9 (nine) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
following and preceding .
Evolution of the Arabic digit
In the
beginning
Beginning may refer to:
* ''Beginning'' (album), by Pakho Chau
* ''Beginning'' (play), a 2017 play by David Eldridge
* ''Beginning'' (film), a Georgian-French drama film
*"Beginning", a song by heavy metal band Kotipelto
*"Beginning", a 2018 trac ...
, various Indians wrote a digit 9 similar in shape to the modern closing
question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase ''a''. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.
While the shape of the glyph for the digit 9 has an
ascender in most modern
typefaces, in typefaces with
text figures the character usually has a
descender, as, for example, in
.
The modern digit resembles an inverted ''6''. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q.
In a
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
, the number 9 can be constructed in two ways, either with a hook at the end of its stem or without one. Most
LCD calculators use the former, but some
VFD
VFD may refer to:
* Factory Workers' Union of Germany, (german: Verband der Fabrikarbeiter Deutschlands, link=no), a former trade union in Germany
* Vacuum fluorescent display, a display device on consumer electronics equipment
* Variable-freque ...
models use the latter.
Mathematics
Nine is the fourth
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
, and the first composite number that is odd. 9 is the highest single-digit number in the
decimal system. It is the third
square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The usu ...
(3
2), and the second non-unitary square
prime of the form ''p''
2 and first that is odd, with all subsequent squares of this form odd as well.
By
Mihăilescu's theorem, 9 is the only positive
perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...
that is one more than another positive perfect power, since the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of 3 is one more than the
cube of 2.
A number that is 4 or 5
modulo 9 cannot be represented as the
sum of three cubes.
9 is a
Motzkin number
In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have d ...
, for the number of ways of drawing non-intersecting
chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
between four points on a
circle.
Since , 9 is an
exponential factorial.
Six recurring nines appear in the decimal places 762 through 767 of
. (See
six nines in pi
A sequence of six consecutive nines occurs in the decimal representation of the number pi (), starting at the 762nd decimal place.. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the d ...
).
The first non-trivial
magic square is a
x
magic square made of nine cells, with a
magic constant of 15; there are no
x
magic squares with four cells. Meanwhile, a
x
magic square has a magic constant of
369.
A
polygon with nine sides is called a
nonagon
In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon.
The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
. Also an ''enneagon'', it is able to fill a
plane-vertex alongside an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
and an regular
octadecagon
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.
Regular octadecagon
A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates tw ...
, or 18-sided
polygon.
There are nine distinct
uniform coloring
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following differ ...
s of the
triangular tiling and the
square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of th ...
, which are the two simplest
regular tilings
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli sy ...
; the
hexagonal tiling, on the other hand, has three distinct uniform colorings.
There are nine
edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...
convex polyhedra in
three dimensions:
*the five
regular Platonic solids: the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
,
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
,
cube,
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
and
icosahedron;
*the two
quasiregular Archimedean solids: the
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
and the
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
; and
*two
Catalan solids: the
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahed ...
and the
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...
, which are
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...
to the only two quasiregular polyhedra.
In
four-dimensional space, there are nine
paracompact hyperbolic honeycomb
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s, as well as nine
regular compact hyperbolic honeycombs from regular
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and
star ''
polychora''. There are also nine uniform
demitesseractic (
)
Euclidean honeycombs in the fourth dimension.
There are only three types of
Coxeter groups of
uniform figures in dimensions
nine
9 is a number, numeral, and glyph.
9 or nine may also refer to:
Dates
* AD 9, the ninth year of the AD era
* 9 BC, the ninth year before the AD era
* 9, numerical symbol for the month of September
Places
* Nine, Portugal, a parish in the ...
and thereafter, aside from the many families of
prisms and
proprism
In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for ''product prism''. The dimension of the ...
s: the
simplex groups, the
hypercube groups, and the
demihypercube groups. The
ninth dimension is also the final dimension that contains
Coxeter-Dynkin diagrams as
uniform solutions in
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
. Inclusive of compact hyperbolic solutions, there are a total of 238
compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last
paracompact groups is the group
with
1023 total honeycombs, the simplest of which is
621 whose
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is the
521 honeycomb: the vertex arrangement of the densest-possible packing of spheres in
8 dimensions which forms the
lattice. The 6
21 honeycomb is made of
9-simplex
In geometry, a 9- simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-fa ...
es and
9-orthoplexes, with 1023 total
polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k
21 family of
semiregular polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...
s, first defined by
Thorold Gosset in 1900.
There are nine
Heegner numbers, or
square-free positive integers
that yield an imaginary
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...