14 (number) explained

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

Mathematics

Fourteen is the seventh composite number.

Properties

14 is the third distinct semiprime,^[1] being the third of the form

(where 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.

14 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.

14 is the third companion Pell number and the fourth Catalan number.^{[2] [3]} It is the lowest even

for which the Euler totient has no solution, making it the first even nontotient.^[4]

According to the Shapiro inequality, 14 is the least number

such that there exist , , , where:^[5] with and

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.

There are fourteen even numbers that cannot be expressed as the sum of two odd composite numbers:

where 14 is the seventh such number.^[7]

Polygons

14 is the number of equilateral triangles that are formed by the sides and diagonals of a regular six-sided hexagon.^[8] In a hexagonal lattice, 14 is also the number of fixed two-dimensional triangular-celled polyiamonds with four cells.^[9]

14 is the number of elements in a regular heptagon (where there are seven vertices and edges), and the total number of diagonals between all its vertices.

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.^{[10] [11]}

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

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.^[12]
• The rhombic dodecahedron, dual to the cuboctahedron, contains 14 vertices and is the only Catalan solid that can tessellate space.^[13]
• 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.^{[14] [15]}
• Steffen's polyhedron, the simplest flexible polyhedron without self-crossings, has 14 triangular faces.^[16]

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.^[17]

^pp.10-11,14 This is the second simplest known triangular flexible polyhedron, after Steffen's polyhedron.^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.^p.139 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).^{[20] [21]}

Fourteen possible Bravais lattices exist that fill three-dimensional space.^[22]

G₂

, and holds a compact form homeomorphic to the zero divisors with entries of unit norm in the sedenions, .^[23]

Riemann zeta function

The floor of the imaginary part of the first non-trivial zero in the Riemann zeta function is

,^[24] in equivalence with its nearest integer value,^[25] from an approximation of ^{[26] [27]}

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).

It can also signify the Fourteen Holy Helpers.

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.

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.^[28]
• The Piano Sonata No. 14, also known as Moonlight Sonata, is one of the most famous piano sonatas composed by Ludwig van Beethoven.
• Auto racing legend A.J. Foyt most often carried the #14 on his cars, and as his primary number since the early 1970s.
• MLB Hall of Famer Ernie Banks wore the #14 while playing for the Chicago Cubs, with the team retiring the number in his honor in 1982.

References

Bibliography

Notes and References

1. A001358.
2. Web site: Sloane's A002203 : Companion Pell numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-01.
3. Web site: Sloane's A000108 : Catalan numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-01.
4. Web site: Sloane's A005277 : Nontotients. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-01.
5. Troesch . B. A. . On Shapiro's Cyclic Inequality for N = 13 . . 45 . 171 . July 1975 . 199 . 10.1090/S0025-5718-1985-0790653-0 . free . 790653 . 51803624 . 0593.26012 .
6. Book: Kelley, John . John L. Kelley

   . John L. Kelley . General Topology . registration . Van Nostrand . New York . 1955 . 57 . 9780387901251 . 10277303 .

7. 2024-08-03 .
8. 2024-05-05 .
9. 2024-05-15 .
10. Branko . Grünbaum . Branko Grünbaum . Geoffrey . Shepard . G.C. Shephard . Tilings by Regular Polygons . November 1977 . . 50 . 5 . Taylor & Francis, Ltd.. 231 . 10.2307/2689529 . 2689529 . 123776612 . 0385.51006 .
11. Web site: Baez . John C. . John Carlos Baez . Pentagon-Decagon Packing . AMS Blogs . . February 2015 . 2023-01-18 .
12. Book: Coxeter, H.S.M. . Harold Scott MacDonald Coxeter . Regular Polytopes . registration . Chapter 2: Regular polyhedra . Dover . 3rd . New York . 1973 . 18–19 . 0-486-61480-8 . 798003 .
13. Book: Williams, Robert . https://archive.org/details/geometricalfound0000will/page/168/mode/2up . registration . The Geometrical Foundation of Natural Structure: A Source Book of Design . Chapter 5: Polyhedra Packing and Space Filling . . New York . 1979 . 168 . 9780486237299 . 5939651 . 108409770 .
14. Szilassi . Lajos . Structural Topology . 69–80 . Regular toroids . 13 . 1986 . 0605.52002 .
15. Császár . Ákos . Ákos Császár . A polyhedron without diagonals . https://web.archive.org/web/20170918064243/http://www.diale.org/pdf/csaszar.pdf . 2017-09-18 . Acta Scientiarum Mathematicarum (Szeged) . 140–142 . 13 . 1949.
16. Lijingjiao . Iila . Tachi . Tomohiro . Guest . Simon D. . 1 . Optimizing the Steffen flexible polyhedron . Proceedings of the International Association for Shell and Spatial Structures (Future Visions Symposium) . Amsterdam . IASS . 2015 . 10.17863/CAM.26518 . 125747070 .
17. Li . Jingjiao . 2018 . Flexible Polyhedra: Exploring finite mechanisms of triangulated polyhedra . Ph.D. Thesis . University of Cambridge, Department of Engineering . xvii, 1-156 . 10.17863/CAM.18803 . free . 204175310 .
18. 2023-01-18 .
19. 2023-01-18 .
20. Grünbaum . Branko . Branko Grünbaum . An enduring error . . 64 . 3 . 89–101 . . Helsinki . 2009 . 10.4171/EM/120 . free . 2520469 . 1176.52002 . 119739774 .
21. Hartley . Michael I. . Williams . Gordon I. . Representing the sporadic Archimedean polyhedra as abstract polytopes . . 310 . 12 . . Amsterdam . 1835–1844 . 2010 . 10.1016/j.disc.2010.01.012 . free . 0910.2445 . 2009arXiv0910.2445H . 2610288 . 1192.52018 . 14912118 .
22. 2023-01-18 .
23. Baez . John C. . John Baez . The Octonions . . New Series . 39 . 2 . 186 . 2002 . 10.1090/S0273-0979-01-00934-X . math/0105155. 1886087 . 586512 . 1026.17001 .
24. 2024-01-16 .
25. 2024-01-16 .
26. 2024-01-16 .
27. Web site: Odlyzko . Andrew . Andrew Odlyzko . The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]]. Andrew Odlyzko: Home Page . . 2024-01-16 .
28. Web site: Bowley. Roger. 14 and Shakespeare the Numbers Man. Numberphile. Brady Haran. 2016-01-03. https://web.archive.org/web/20160201025743/http://numberphile.com/videos/14_shakespeare.html. 2016-02-01. dead.