Natural number, composite number
"14th" redirects here. For the electronic music duo, see
14th (band) .
Natural number
14 (fourteen ) is the
natural number following
13 and preceding
15 .
Look up
fourteen in Wiktionary, the free dictionary.
Mathematics
Fourteen is the seventh
composite number .
Properties
14 is the third distinct
semiprime ,
[1] being the third of the form
2
×
q
{\displaystyle 2\times q}
(where
q
{\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 .
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
n
{\displaystyle n}
for which the
Euler totient
φ
(
x
)
=
n
{\displaystyle \varphi (x)=n}
has no solution, making it the first even
nontotient .
[4]
According to the
Shapiro inequality , 14 is the least number
n
{\displaystyle n}
such that there exist
x
1
{\displaystyle x_{1}}
,
x
2
{\displaystyle x_{2}}
, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x_{3}}
, where:
[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}},}
with
x
n
+
1
=
x
1
{\displaystyle x_{n+1}=x_{1}}
and
x
n
+
2
=
x
2
.
{\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 .
After
11 (the third
super-prime ), 14 is the first of only two
composite numbers whose
arithmetic mean of divisors is the first
perfect number and
unitary perfect number ,
6 (the other number is 15).
[7]
[8]
14 is also the sum of the first three non-zero
squares :
1
2
+
2
2
+
3
2
{\displaystyle 1^{2}+2^{2}+3^{2}}
.
Polygons
14 is the number of
equilateral triangles that are formed by the
sides and
diagonals of a
regular six-sided
hexagon .
[9] In a
hexagonal lattice , 14 is also the number of fixed two-dimensional
triangular -celled
polyiamonds with four cells.
[10]
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.
[11]
[12]
The fundamental domain of the
Klein quartic is a regular hyperbolic 14-sided
tetradecagon , with an area of
8
π
{\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
8
π
{\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 .
[13]
The
rhombic dodecahedron ,
dual to the cuboctahedron, contains 14 vertices and is the only
Catalan solid that can
tessellate space.
[14]
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.
[15]
[16]
Steffen's polyhedron , the simplest
flexible polyhedron without self-crossings, has 14 triangular faces.
[17]
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.
[18] pp.10-11,14 This is the second simplest known triangular flexible polyhedron, after Steffen's polyhedron.
[18] 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.
[18] 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 );
[19]
[20] the simplest of the ninety-two
Johnson solids is the
square pyramid
J
1
.
{\displaystyle J_{1}.}
[a] 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).
[21]
[22]
[b]
Fourteen possible
Bravais lattices exist that fill three-dimensional space.
[23]
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
O
{\displaystyle \mathbb {O} }
, and holds a compact form
homeomorphic to the
zero divisors with entries of
unit norm in the
sedenions ,
S
{\displaystyle \mathbb {S} }
.
[24]
[25]
Riemann zeta function
The
floor of the
imaginary part of the first non-trivial zero in the
Riemann zeta function is
14
{\displaystyle 14}
,
[26] in equivalence with its
nearest integer value,
[27] from an approximation of
14.1347251417
…
{\displaystyle 14.1347251417\ldots }
[28]
[29]
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 )
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:
Notes
^ Furthermore, the square pyramid can be attached to uniform and non-uniform polyhedra (such as other Johnson solids) to generate
fourteen 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 , and
J 87 .
^ 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 (
U 09 ,
U 76i ,
U 08 ,
U 77c ,
U 07 ), vertices (
U 76d ,
U 77d ,
U 78b ,
U 78c ,
U 79b ,
U 79c ,
U 80b ) or edges (
U 19 ).
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^
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^
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^
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^
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