14 (number)

"14th" redirects here. For the electronic music duo, see 14th (band).

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Cardinal	fourteen
Ordinal	14th (fourteenth)
Numeral system	tetradecimal
Factorization	2 × 7
Divisors	1, 2, 7, 14
Greek numeral	ΙΔ´
Roman numeral	XIV
Greek prefix	tetrakaideca-
Latin prefix	quattuordec-
Binary	11102
Ternary	1123
Senary	226
Octal	168
Duodecimal	1212
Hexadecimal	E16

14 (fourteen) is a natural number following 13 and preceding 15.

In relation to the word "four" (4), 14 is spelled "fourteen".

Mathematics

Fourteen is the seventh composite number. It is specifically, the third distinct semiprime, ^[1] being the third of the form ${\displaystyle 2\times q}$ (where ${\displaystyle q}$ is a higher prime). More specifically, it is the first member of the second cluster of two discrete semiprimes (14, 15); the next such cluster is (21, 22), members whose sum is the fourteenth prime number, 43. It has an aliquot sum of 8, within an aliquot sequence of two composite numbers (14, 8, 7, 1, 0) in the prime 7-aliquot tree.

Properties

14 is the third Companion Pell number, and the fourth Catalan number. ^{[2] [3]} It is the lowest even ${\displaystyle n}$ for which the Euler totient ${\displaystyle \varphi (x)=n}$ has no solution, making it the first even nontotient. ^[4]

According to the Shapiro inequality, 14 is the least number ${\displaystyle n}$ such that there exist ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, ${\displaystyle x_{3}}$, where: ^[5]

${\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}<{\frac {n}{2}},}$

with ${\displaystyle x_{n+1}=x_{1}}$ and ${\displaystyle x_{n+2}=x_{2}.}$

A set of real numbers to which it is applied closure and complement operations in any possible sequence generates 14 distinct sets. ^[6] This holds even if the reals are replaced by a more general topological space; see Kuratowski's closure-complement problem.

Polygons

There are fourteen polygons that can fill a plane-vertex tiling, where five polygons tile the plane uniformly, and nine others only tile the plane alongside irregular polygons. ^{[7] [8]}

The fundamental domain of the Klein quartic is a regular hyperbolic 14-sided tetradecagon, with an area of ${\displaystyle 8\pi }$.

The Klein quartic is a compact Riemann surface of genus 3 that has the largest possible automorphism group order of its kind (of order 168) whose fundamental domain is a regular hyperbolic 14-sided tetradecagon, with an area of ${\displaystyle 8\pi }$ by the Gauss-Bonnet theorem.

Solids

Several distinguished polyhedra in three dimensions contain fourteen faces or vertices as facets:

• The cuboctahedron, one of two quasiregular polyhedra, has 14 faces and is the only uniform polyhedron with radial equilateral symmetry. ^[9]
• The rhombic dodecahedron, dual to the cuboctahedron, contains 14 vertices and is the only Catalan solid that can tessellate space. ^[10]
• The truncated octahedron contains 14 faces, is the permutohedron of order four, and the only Archimedean solid to tessellate space.
• The dodecagonal prism, which is the largest prism that can tessellate space alongside other uniform prisms, has 14 faces.
• The Szilassi polyhedron and its dual, the Császár polyhedron, are the simplest toroidal polyhedra; they have 14 vertices and 14 triangular faces, respectively. ^{[11] [12]}
• Steffen's polyhedron, the simplest flexible polyhedron without self-crossings, has 14 triangular faces. ^[13]

A regular tetrahedron cell, the simplest uniform polyhedron and Platonic solid, is made up of a total of 14 elements: 4 edges, 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 continuous manifold boundary that do not contain any diagonals.
• Two tetrahedra that are joined by a common edge whose four adjacent and opposite faces are replaced with two specific seven-faced crinkles will create a new flexible polyhedron, with a total of 14 possible clashes where faces can meet. ^{[14] pp.10-11,14} This is the second simplest known triangular flexible polyhedron, after Steffen's polyhedron. ^{[14] p.16} If three tetrahedra are joined at two separate opposing edges and made into a single flexible polyhedron, called a 2-dof flexible polyhedron, each hinge will only have a total range of motion of 14 degrees. ^{[14] p.139}

14 is also the root (non-unitary) trivial stella octangula number, where two self-dual tetrahedra are represented through figurate numbers, while also being the first non-trivial square pyramidal number (after 5); ^{[15] [16]} the simplest of the ninety-two Johnson solids is the square pyramid ${\displaystyle J_{1}.}$ ^{[lower-alpha 1]} There are a total of fourteen semi-regular polyhedra, when the pseudorhombicuboctahedron is included as a non-vertex transitive Archimedean solid (a lower class of polyhedra that follow the five Platonic solids). ^{[17] [18] [lower-alpha 2]}

Fourteen possible Bravais lattices exist that fill three-dimensional space. ^[19]

G₂

The exceptional Lie algebra G₂ is the simplest of five such algebras, with a minimal faithful representation in fourteen dimensions. It is the automorphism group of the octonions ${\displaystyle \mathbb {O} }$, and holds a compact form homeomorphic to the zero divisors with entries of unit norm in the sedenions, ${\displaystyle \mathbb {S} }$. ^{[20] [21]}

Riemann zeta function

The floor of the imaginary part of the first non-trivial zero in the Riemann zeta function is ${\displaystyle 14}$, ^[22] in equivalence with its nearest integer value, ^[23] from an approximation of ${\displaystyle 14.1347251417\ldots }$ ^{[24] [25]}

In science

Chemistry

14 is the atomic number of silicon, and the approximate atomic weight of nitrogen. The maximum number of electrons that can fit in an f sublevel is fourteen.

