Natural number, composite number
Natural number
14 (fourteen ) is thenatural number following13 and preceding15 .
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14 is asquare pyramidal number . Fourteen is the seventhcomposite number .
14 is the third distinctsemiprime ,[ 1] 2 × q {\displaystyle 2\times q} q {\displaystyle q} semiprimes (14,15 ); the next such cluster is (21 ,22 ), members whose sum is the fourteenth prime number,43 .
14 has analiquot sum of10 , within analiquot sequence of two composite numbers (14,10 ,8 ,7 ,1 , 0) in the prime7 -aliquot tree.
14 is the thirdcompanion Pell number and the fourthCatalan number .[ 2] [ 3] n {\displaystyle n} Euler totient φ ( x ) = n {\displaystyle \varphi (x)=n} nontotient .[ 4]
According to theShapiro inequality , 14 is the least numbern {\displaystyle n} x 1 {\displaystyle x_{1}} x 2 {\displaystyle x_{2}} x 3 {\displaystyle x_{3}} [ 5]
∑ i = 1 n x i x i + 1 + x i + 2 < n 2 , {\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}<{\frac {n}{2}},} withx n + 1 = x 1 {\displaystyle x_{n+1}=x_{1}} x n + 2 = x 2 . {\displaystyle x_{n+2}=x_{2}.}
Aset ofreal numbers to which it is appliedclosure andcomplement operations in any possible sequence generates14 distinct sets.[ 6] topological space ; seeKuratowski's closure-complement problem .
There are fourteeneven numbers that cannot be expressed as the sum of two oddcomposite numbers :
{ 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 20 , 22 , 26 , 28 , 32 , 38 } {\displaystyle \{2,4,6,8,10,12,14,16,20,22,26,28,32,38\}} where 14 is the seventh such number.[ 7]
14 is the number ofequilateral triangles that are formed by thesides anddiagonals of aregular six-sidedhexagon .[ 8] hexagonal lattice , 14 is also the number of fixed two-dimensionaltriangular -celledpolyiamonds with four cells.[ 9]
14 is the number ofelements in aregular heptagon (where there are sevenvertices and edges), and the total number ofdiagonals between all its vertices.
There are fourteen polygons that can fill aplane-vertex tiling , where five polygons tile the planeuniformly , and nine others only tile the plane alongside irregular polygons.[ 10] [ 11]
The fundamental domain of theKlein quartic is a regular hyperbolic 14-sidedtetradecagon , with an area of8 π {\displaystyle 8\pi } TheKlein quartic is a compactRiemann surface of genus 3 that has the largest possibleautomorphism group order of its kind (of order168 ) whose fundamental domain is a regular hyperbolic 14-sidedtetradecagon , with an area of8 π {\displaystyle 8\pi } Gauss-Bonnet theorem .
Several distinguishedpolyhedra inthree dimensions contain fourteenfaces orvertices asfacets :
Thecuboctahedron , one of twoquasiregular polyhedra , has 14 faces and is the onlyuniform polyhedron withradial equilateral symmetry .[ 12] Therhombic dodecahedron ,dual to the cuboctahedron, contains 14 vertices and is the onlyCatalan solid that cantessellate space.[ 13] Thetruncated octahedron contains 14 faces, is thepermutohedron of order four, and the onlyArchimedean solid to tessellate space. Thedodecagonal prism , which is the largestprism that can tessellate space alongside other uniform prisms, has 14 faces. TheSzilassi polyhedron and its dual, theCsászár polyhedron , are the simplesttoroidal polyhedra ; they have 14 vertices and 14 triangular faces, respectively.[ 14] [ 15] Steffen's polyhedron , the simplestflexible polyhedron without self-crossings, has 14 triangular faces.[ 16] A regulartetrahedron cell , the simplestuniform polyhedron andPlatonic solid , is made up of a total of14 elements : 4edges , 6 vertices, and 4 faces.
