214th Engineering Installation Squadron

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

Mathematics

Fourteen is the seventh composite number.

Properties

14 is the third distinct semiprime, being the third of the form $2 \times q$ (where $q$ is a higher prime). More specifically, it is the first member of the second cluster of two discrete semiprimes (14, 15 (number), 15); the next such cluster is (21 (number), 21, 22 (number), 22), members whose sum is the fourteenth prime number, 43 (number), 43.

14 has an aliquot sum of 10 (number), 10, within an aliquot sequence of two composite numbers (14, 10 (number), 10, 8 (number), 8, 7 (number), 7, 1 (number), 1, 0) in the prime 7-aliquot tree.

14 is the third Pell number, companion Pell number and the fourth Catalan number. It is the lowest even $n$ for which the Euler totient $\varphi(x) = n$ has no solution, making it the first even nontotient.

According to the Shapiro inequality, 14 is the least number $n$ such that there exist $x_$, $x_$, $x_$, where:
:$\sum_^ \frac < \frac,$
with $x_ = x_$ and $x_ = x_.$

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

There are fourteen Parity (mathematics), even numbers that cannot be expressed as the sum of two odd composite numbers:
:$\$
where 14 is the seventh such number.

Polygons

14 is the number of equilateral triangles that are formed by the Edge (geometry), sides and diagonals of a Regular polygon, regular six-sided hexagon. In a hexagonal lattice, 14 is also the number of fixed two-dimensional Triangle, triangular-celled polyiamonds with four cells.

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

There are fourteen polygons that can fill a Euclidean tilings by convex regular polygons#Plane-vertex tilings, plane-vertex tiling, where five polygons tile the plane Euclidean tilings by convex regular polygons, uniformly, and nine others only tile the plane alongside irregular polygons.

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 (number), 168) whose fundamental domain is a regular hyperbolic 14-sided tetradecagon, with an area of $8\pi$ by the Gauss-Bonnet theorem.

Solids

Several distinguished polyhedra in Three-dimensional space, three dimensions contain fourteen Face (geometry), faces or Vertex (geometry), vertices as Facet (geometry), facets:
* The cuboctahedron, one of two quasiregular polyhedra, has 14 faces and is the only uniform polyhedron with radial equilateral symmetry.
* The rhombic dodecahedron, Dual polyhedron, dual to the cuboctahedron, contains 14 vertices and is the only Catalan solid that can Tessellation, tessellate space.
* 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 (geometry), 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 polyhedron, toroidal polyhedra; they have 14 vertices and 14 triangular faces, respectively.
* Steffen's polyhedron, the simplest flexible polyhedron without self-crossings, has 14 triangular faces.

A regular tetrahedron Cell (geometry), cell, the simplest uniform polyhedron and Platonic solid, is made up of a total of 14 Simplex#Elements, elements: 4 Edge (geometry), 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.^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

14 is also the root (non-unitary) trivial stella octangula number, where two Stella octangula, self-dual tetrahedra are represented through figurate numbers, while also being the first non-trivial square pyramidal number (after 5); the simplest of the ninety-two Johnson solids is the square pyramid $J_.$ There are a total of fourteen Semi-regular polytope, 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).

Fourteen possible Bravais lattices exist that fill three-dimensional space.

G₂

The exceptional Lie algebra G2 (mathematics), G₂ is the simplest of five such algebras, with a minimal faithful representation in fourteen dimensions. It is the automorphism group of the octonions $\mathbb$, and holds a compact form homeomorphic to the zero divisors with entries of Norm (mathematics), unit norm in the sedenions, $\mathbb$.

Riemann zeta function

The Floor function, floor of the imaginary part of the first non-trivial zero in the Riemann zeta function is $14$, in equivalence with its Rounding, nearest integer value, from an approximation of $14.1347251417\ldots$

In religion and mythology

Christianity

There is a fourteen-point silver star marking the traditional spot of Jesus’ Nativity of Jesus, birth in the Church of the Nativity, Basilica of the Nativity in Bethlehem. According to the genealogy of Jesus in the Gospel of Matthew, “…there were fourteen generations in all from Abraham to David, fourteen generations from David to the Babylonian captivity, exile to Babylon, and fourteen from the exile to the Christ, Messiah” (s:Bible (American Standard)/Matthew#1:17, Matthew 1:17).

Islam

In Islam, 14 has a special significance because of the Fourteen Infallibles who are especially revered and important in Twelver Shi'ism. They are all considered to be infallible by Twelvers alongside Prophets and messengers in Islam, the Prophets of Islam, however these fourteen are said to have a greater significance and closeness to Allah, God.

These fourteen include:

# Muhammad, Prophet Muhammad (SAWA)
# His daughter, Fatima, Lady Fatima (SA)
# Her husband, Ali, Imam Ali (AS)
# His son, Hasan ibn Ali, Imam Hasan (AS)
# His brother, Husayn ibn Ali, Imam Husayn (AS)
# His son, Ali al-Sajjad, Imam Ali al-Sajjad (AS)
# His son, Muhammad al-Baqir, Imam Muhammad al-Baqir (AS)
# His son, Ja'far al-Sadiq, Imam Ja'far al-Sadiq (AS)
# His son, Musa al-Kazim, Imam Musa al-Kazim (AS)
# His son, Imam Reza, Imam Ali al-Rida (AS)
# His son, Muhammad al-Jawad, Imam Muhammad al-Jawad (AS)
# His son, Ali al-Hadi, Imam Ali al-Hadi (AS)
# His son, Hasan al-Askari, Imam Hasan al-Askari (AS)
# His son, Muhammad al-Mahdi, Imam Muhammad al-Mahdi (AJTFS)

Mythology

The number 14 was linked to Šumugan and Nergal.

In other fields

Fourteen is:
* The number of days in a fortnight.

Notes

References

Bibliography

*
{{Integers, zero

Integers