In religion and mythology

Christianity

According to the Gospel of Matthew "there were fourteen generations in all from Abraham to David, fourteen generations from David to the exile to Babylon, and fourteen from the exile to the Messiah". (Matthew 1, 17)

Islam

The number of Muqattaʿat in the Quran.

Mythology

The number of pieces the body of Osiris was torn into by his fratricidal brother Set.

The number 14 was regarded as connected to Šumugan and Nergal. ^[26]

In other fields

Fourteen is:

• The number of days in a fortnight.
• The Fourteenth Amendment to the United States Constitution gave citizenship to those of African descent, in a post-Civil War (Reconstruction) measure aimed at restoring the rights of slaves.
• The number of lines in a sonnet. ^[27]
• The Piano Sonata No. 14, also known as Moonlight Sonata, is one of the most famous piano sonatas composed by Ludwig van Beethoven.

Notes

1. Furthermore, the square pyramid can be attached to uniform and non-uniform polyhedra (such as other Johnson solids) to generate fourteen other Johnson solids: J₈, J₁₀, J₁₅, J₁₇, J₄₉, J₅₀, J₅₁, J₅₂, J₅₃, J₅₄, J₅₅, J₅₆, J₅₇, and J₈₇.
2. Where the tetrahedron — which is self-dual, inscribable inside all other Platonic solids, and vice-versa — contains fourteen elements, there exist thirteen uniform polyhedra that contain fourteen faces (U₀₉, U_76i, U₀₈, U_77c, U₀₇), vertices (U_76d, U_77d, U_78b, U_78c, U_79b, U_79c, U_80b) or edges (U₁₉).

References

1. Sloane, N. J. A. (ed.). "SequenceA001358". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. "Sloane's A002203 : Companion Pell numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
3. "Sloane's A000108 : Catalan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
4. "Sloane's A005277 : Nontotients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
5. 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.
6. Kelley, John (1955). General Topology . New York: Van Nostrand. p. 57. ISBN   9780387901251. OCLC   10277303.
7. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine . Taylor & Francis, Ltd. 50 (5): 231. doi:10.2307/2689529. JSTOR   2689529. S2CID   123776612. Zbl   0385.51006.
8. Baez, John C. (February 2015). "Pentagon-Decagon Packing". AMS Blogs. American Mathematical Society . Retrieved 2023-01-18.
9. Coxeter, H.S.M. (1973). "Chapter 2: Regular polyhedra". Regular Polytopes (3rd ed.). New York: Dover. pp. 18–19. ISBN   0-486-61480-8. OCLC   798003.
10. 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.
11. Szilassi, Lajos (1986). "Regular toroids" (PDF). Structural Topology. 13: 69–80. Zbl   0605.52002.
12. Császár, Ákos (1949). "A polyhedron without diagonals" (PDF). Acta Scientiarum Mathematicarum (Szeged). 13: 140–142. Archived from the original (PDF) on 2017-09-18.
13. 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.
14. 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.
15. Sloane, N. J. A. (ed.). "SequenceA007588(Stella octangula numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-18.
16. Sloane, N. J. A. (ed.). "SequenceA000330(Square pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-18.
17. Grünbaum, Branko (2009). "An enduring error". Elemente der Mathematik . Helsinki: European Mathematical Society. 64 (3): 89–101. doi: 10.4171/EM/120 . MR   2520469. S2CID   119739774. Zbl   1176.52002.
18. Hartley, Michael I.; Williams, Gordon I. (2010). "Representing the sporadic Archimedean polyhedra as abstract polytopes". Discrete Mathematics . Amsterdam: Elsevier. 310 (12): 1835–1844. arXiv: 0910.2445 . Bibcode:2009arXiv0910.2445H. doi: 10.1016/j.disc.2010.01.012 . MR   2610288. S2CID   14912118. Zbl   1192.52018.
19. Sloane, N. J. A. (ed.). "SequenceA256413(Number of n-dimensional Bravais lattices.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-18.
20. Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society . New Series. 39 (2): 186. arXiv: math/0105155 . doi:10.1090/S0273-0979-01-00934-X. MR   1886087. S2CID   586512. Zbl   1026.17001.
21. 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
22. Sloane, N. J. A. (ed.). "SequenceA013629(Floor of imaginary parts of nontrivial zeros of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-16.
23. Sloane, N. J. A. (ed.). "SequenceA002410(Nearest integer to imaginary part of n-th zero of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-16.
24. Sloane, N. J. A. (ed.). "SequenceA058303(Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-16.
25. Odlyzko, Andrew. "The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]". Andrew Odlyzko: Home Page. UMN CSE . Retrieved 2024-01-16.
26. Wiggermann 1998, p. 222.
27. Bowley, Roger. "14 and Shakespeare the Numbers Man". Numberphile. Brady Haran. Archived from the original on 2016-02-01. Retrieved 2016-01-03.

Bibliography

• Wiggermann, Frans A. M. (1998), "Nergal A. Philological", Reallexikon der Assyriologie, retrieved 2022-03-06