Szilassi's polyhedron and the tetrahedron are the only two known polyhedra where each face shares an edge with each other face, while Császár's polyhedron and the tetrahedron are the only two known polyhedra with a continuousmanifold boundary that do not contain anydiagonals . Two tetrahedra that are joined by a common edge whose four adjacent and opposite faces are replaced with two specific seven-facedcrinkles will create a new flexible polyhedron, with a total of 14 possibleclashes where faces can meet.[ 17] pp.10-11,14 [ 17] p.16 2-dof flexible polyhedron , each hinge will only have a total range of motion of 14 degrees.[ 17] p.139 14 is also the root (non-unitary) trivialstella octangula number , where twoself-dual tetrahedra are represented throughfigurate numbers , while also being the first non-trivialsquare pyramidal number (after5 );[ 18] [ 19] Johnson solids is thesquare pyramid J 1 . {\displaystyle J_{1}.} [ a] semi-regular polyhedra , when thepseudorhombicuboctahedron is included as a non-vertex transitive Archimedean solid (a lower class of polyhedra that follow the five Platonic solids).[ 20] [ 21] [ b]
Fourteen possibleBravais lattices exist that fill three-dimensional space.[ 22]
Theexceptional Lie algebra G2 faithful representation in fourteen dimensions. It is theautomorphism group of theoctonions O {\displaystyle \mathbb {O} } homeomorphic to thezero divisors with entries ofunit norm in thesedenions ,S {\displaystyle \mathbb {S} } [ 23] [ 24]
Riemann zeta function [ edit ] Thefloor of theimaginary part of the first non-trivial zero in theRiemann zeta function is14 {\displaystyle 14} [ 25] nearest integer value,[ 26] 14.1347251417 … {\displaystyle 14.1347251417\ldots } [ 27] [ 28]
In religion and mythology [ edit ] There is a fourteen-point silver star marking the traditional spot of Jesus’birth in theBasilica of the Nativity inBethlehem . According to thegenealogy of Jesus in theGospel of Matthew , “...there were fourteen generations in all fromAbraham toDavid , fourteen generations from David to theexile to Babylon , and fourteen from the exile to theMessiah ” (Matthew 1:17 ).
In Islam, 14 has a special significance because ofthe Fourteen Infallibles who are especially revered and important inTwelver Shi'ism . They are all considered to be infallible by Twelvers alongsidethe Prophets of Islam , however these fourteen are said to have a greater significance and closeness toGod.
These fourteen include:
Prophet Muhammad (SAWA) His daughter,Lady Fatima (SA) Her husband,Imam Ali (AS) His son,Imam Hasan (AS) His brother,Imam Husayn (AS) His son,Imam Ali al-Sajjad (AS) His son,Imam Muhammad al-Baqir (AS) His son,Imam Ja'far al-Sadiq (AS) His son,Imam Musa al-Kazim (AS) His son,Imam Ali al-Rida (AS) His son,Imam Muhammad al-Jawad (AS) His son,Imam Ali al-Hadi (AS) His son,Imam Hasan al-Askari (AS) His son,Imam Muhammad al-Mahdi (AJTFS) The number 14 was linked toŠumugan andNergal .[ 29]
Fourteen is:
^ Furthermore, the square pyramid can be attached to uniform and non-uniform polyhedra (such as other Johnson solids) to generatefourteen other Johnson solids :J 8 J 10 J 15 J 17 J 49 J 50 J 51 J 52 J 53 J 54 J 55 J 56 J 57 J 87 ^ Where the tetrahedron — which isself-dual ,inscribable inside all other Platonic solids, and vice versa — contains fourteen elements, there exist thirteen uniform polyhedra that contain fourteen faces (U 09 U 76i U 08 U 77c U 07 U 76d U 77d U 78b U 78c U 79b U 79c U 80b U 19 ^ Sloane, N. J. A. (ed.)."Sequence A001358" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ "Sloane's A002203 : Companion Pell numbers" .The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2016-06-01 .^ "Sloane's A000108 : Catalan numbers" .The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2016-06-01 .^ "Sloane's A005277 : Nontotients" .The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2016-06-01 .^ Troesch, B. A. (July 1975)."On Shapiro's Cyclic Inequality for N = 13" (PDF) .Mathematics of Computation 45 (171): 199.doi :10.1090/S0025-5718-1985-0790653-0 MR 0790653 .S2CID 51803624 .Zbl 0593.26012 . ^ Kelley, John (1955).General Topology ISBN 9780387901251 OCLC 10277303 .