[Pyramid (geometry)]]) |
* [[Tridecagonal pyramid]] (13 triangles, 1 regular [[tridecagon]]) (see ''[[Pyramid (geometry)]]'') |
||
*[[Dissected regular icosahedron]] (the vertex figure of the [[grand antiprism]]) (12 equilateral triangles and 2 [[trapezoid]]s) |
* [[Dissected regular icosahedron]] (the vertex figure of the [[grand antiprism]]) (12 equilateral triangles and 2 [[trapezoid]]s) |
||
*[[Hexagonal truncated trapezohedron]]: (12 [[pentagon]]s, 2 hexagons)<br>Includes an optimal space-filling shape in foams (see [[Weaire–Phelan structure]]) and in the crystal structure of [[ |
* [[Hexagonal truncated trapezohedron]]: (12 [[pentagon]]s, 2 hexagons)<br>Includes an optimal space-filling shape in foams (see ''[[Weaire–Phelan structure]]'') and in the crystal structure of [[clathrate hydrate]] (see illustration, next to label 5<sup>12</sup>6<sup>2</sup>) |
||
* [[Hexagonal bifrustum]] (12 trapezoids, 2 hexagons) |
|||
* The [[One pound (British_coin)|British £1 coin]] in circulation from 2017 – with twelve edges and two faces – is an irregular dodecagonal prism, when one disregards the edging and relief features.<ref>{{Cite web|url=https://www.royalmint.com/new-pound-coin/|title = New Pound Coin | the Royal Mint}}</ref> |
|||
==See also== |
== See also == |
||
* [[Császár polyhedron]] – A nonconvex tetradecahedron of all triangle faces |
* [[Császár polyhedron]] – A nonconvex tetradecahedron of all triangle faces |
||
* [[Steffen's polyhedron]] – A [[flexible polyhedron|flexible]] tetradecahedron |
|||
*[[Permutohedron]] |
|||
* [[Permutohedron]] – A polyhedron that can be defined in any dimension and equals the truncated octahedron in three dimensions |
|||
== References== |
== References == |
||
{{reflist}} |
{{reflist}} |
||
* |
* {{webarchive | url=https://web.archive.org/web/20050212114016/http://members.aol.com/Polycell/what.html | date=12 February 2005 | title="What Are Polyhedra?"}}, with Greek Numerical Prefixes |
||
==External links== |
== External links == |
||
*{{MathWorld | urlname=Tetradecahedron | title=Tetradecahedron}} |
* {{MathWorld | urlname=Tetradecahedron | title=Tetradecahedron}} |
||
*[http://dmccooey.com/polyhedra/SelfDualTetradecahedron1.html Self-dual tetradecahedra] |
* [http://dmccooey.com/polyhedra/SelfDualTetradecahedron1.html Self-dual tetradecahedra] |
||
{{Polyhedra}} |
{{Polyhedra}} |
||
Latest revision as of 01:26, 6 August 2024
A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.
A tetradecahedron is sometimes called a tetrakaidecahedron.[1][2] No difference in meaning is ascribed.[3][4] The Greek word kai means 'and'. There is evidence that mammalian epidermal cells are shaped like flattened tetrakaidecahedra, an idea first suggested by Lord Kelvin.[5] The polyhedron can also be found in soap bubbles and in sintered ceramics, due to its ability to tesselate in 3D space.[6][7]
Convex
[edit]There are 1,496,225,352 topologically distinct convex tetradecahedra, excluding mirror images, having at least 9 vertices.[8] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Examples
[edit]An incomplete list of forms includes:
Tetradecahedra having all regular polygonal faces (all exist in irregular-faced forms as well):
- Archimedean solids:
- Cuboctahedron (8 triangles, 6 squares)
- Truncated cube (8 triangles, 6 octagons)
- Truncated octahedron (6 squares, 8 hexagons)
- Prisms and antiprisms:
- Dodecagonal prism (12 squares, 2 dodecagons)
- Hexagonal antiprism (12 triangles, 2 hexagons)
- Johnson solids:
- J18: Elongated triangular cupola (4 triangles, 9 squares, 1 hexagon)
- J27: Triangular orthobicupola (8 triangles, 6 squares)
- J51: Triaugmented triangular prism (14 triangles)
- J55: Parabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
- J56: Metabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
- J65: Augmented truncated tetrahedron (8 triangles, 3 squares, 3 hexagons)
- J86: Sphenocorona (12 triangles, 2 squares)
- J91: Bilunabirotunda (8 triangles, 2 squares, 4 pentagons)
Tetradecahedra having at least one irregular face:
- Heptagonal bipyramid (14 triangles) (see Dipyramid)
- Heptagonal trapezohedron (14 kites) (see Trapezohedron)
- Tridecagonal pyramid (13 triangles, 1 regular tridecagon) (see Pyramid (geometry))
- Dissected regular icosahedron (the vertex figure of the grand antiprism) (12 equilateral triangles and 2 trapezoids)
- Hexagonal truncated trapezohedron: (12 pentagons, 2 hexagons)
Includes an optimal space-filling shape in foams (see Weaire–Phelan structure) and in the crystal structure of clathrate hydrate (see illustration, next to label 51262) - Hexagonal bifrustum (12 trapezoids, 2 hexagons)
- The British £1 coin in circulation from 2017 – with twelve edges and two faces – is an irregular dodecagonal prism, when one disregards the edging and relief features.[9]
See also
[edit]- Császár polyhedron – A nonconvex tetradecahedron of all triangle faces
- Steffen's polyhedron – A flexible tetradecahedron
- Permutohedron – A polyhedron that can be defined in any dimension and equals the truncated octahedron in three dimensions
References
[edit]- ^ Weisstein, Eric W. "Tetradecahedron". MathWorld. Retrieved 28 December 2024.
- ^ "Tetradecahedron". Archived from the original on 18 July 2011. Retrieved 29 October 2007.
- ^ Weisstein, Eric W. "Tetrakaidecahedron". MathWorld. Retrieved 28 December 2024.
- ^ "Tetrakaidecahedron". Archived from the original on 28 September 2011. Retrieved 29 October 2007.
- ^ Yokouchi, Mariko; Atsugi, Toru; Logtestijn, Mark van; Tanaka, Reiko J.; Kajimura, Mayumi; Suematsu, Makoto; Furuse, Mikio; Amagai, Masayuki; Kubo, Akiharu (2016). "Epidermal cell turnover across tight junctions based on Kelvin's tetrakaidecahedron cell shape". eLife. 5. doi:10.7554/eLife.19593. PMC 5127639. PMID 27894419.
- ^ "Most space Filling Structure in the World! – Tetradecahedron". Ardent Metallurgist. 2020-07-26. Retrieved 2022-11-15.
- ^ Wey, Ming-Yen; Tseng, Hui-Hsin; Chiang, Chian-kai (2014-03-01). "Improving the mechanical strength and gas separation performance of CMS membranes by simply sintering treatment of α-Al2O3 support". Journal of Membrane Science. 453: 603–613. doi:10.1016/j.memsci.2013.11.039. ISSN 0376-7388.
- ^ Counting polyhedra
- ^ "New Pound Coin | the Royal Mint".
- "What Are Polyhedra?" at the Wayback Machine (archived 12 February 2005), with Greek Numerical Prefixes