{{cite book }}:ISBN / Date incompatibility (help ) ^ Sloane, N. J. A. (ed.)."Sequence A118081 (Even numbers that can't be represented as the sum of two odd composite numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-08-03 .^ Sloane, N. J. A. (ed.)."Sequence A238822 (Number of equilateral triangles bounded by the sides and diagonals of a regular 3n-gon.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-05-05 .^ Sloane, N. J. A. (ed.)."Sequence A001420 (Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-05-15 .^ Grünbaum, Branko ;Shepard, Geoffrey (November 1977)."Tilings by Regular Polygons" (PDF) .Mathematics Magazine 50 (5). Taylor & Francis, Ltd.: 231.doi :10.2307/2689529 .JSTOR 2689529 .S2CID 123776612 .Zbl 0385.51006 . Archived fromthe original (PDF) on 2016-03-03. Retrieved2023-01-18 .^ Baez, John C. (February 2015)."Pentagon-Decagon Packing" .AMS Blogs .American Mathematical Society . Retrieved2023-01-18 .^ Coxeter, H.S.M. (1973). "Chapter 2: Regular polyhedra".Regular Polytopes 18– 19.ISBN 0-486-61480-8 OCLC 798003 .^ Williams, Robert (1979)."Chapter 5: Polyhedra Packing and Space Filling" The Geometrical Foundation of Natural Structure: A Source Book of Design . New York:Dover Publications, Inc. p. 168.ISBN 9780486237299 OCLC 5939651 .S2CID 108409770 . ^ Szilassi, Lajos (1986)."Regular toroids" (PDF) .Structural Topology .13 :69– 80.Zbl 0605.52002 . ^ Császár, Ákos (1949)."A polyhedron without diagonals" (PDF) .Acta Scientiarum Mathematicarum (Szeged) .13 :140– 142. Archived fromthe original (PDF) on 2017-09-18.^ Lijingjiao, Iila; et al. (2015)."Optimizing the Steffen flexible polyhedron" (PDF) .Proceedings of the International Association for Shell and Spatial Structures (Future Visions Symposium) . Amsterdam: IASS.doi :10.17863/CAM.26518 .S2CID 125747070 . ^a b c Li, Jingjiao (2018).Flexible Polyhedra: Exploring finite mechanisms of triangulated polyhedra (PDF) (Ph.D. Thesis).University of Cambridge , Department of Engineering. pp. xvii,1– 156.doi :10.17863/CAM.18803 S2CID 204175310 . ^ Sloane, N. J. A. (ed.)."Sequence A007588 (Stella octangula numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-01-18 .^ Sloane, N. J. A. (ed.)."Sequence A000330 (Square pyramidal numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-01-18 .^ Grünbaum, Branko (2009)."An enduring error" .Elemente der Mathematik 64 (3). Helsinki:European Mathematical Society :89– 101.doi :10.4171/EM/120 MR 2520469 .S2CID 119739774 .Zbl 1176.52002 .^ Hartley, Michael I.; Williams, Gordon I. (2010)."Representing the sporadic Archimedean polyhedra as abstract polytopes" .Discrete Mathematics 310 (12). Amsterdam:Elsevier :1835– 1844.arXiv :0910.2445 Bibcode :2009arXiv0910.2445H .doi :10.1016/j.disc.2010.01.012 MR 2610288 .S2CID 14912118 .Zbl 1192.52018 . ^ Sloane, N. J. A. (ed.)."Sequence A256413 (Number of n-dimensional Bravais lattices.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-01-18 .^ Baez, John C. (2002)."The Octonions" .Bulletin of the American Mathematical Society 39 (2): 186.arXiv :math/0105155 doi :10.1090/S0273-0979-01-00934-X .MR 1886087 .S2CID 586512 .Zbl 1026.17001 .^ Moreno, Guillermo (1998), "The zero divisors of the Cayley–Dickson algebras over the real numbers",Bol. Soc. Mat. Mexicana , Series 3,4 (1):13– 28,arXiv :q-alg/9710013 Bibcode :1997q.alg....10013G ,MR 1625585 ,S2CID 20191410 ,Zbl 1006.17005 ^ Sloane, N. J. A. (ed.)."Sequence A013629 (Floor of imaginary parts of nontrivial zeros of Riemann zeta function.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-16 .^ Sloane, N. J. A. (ed.)."Sequence A002410 (Nearest integer to imaginary part of n-th zero of Riemann zeta function.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-16 .^ Sloane, N. J. A. (ed.)."Sequence A058303 (Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-16 .^ Odlyzko, Andrew ."The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]" .Andrew Odlyzko: Home Page .UMN CSE . Retrieved2024-01-16 .^ Wiggermann 1998 , p. 222.
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